Chapter 12 Resource Masters
Consumable Workbooks Many of the worksheets contained in the Chapter Resource Masters
are available as consumable workbooks in both English and Spanish.
Study Guide and Intervention Workbook
Homework Practice Workbook
ISBN10
0-07-890848-5
0-07-890849-3
ISBN13
978-0-07-890848-4
978-0-07-890849-1
Spanish Version
Homework Practice Workbook
0-07-890853-1
978-0-07-890853-8
Answers for Workbooks The answers for Chapter 12 of these workbooks can be found in the
back of this Chapter Resource Masters booklet.
StudentWorks PlusTM This CD-ROM includes the entire Student Edition text along with the English
workbooks listed above.
TeacherWorks PlusTM All of the materials found in this booklet are included for viewing, printing,
and editing in this CD-ROM.
Spanish Assessment Masters (ISBN10: 0-07-890856-6, ISBN13: 978-0-07-890856-9)
These masters contain a Spanish version of Chapter 12 Test Form 2A and Form 2C.
Copyright © by The McGraw-Hill Companies, Inc. All rights reserved. Permission is
granted to reproduce the material contained herein on the condition that such materials
be reproduced only for classroom use; be provided to students, teachers, and families
without charge; and be used solely in conjunction with the Glencoe Geometry program.
Any other reproduction, for sale or other use, is expressly prohibited.
Send all inquiries to:
Glencoe/McGraw-Hill
8787 Orion Place
Columbus, OH 43240-4027
ISBN: 978-0-07-890521-6
MHID: 0-07-890521-4
Printed in the United States of America.
6 7 8 9 10 11 12 REL 19 18 17 16 15 14 13
Contents
Teacher’s Guide to Using the Chapter 12
Resource Masters .............................................iv
Lesson 12-5
Chapter 12 Student-Built Glossary .................... 1
Chapter 12 Anticipation Guide (English) ........... 3
Chapter 12 Anticipation Guide (Spanish) .......... 4
Volumes of Pyramids and Cones
Study Guide and Intervention .......................... 31
Skills Practice .................................................. 33
Practice............................................................ 34
Word Problem Practice ................................... 35
Enrichment ...................................................... 36
Lesson 12-1
Lesson 12-6
Chapter Resources
Representations of Three-Dimensional
Figures
Study Guide and Intervention ............................ 5
Skills Practice .................................................... 7
Practice.............................................................. 8
Word Problem Practice ..................................... 9
Enrichment ...................................................... 10
Graphing Calculator Activity ............................ 11
Surface Areas and Volumes of Spheres
Study Guide and Intervention .......................... 37
Skills Practice .................................................. 39
Practice............................................................ 40
Word Problem Practice ................................... 41
Enrichment ...................................................... 42
Lesson 12-7
Spherical Geometry
Study Guide and Intervention .......................... 43
Skills Practice .................................................. 45
Practice............................................................ 46
Word Problem Practice ................................... 47
Enrichment ...................................................... 48
Lesson 12-2
Surface Area of Prisms and Cylinders
Study Guide and Intervention .......................... 12
Skills Practice .................................................. 14
Practice............................................................ 15
Word Problem Practice ................................... 16
Enrichment ...................................................... 17
Lesson 12-8
Congruent and Similar Solids
Study Guide and Intervention .......................... 49
Skills Practice .................................................. 51
Practice............................................................ 52
Word Problem Practice ................................... 53
Enrichment ...................................................... 54
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Lesson 12-3
Surface Area of Pyramids and Cones
Study Guide and Intervention .......................... 18
Skills Practice .................................................. 20
Practice............................................................ 21
Word Problem Practice ................................... 22
Enrichment ...................................................... 23
Spreadsheet Activity ........................................ 24
Assessment
Student Recording Sheet ................................ 55
Rubric for Extended-Response ....................... 56
Chapter 12 Quizzes 1 and 2 ........................... 57
Chapter 12 Quizzes 3 and 4 ........................... 58
Chapter 12 Mid-Chapter Test .......................... 59
Chapter 12 Vocabulary Test ........................... 60
Chapter 12 Test, Form 1 ................................. 61
Chapter 12 Test, Form 2A............................... 63
Chapter 12 Test, Form 2B............................... 65
Chapter 12 Test, Form 2C .............................. 67
Chapter 12 Test, Form 2D .............................. 69
Chapter 12 Test, Form 3 ................................. 71
Chapter 12 Extended-Response Test ............. 73
Standardized Test Practice ............................. 74
Lesson 12-4
Volumes of Prisms and Cylinders
Study Guide and Intervention .......................... 25
Skills Practice .................................................. 27
Practice............................................................ 28
Word Problem Practice ................................... 29
Enrichment ...................................................... 30
Answers ........................................... A1–A36
iii
Teacher’s Guide to Using the
Chapter 12 Resource Masters
The Chapter 12 Resource Masters includes the core materials needed for Chapter 12. These
materials include worksheets, extensions, and assessment options. The answers for these
pages appear at the back of this booklet.
All of the materials found in this booklet are included for viewing and printing on the
TeacherWorks PlusTM CD-ROM.
Chapter Resources
Practice This master closely follows
the types of problems found in the
Exercises section of the Student Edition
and includes word problems. Use as an
additional practice option or as homework
for second-day teaching of the lesson.
Student-Built Glossary (pages 1–2) These
masters are a student study tool that
presents up to twenty of the key vocabulary
terms from the chapter. Students are to
record definitions and/or examples for each
term. You may suggest that students
highlight or star the terms with which they
are not familiar. Give this to students before
beginning Lesson 12–1. Encourage them to
add these pages to their mathematics study
notebooks. Remind them to complete the
appropriate words as they study each
lesson.
Word Problem Practice This master
includes additional practice in solving word
problems that apply the concepts of the
lesson. Use as an additional practice or as
homework for second-day teaching of the
lesson.
Anticipation Guide (pages 3–4) This
master, presented in both English and
Spanish, is a survey used before beginning
the chapter to pinpoint what students may
or may not know about the concepts in the
chapter. Students will revisit this survey
after they complete the chapter to see if
their perceptions have changed.
Graphing Calculator, TI-Nspire, or
Spreadsheet Activities These activities
present ways in which technology can be
used with the concepts in some lessons of
this chapter. Use as an alternative
approach to some concepts or as an integral
part of your lesson presentation.
Lesson Resources
Study Guide and Intervention These
masters provide vocabulary, key concepts,
additional worked-out examples and
Check Your Progress exercises to use as a
reteaching activity. It can also be used in
conjunction with the Student Edition as an
instructional tool for students who have
been absent.
Skills Practice This master focuses more
on the computational nature of the lesson.
Use as an additional practice option or as
homework for second-day teaching of the
lesson.
iv
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Enrichment These activities may extend
the concepts of the lesson, offer a historical
or multicultural look at the concepts, or
widen students’ perspectives on the
mathematics they are learning. They are
written for use with all levels of students.
Assessment Options
Leveled Chapter Tests
• Form 1 contains multiple-choice
questions and is intended for use with
approaching grade level students.
• Forms 2A and 2B contain multiplechoice questions aimed at on grade level
students. These tests are similar in
format to offer comparable testing
situations.
• Forms 2C and 2D contain free-response
questions aimed at on grade level
students. These tests are similar in
format to offer comparable testing
situations.
• Form 3 is a free-response test for use
with beyond grade level students.
The assessment masters in the Chapter 12
Resource Masters offer a wide range of
assessment tools for formative (monitoring)
assessment and summative (final)
assessment.
Student Recording Sheet This master
corresponds with the standardized test
practice at the end of the chapter.
Extended-Response Rubric This master
provides information for teachers and
students on how to assess performance on
open-ended questions.
Quizzes Four free-response quizzes offer
assessment at appropriate intervals in the
chapter.
All of the above mentioned tests include a
free-response Bonus question.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Mid-Chapter Test This 1-page test
provides an option to assess the first half of
the chapter. It parallels the timing of the
Mid-Chapter Quiz in the Student Edition
and includes both multiple-choice and
free-response questions.
Extended-Response Test Performance
assessment tasks are suitable for all
students. Sample answers and a scoring
rubric are included for evaluation.
Standardized Test Practice These three
pages are cumulative in nature. It includes
three parts: multiple-choice questions
with bubble-in answer format, griddable
questions with answer grids, and
short-answer free-response questions.
Vocabulary Test This test is suitable for
all students. It includes a list of vocabulary
words and 12 questions to assess students’
knowledge of those words. This can also be
used in conjunction with one of the leveled
chapter tests.
Answers
• The answers for the Anticipation Guide
and Lesson Resources are provided as
reduced pages.
• Full-size answer keys are provided for
the assessment masters.
v
NAME
DATE
12
PERIOD
This is an alphabetical list of the key vocabulary terms you will learn in
Chapter 12. As you study the chapter, complete each term’s definition or
description. Remember to add the page number where you found the term. Add
these pages to your Geometry Study Notebook to review vocabulary at the end of
the chapter.
Vocabulary Term
Found
on Page
Definition/Description/Example
altitude
axis
congruent solids
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
cross section
Euclidean geometry
great circle
isometric view
lateral area
lateral edge
lateral face
(continued on the next page)
Chapter 12
1
Glencoe Geometry
Chapter Resources
Student-Built Glossary
NAME
DATE
12
PERIOD
Student-Built Glossary (continued)
Vocabulary Term
Found
on Page
Definition/Description/Example
oblique cone
oblique cylinder
oblique prism
regular pyramid
right cone
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
right cylinder
right prism (PRIZ·uhm)
similar solids
slant height
spherical geometry
Chapter 12
2
Glencoe Geometry
NAME
12
DATE
PERIOD
Anticipating Guide
Step 1
Before you begin Chapter 12
•
Read each statement.
•
Decide whether you Agree (A) or Disagree (D) with the statement.
•
Write A or D in the first column OR if you are not sure whether you agree or
disagree, write NS (Not Sure).
STEP 1
A, D, or NS
STEP 2
A or D
Statement
1. The shape of a horizontal cross section of a square pyramid
is a triangle.
2. The lateral area of a prism is equal to the sum of the
areas of each face.
3. The axis of an oblique cylinder is different than the
height of the cylinder.
4. The slant height and height of a regular pyramid are
the same.
5. The lateral area of a cone equals the product of π, the
radius, and the height of the cone.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
6. The volume of a right cylinder with radius r and
height h is πr2h.
7. The volume of a pyramid or a cone is found by multiplying the
area of the base by the height.
8. To find the surface area of a sphere with radius r,
multiply πr2 by 4.
9. All postulates and properties of Euclidean geometry are true in
spherical geometry.
10. All spheres and all cubes are similar solids.
•
After you complete Chapter 12
Step 2
Reread each statement and complete the last column by entering an A or a D.
•
Did any of your opinions about the statements change from the first column?
•
For those statements that you mark with a D, use a piece of paper to write an
example of why you disagree.
Chapter 12
3
Glencoe Geometry
Chapter Resources
Extending Surface Area and Volume
NOMBRE
12
FECHA
PERÍODO
Ejercicios Preparations
Extiende el Área de Superficie y volumen
Paso 1
Antes de comenzar el Capítulo 12
• Lee cada enunciado.
• Decide si estás de acuerdo (A) o en desacuerdo (D) con el enunciado.
• Escribe A o D en la primera columna O si no estás seguro(a) de la respuesta,
escribe NS (No estoy seguro(a).
PASO 1
A, D o NS
PASO 2
AoD
Enunciado
1. La forma de un corte transversal horizontal de una pirámide
cuadrada es un triángulo.
2. El área lateral de un prisma es igual a la suma de las
áreas de cada cara.
3. El eje de un cilindro oblicuo es diferente a la altura del
cilindro.
4. La altura oblicua y la altura de una pirámide regular
son las mismas.
5. El área lateral de un cono es igual al producto de π, el
radio, por la altura del cono.
6. El volumen de un cilindro recto con radio r y altura h es πr2h.
8. Para calcular el área de superficie de una esfera con
radio r, multiplica πr2 por 4.
9. Todos los postulados y propiedades de la geometría euclidiana son
verdaderos en geometría esférica.
10. Todas las esferas y todos los cubos son sólidos semejantes.
Paso 2
Después de completar el Capítulo 12
• Vuelve a leer cada enunciado y completa la última columna con una A o una D.
• ¿Cambió cualquiera de tus opiniones sobre los enunciados de la primera columna?
• En una hoja de papel aparte, escribe un ejemplo de por qué estás en desacuerdo con los
enunciados que marcaste con una D.
Capítulo 12
4
Geometría de Glencoe
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
7. El volumen de una pirámide o un cono se calcula multiplicando el
área de la base por la altura.
NAME
DATE
12-1
PERIOD
Study Guide and Intervention
Representations of Three-Dimensional Figures
Draw Isometric Views Isometric dot paper can be used to draw isometric views, or
corner views, of a three-dimensional object on two-dimensional paper.
A
B
E
Example 2
Use isometric dot paper and the
orthographic drawing to sketch a solid.
• The top view indicates two columns.
top view
left view
• The right and left views indicate that the height of figure is
three blocks.
• The front view indicates that the columns have heights 2 and 3 blocks.
F
front view
Connect the dots on the isometric dot paper to represent the edges of the
solid. Shade the tops of each column.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
C
D
right view
object
Exercises
Sketch each solid using isometric dot paper.
1. cube with 4 units on each side
2. rectangular prism 1 unit high, 5 units
long, and 4 units wide
Use isometric dot paper and each orthographic drawing to sketch a solid.
3.
4.
top view
Chapter 12
left view
front view
top view
right view
5
left view
front view
right view
Glencoe Geometry
Lesson 12 -1
Example 1
Use isometric dot paper to sketch a triangular
prism 3 units high, with two sides of the base that are 3 units
long and 4 units long.
−−
−−
Step 1 Draw AB at 3 units and draw AC at 4 units.
−−− −−−
−−
Step 2 Draw AD, BE, and CF, each at 3 units.
−−−
Step 3 Draw BC and DEF.
NAME
12-1
DATE
PERIOD
Study Guide and Intervention (continued)
Representations of Three-Dimensional Figures
Cross Sections The intersection of a solid and a plane is called a cross section of the
solid. The shape of a cross section depends upon the angle of the plane.
Example
There are several interesting shapes that are cross sections of a cone. Determine the shape
resulting from each cross section of the cone.
If the plane is parallel to the base of the cone,
then the resulting cross section will be a
circle.
b.
If the plane cuts through the cone
perpendicular to the base and through the
center of the cone, then the resulting
cross section will be a triangle.
c.
If the plane cuts across the entire cone, then
the resulting cross section will be an ellipse.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
a.
Exercises
Describe each cross section.
1.
Chapter 12
2.
3.
6
Glencoe Geometry
NAME
12-1
DATE
PERIOD
Skills Practice
Representations of Three-Dimensional Figures
Use isometric dot paper to sketch each prism.
2. rectangular prism 2 units high,
5 units long, and 2 units wide
Use isometric dot paper and each orthographic drawing to sketch a solid.
3.
4.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
top view
left view
front view
right view
top view
left view
front view
right view
Describe each cross section.
5.
6.
7.
8.
Chapter 12
7
Glencoe Geometry
Lesson 12-1
1. cube 2 units on each edge
NAME
DATE
12-1
PERIOD
Practice
Representations of Three-Dimensional Figures
Use isometric dot paper to sketch each prism.
1. rectangular prism 3 units high,
3 units long, and 2 units wide
2. triangular prism 3 units high, whose bases
are right triangles with legs 2 units and
4 units long
Use isometric dot paper and each orthographic drawing to sketch a solid.
3.
4.
top view
left view
front view
right view
top view
left view
front view
right view
5.
6.
7. SPHERES Consider the sphere in Exercise 5. Based on the cross section resulting from
a horizontal and a vertical slice of the sphere, make a conjecture about all spherical
cross sections.
8. MINERALS Pyrite, also known as fool’s gold, can form crystals that are perfect cubes.
Suppose a gemologist wants to cut a cube of pyrite to get a square and a rectanglar face.
What cuts should be made to get each of the shapes? Illustrate your answers.
Chapter 12
8
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Sketch the cross section from a vertical slice of each figure.
NAME
12-1
DATE
PERIOD
Word Problem Practice
Representations of Three-Dimensional Figures
4. ENGINEERING Stephanie needs an
object whose top view is a circle and
whose left and front views are squares.
Describe an object that will satisfy these
conditions.
2. BLOCKS Margot’s three-year-old son
made the magnetic block sculpture
shown below in corner view.
5. DESK SUPPORTS The figure shows the
support for a desk.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Draw the right view of the sculpture.
a. Draw the top view.
b. Draw the front view.
3. CUBES Nathan marks the midpoints of
three edges of a
cube as shown.
He then slices the
cube along a plane
that contains these
three points.
Describe the
resulting cross section.
Chapter 12
c. Draw the right view.
9
Glencoe Geometry
Lesson 12-1
1. LABELS Jamal removes the label from a
cylindrical soup can to earn points for
his school. Sketch the shape of the label.
NAME
12-1
DATE
PERIOD
Enrichment
Drawing Solids on Isometric Dot Paper
Isometric dot paper is helpful for drawing solids. Remember to use
dashed lines for hidden edges.
For each solid shown, draw another solid whose dimensions
are twice as large.
2.
3.
4.
5.
6.
Chapter 12
10
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
1.
Glencoe Geometry
NAME
12-1
DATE
PERIOD
Graphing Calculator Activity
Perspective Drawings
The science of perspective drawing studies how to draw a threedimensional object on a two-dimensional page. This science became highly
refined during the Renaissance with the work of artists such as Albrecht
Dürer and Leonardo da Vinci.
Lesson 12-1
Today, computers are often used to make perspective drawings, particularly
elaborate graphics used in television and movies. The three-dimensional
coordinates of objects are figured. Then algebra is used to transform these
into two-dimensional coordinates. The graph of these new coordinates is
called a projection.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
The formulas below will draw one type of projection in which the y-axis is
drawn horizontally, the z-axis vertically, and the x-axis at an angle of a˚
with the y-axis. If the three-dimensional coordinates of a point are (x, y, z),
then the projection coordinates (X, Y) are given by
X = x(-cos a) + y and Y = x(-sin a) + z.
Although this type of projection gives a fairly good perspective drawing, it
does distort some lengths.
1. The drawing with the coordinates given below is a cube.
A(5, 0, 5), B(5, 5, 5), C(5, 5, 0), D(5, 0, 0),
E(0, 0, 5), F(0, 5, 5), G(0, 5, 0), H(0, 0, 0)
Use the formulas above to find the projection coordinates of each A
point, using a = 45. Round projection coordinates to the nearest
integer. Graph the cube on a graphing calculator. Make a sketch
of the display.
A'(__, __) B'(__, __) C(__, __) D(__, __) E'(__, __) F'(__, __)
D
G(__, __) H(__, __)
E
F
B
H
G
C
2. The points A(10, 2, 0), B(10, 10, 0), C(2, 10, 0), and D(3, 3, 4) are vertices of a pyramid.
Find the projection coordinates, using a = 25. Round coordinates to the nearest integer.
−−− −−− −−− −−−
Then graph the pyramid on a graphing calculator by drawing AB, BC, CD, DA,
−−−
and DB. Make a sketch of the display.
A(__, __) B(__, __) C(__, __) D(__, __)
Chapter 12
11
Glencoe Geometry
NAME
DATE
12-2
PERIOD
Study Guide and Intervention
Surface Areas of Prisms and Cylinders
Lateral and Surface Areas of Prisms In a solid figure,
faces that are not bases are lateral faces. The lateral area is
the sum of the area of the lateral faces. The surface area is
the sum of the lateral area and the area of the bases.
altitude
lateral
edge
Lateral Area
of a Prism
If a prism has a lateral area of L square units, a height of h units,
and each base has a perimeter of P units, then L = Ph.
Surface Area
of a Prism
If a prism has a surface area of S square units, a lateral area of L
square units, and each base has an area of B square units, then
S = L + 2B or S = Ph + 2B
lateral
face
pentagonal prism
Example
Find the lateral and surface area of the regular pentagonal
prism above if each base has a perimeter of 75 centimeters and the height is
10 centimeters.
L = Ph
= 75(10)
= 750
Lateral area of a prism
P = 75, h = 10
Multiply.
S = L + 2B
1
= 750 + 2 −
aP
(2 )
(
36°
)
7.5
= 750 + −
(75)
tan 36°
a
15 cm
≈ 1524.2
7.5
tan 36° = −
a
7.5
a=−
tan 36°
Exercises
Find the lateral area and surface area of each prism. Round to the nearest tenth
if necessary.
2.
1.
3m
10 in.
10 m
4m
15 in.
3.
4.
10 cm
9 cm
6 in.
18 in.
5.
Chapter 12
10 cm
8 cm
12 cm
20 cm
4 in.
4 in.
8 in.
6.
12 in.
4m
12
16 m
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
The lateral area is 750 square centimeters and the surface area is about 1524.2 square
centimeters.
NAME
DATE
12-2
PERIOD
Study Guide and Intervention (continued)
Surface Areas of Prisms and Cylinders
Lateral and Surface Areas of Cylinders A cylinder is a
solid with bases that are congruent circles lying in parallel planes.
The axis of a cylinder is the segment with endpoints at the centers
of these circles. For a right cylinder, the axis is also the altitude
of the cylinder.
base
height
axis
base
radius of base
Lateral Area
of a Cylinder
If a cylinder has a lateral area of L square units, a height of h units, and a base
has a radius of r units, then L = 2πrh.
Surface Area
of a Cylinder
If a cylinder has a surface area of S square units, a height of h units, and a
base has a radius of r units, then S = L + 2B or 2πrh + 2πr2.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Find the lateral and surface area of the cylinder. Round to the
nearest tenth.
If d = 12 cm, then r = 6 cm.
L = 2πrh
Lateral area of a cylinder
= 2π(6)(14)
r = 6, h = 14
12 cm
≈ 527.8
Use a calculator.
S = 2πrh + 2πr2
Surface area of a cylinder
2
≈ 527.8 + 2π(6)
2πrh ≈ 527.8, r = 6
≈ 754.0
Use a calculator.
The lateral area is about 527.8 square centimeters and the surface area is about
754.0 square centimeters.
14 cm
Exercises
Find the lateral area and surface area of each cylinder. Round to the nearest
tenth.
1.
2.
4 cm
10 in.
6 in.
12 cm
3.
4.
3 cm
8 cm
3 cm
6 cm
5.
20 cm
6.
2m
1m
12 m
4m
Chapter 12
13
Glencoe Geometry
Lesson 12-2
Example
NAME
DATE
12-2
PERIOD
Skills Practice
Surface Areas of Prisms and Cylinders
Find the lateral area and surface area of each prism. Round to the nearest tenth
if necessary.
1.
2.
6m
12 yd
12 m
10 yd
8m
12 yd
3.
4.
6 in.
9 cm
7.8 cm
8 in.
9 cm
12 cm
5 in.
9 cm
10 in.
5.
6.
2m
10 in.
2m
12 in.
7. 3 yd
8.
8 in.
12 in.
2 yd
Chapter 12
14
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Find the lateral area and surface area of each cylinder. Round to the nearest
tenth.
NAME
DATE
12-2
PERIOD
Practice
Surface Areas of Prisms and Cylinders
Find the lateral and surface area of each prism. Round to the nearest tenth if
necessary.
2.
1.
15 cm
32 cm
5 ft
10 ft
8 ft
15 cm
4.
4 yd
4 yd
2m
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
9.5 yd
11 m
Lesson 12-2
3.
5 yd
Find the lateral area and surface area of each cylinder. Round to the nearest
tenth.
5.
5 ft
6.
4m
7 ft
8.5 m
7.
19 in.
8.
17 in.
12 m
Chapter 12
15
30 m
Glencoe Geometry
NAME
12-2
DATE
PERIOD
Word Problem Practice
Surface Areas of Prisms and Cylinders
1. LOGOS The Z company specializes in
caring for zebras. They want to make a
3-dimensional “Z” to put in front of their
company headquarters. The “Z” is
15 inches thick and the perimeter of the
base is 390 inches.
4. EXHAUST PIPES An exhaust pipe is
shaped like a cylinder with a height of
50 inches and a radius of 2 inches. What
is the lateral surface area of the exhaust
pipe? Round your answer to the nearest
hundredth.
15"
5. TOWERS A circular tower is made by
placing one cylinder on top of another.
Both cylinders have a height of 18
inches. The top cylinder has a radius of
18 inches and the bottom cylinder has a
radius of 36 inches.
What is the lateral surface area of
this “Z”?
2. STAIRWELLS Management decides to
enclose stairs connecting the first and
second floors of a parking garage in a
stairwell shaped like an oblique
rectangular prism.
18 in.
18 in.
9 ft
16 ft
15 ft
What is the lateral surface area of the
stairwell?
b. Another tower is constructed by
placing the original tower on top of
another cylinder with a height of
18 inches and a radius of 54 inches.
What is the total surface area of the
new tower? Round your answer to the
nearest hundredth.
3. CAKES A cake is a rectangular prism
with height 4 inches and base 12 inches
by 15 inches. Wallace wants to apply
frosting to the sides and the top of the
cake. What is the surface area of the
part of the cake that will have frosting?
Chapter 12
16
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
a. What is the total surface area of the
tower? Round your answer to the
nearest hundredth.
20 ft
NAME
DATE
12-2
PERIOD
Enrichment
Minimizing Cost in Manufacturing
Suppose that a manufacturer wants to make a can that has a volume of
40 cubic inches. The cost to make the can is 3 cents per square inch for
the top and bottom and 1 cent per square inch for the side.
r
h
1. Write the value of h in terms of r, given v = πr 2 h.
2. Write a formula for the cost in terms of r.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
4. Repeat the procedure using 2 cents per square inch for the top and bottom and
4 cents per square inch for the top and bottom.
5. What would you expect to happen as the cost of the top and bottom increases?
6. Compute the table for the cost value given. What happens to the height of the can as the
cost of the top and bottom increases?
Cost Top
Cost
Minimum
& Bottom
Cylinder
h
2 cents
1 cent
3 cents
1 cent
4 cents
1 cent
5 cents
1 cent
6 cents
1 cent
Chapter 12
17
Glencoe Geometry
Lesson 12-2
3. Use a graphing calculator to graph the formula, letting Y1 represent the cost and X
represent r. Use the graph to estimate the point at which the cost is minimized.
NAME
12-3
DATE
PERIOD
Study Guide and Intervention
Surface Areas of Pyramids and Cones
Lateral and Surface Areas of Pyramids A pyramid is a
solid with a polygon base. The lateral faces intersect in a common
slant height
point known as the vertex. The altitude is the segment from the
vertex that is perpendicular to the base. For a regular pyramid,
the base is a regular polygon and the altitude has an endpoint at
the center of the base. All the lateral edges are congruent and all
the lateral faces are congruent isosceles triangles. The height of each
lateral face is called the slant height.
Lateral Area of
a Regular Pyramid
Surface Area of
a Regular Pyramid
lateral edge
height
base
1
The lateral area L of a regular pyramid is L = −
Pℓ, where ℓ
2
is the slant height and P is the perimeter of the base.
1
The surface area S of a regular pyramid is S = −
Pℓ + B,
2
where ℓ is the slant height, P is the perimeter of the base,
and B is the area of the base.
Example
For the regular square pyramid above, find the lateral area and
surface area if the length of a side of the base is 12 centimeters and the height is
8 centimeters. Round to the nearest tenth if necessary.
Find the slant height.
ℓ2 = 62 + 82
Pythagorean Theorem
2
ℓ = 100
Simplify.
ℓ = 10
Take the positive square root of each side.
Lateral area of a regular pyramid
2
1
= − (48)(10)
2
= 240
1
S=−
Pℓ + B
Surface area of a regular pyramid
2
P = 4 12 or 48, ℓ = 10
= 240 + 144
Simplify.
= 384
1
−
Pℓ = 240, B = 12 · 12 or 144
2
The lateral area is 240 square centimeters, and the surface area is 384 square centimeters.
Exercises
Find the lateral area and surface area of each regular pyramid. Round to the
nearest tenth if necessary.
2.
1.
8 ft
20 cm
45°
15 cm
3. 10 cm
4.
8.7 in.
6 in.
60°
Chapter 12
18
15 in.
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
1
L=−
Pℓ
NAME
DATE
12-3
PERIOD
Study Guide and Intervention (continued)
Surface Areas of Pyramids and Cones
Lateral and Surface Areas of Cones A cone has
V
V
altitude
a circular base and a vertex. The axis of the cone is the
segment with endpoints at the vertex and the center of
the base. If the axis is also the altitude, then the cone is a
right cone. If the axis is not the altitude, then the cone
is an oblique cone.
axis
slant height
base
oblique cone
Lateral Area of
a Cone
The lateral area L of a right circular cone is L = πr, where r is
the radius and is the slant height.
Surface Area of
a Cone
The surface area S of a right cone is S = πr + πr2, where r is
the radius and is the slant height.
base
right cone
Example
For the right cone above, find the lateral area and surface area if
the radius is 6 centimeters and the height is 8 centimeters. Round to the nearest
tenth if necessary.
L = πrℓ
= π(6)(10)
≈ 188.5
Lateral area of a right cone
r = 6, ℓ = 10
Simplify.
S = πrℓ + πr2
≈ 188.5 + π(62)
≈ 301.6
Surface area of a right cone
πrℓ ≈ 188.5, r = 6
Simplify.
The lateral area is about 188.5 square centimeters and the surface area is about
301.6 square centimeters.
Exercises
Find the lateral area and surface area of each cone. Round to the nearest tenth if
necessary.
1.
12 cm
2.
5 ft
9 cm
3.
12 cm
30°
4.
45°
13 cm
4 in.
Chapter 12
19
Glencoe Geometry
Lesson 12-3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Find the slant height.
ℓ2 = 62 + 82
Pythagorean Theorem
2
ℓ = 100
Simplify.
ℓ = 10
Take the positive square root of each side.
NAME
12-3
DATE
PERIOD
Skills Practice
Surface Areas of Pyramids and Cones
Find the lateral area and surface area of each regular pyramid. Round to the
nearest tenth if necessary.
1.
2.
20 in.
7 cm
4 cm
3.
9m
8 in.
4.
12 ft
10 m
14 ft
Find the lateral area and surface area of each cone. Round to the nearest tenth.
5.
6.
10 ft
14 m
25 ft
7.
8.
21 in.
9 mm
17 mm
8 in.
Chapter 12
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
5m
20
Glencoe Geometry
NAME
DATE
12-3
PERIOD
Practice
Surface Areas of Pyramids and Cones
Find the lateral area and surface area of each regular pyramid. Round to the
nearest tenth if necessary.
1.
2.
12 m
10 yd
7m
9 yd
3.
4.
8 cm
13 ft
5 ft
2.5 cm
Find the lateral area and surface area of each cone. Round to the nearest tenth if
necessary.
6.
5m
7 cm
4m
21 cm
7. Find the surface area of a cone if the height is 14 centimeters and the slant height is
16.4 centimeters.
8. Find the surface area of a cone if the height is 12 inches and the diameter is 27 inches.
9. GAZEBOS The roof of a gazebo is a regular octagonal pyramid. If the base of the
pyramid has sides of 0.5 meter and the slant height of the roof is 1.9 meters, find the
area of the roof.
10. HATS Cuong bought a conical hat on a recent trip to central Vietnam. The basic frame
of the hat is 16 hoops of bamboo that gradually diminish in size. The hat is covered in
palm leaves. If the hat has a diameter of 50 centimeters and a slant height of
32 centimeters, what is the lateral area of the conical hat?
Chapter 12
21
Glencoe Geometry
Lesson 12-3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
5.
NAME
12-3
DATE
PERIOD
Word Problem Practice
Surface Areas of Pyramids and Cones
1. PAPER MODELS Patrick is making a
paper model
of a castle.
Part of the
model
involves
20 cm
20 cm
15 cm
cutting out
the net shown
and folding it
into a
pyramid. The
pyramid has a square base. What is
the lateral surface area of the
resulting pyramid?
4. SPRAY PAINT A can of spray paint
shoots out paint in a cone shaped mist.
The lateral surface area of the cone is
65π square inches when the can is held
12 inches from a canvas. What is the
area of the part of the canvas that gets
sprayed with paint? Round your answer
to the nearest hundredth.
5. MEGAPHONES A megaphone is
formed by taking a cone with a radius
of 20 centimeters and an altitude of
60 centimeters and cutting off the tip.
The cut is made along a plane that is
perpendicular to the axis of the cone and
intersects the axis 12 centimeters from
the vertex. Round your answers to the
nearest hundredth.
3. PAPERWEIGHTS Daphne uses a
paperweight shaped like a pyramid with
a regular hexagon for a base. The side
length of the regular hexagon is 1 inch.
The altitude of the pyramid is 2 inches.
a. What is the lateral surface area of the
original cone?
b. What is the lateral surface area of the
tip that is removed?
c. What is the lateral surface area of the
megaphone?
What is the lateral surface area of this
pyramid? Round your answers to the
nearest hundredth.
Chapter 12
22
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
2. TETRAHEDRON Sung Li builds a paper
model of a regular tetrahedron, a
pyramid with an equilateral triangle for
the base and three equilateral triangles
for the lateral faces. One of the faces of
the tetrahedron has an area of 17 square
inches. What is the total surface area of
the tetrahedron?
NAME
DATE
12-3
PERIOD
Enrichment
Cone Patterns
The pattern at the right is made from a
circle. It can be folded to make a cone.
1. Measure the radius of the circle to the
nearest centimeter.
2. The pattern is what fraction of the
complete circle?
3. What is the circumference of the complete
circle?
4. How long is the circular arc that is the
outside of the pattern?
5. Cut out the pattern and tape it together to
form a cone.
6. Measure the diameter of the circular base of the cone.
8. What is the slant height of the cone?
Lesson 12-3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
7. What is the circumference of the base of the cone?
9. Use the Pythagorean Theorem to calculate the height of the cone.
Use a decimal approximation. Check your calculation by measuring
the height with a metric ruler.
10. Find the lateral area.
11. Find the total surface area.
Make a paper pattern for each cone with the given measurements.
Then cut the pattern out and make the cone. Find the measurements.
12.
13.
6 cm
20 cm
120°
diameter of base =
diameter of base =
lateral area =
lateral area =
height of cone =
(to nearest tenth of a centimeter)
height of cone =
(to nearest tenth of a centimeter)
Chapter 12
23
Glencoe Geometry
NAME
DATE
12-3
PERIOD
Spreadsheet Activity
Surface Areas of Cones
You can use a spreadsheet to determine the surface area of a cone.
Example 1
Lucy wants to wrap a Mother’s Day gift. The gift she has bought for
her mother is in a conical box that has a slant height of 6 inches and has a radius
of 3 inches. She must determine the surface area of the box to determine how
much wrapping paper to buy. Use a spreadsheet to determine the surface area of
the box. Round to the nearest tenth.
Step 1
Use cell A1 for the radius of the cone and cell B1 for the height.
Step 2
In cell C1, enter an equals sign followed by PI()*A1*B1 + PI()*A1^2. Then press
ENTER. This will return the surface area of the cone.
The surface area of the conical box is 84.8 in2 to the nearest tenth.
Example 2
Use a spreadsheet to determine
the surface area of a cone that has a radius of
2.5 centimeters and a slant height of 5.2 centimeters.
Round to the nearest tenth.
A
B
C
1
2
Sheet 1
Use cell A2 for the radius of the cone and
cell B2 for the slant height.
Step 2
Click on the bottom right corner of cell C1 and drag it to C2. This returns the
surface area of the cone.
The surface area of the cone is 60.5 cm2 to the nearest tenth.
Exercises
Use a spreadsheet to find the surface area of each cone with the given
dimensions. Round to the nearest tenth.
1. r = 12 m, = 2.3 m
2. r = 6 m, = 2 m
3. r = 3 in., = 7 in.
4. r = 5 in., = 11 in.
5. r = 1 ft, = 3 ft
6. r = 3 ft, = 1.5 ft
7. r = 10 mm, = 20 mm
8. r = 1.5 mm, = 4.5 mm
9. r = 6.2 cm, = 1.2 cm
10. r = 10 cm, = 15 cm
11. r = 10 m, = 2 m
12. r = 11 m, = 13 m
Chapter 12
24
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Step 1
NAME
DATE
12-4
PERIOD
Study Guide and Intervention
Volumes of Prisms and Cylinders
Volumes of Prisms The measure of the amount of space
that a three-dimensional figure encloses is the volume of the
figure. Volume is measured in units such as cubic feet, cubic
yards, or cubic meters. One cubic unit is the volume of a cube
that measures one unit on each edge.
Volume
of a Prism
If a prism has a volume of V cubic units, a height of h units,
and each base has an area of B square units, then V = Bh.
Example 1
of the prism.
Example 2
Find the volume of the
prism if the area of each base is 6.3
square feet.
Find the volume
4 cm
base
3 cm
7 cm
3.5 ft
V = Bh
Volume of a prism
= (7)(3)(4)
B = (7)(3), h = 4
= 84
Multiply.
The volume of the prism is 84 cubic
centimeters.
V = Bh
Volume of a prism
= (6.3)(3.5)
B = 6.3, h = 3.5
= 22.05
Multiply.
The volume is 22.05 cubic feet.
Exercises
Find the volume of each prism.
1.
4 cm
3 cm
4.
12 ft
15 ft
5.
1.5 cm
8 ft
8 ft
3.
2.
8 ft
12 ft
10 ft
30°
15 ft
6.
2 cm
3 yd
1.5 cm
6 cm
4 cm
Chapter 12
Lesson 12-4
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
cubic foot
cubic yard
27 cubic feet = 1 cubic yard
7 yd
25
4 yd
Glencoe Geometry
NAME
DATE
12-4
PERIOD
Study Guide and Intervention (continued)
Volumes of Prisms and Cylinders
Volumes of Cylinders The volume of a cylinder is the product of the
height and the area of the base. When a solid is not a right solid, use
Cavalieri’s Priniciple to find the volume. The principle states that if two
solids have the same height and the same cross sectional area at every
level, then they have the same volume.
Volume of
a Cylinder
r
h
If a cylinder has a volume of V cubic units, a height of h units,
and the bases have a radius of r units, then V = πr 2h.
Example 1
Find the volume
of the cylinder.
Example 2
Find the volume of the
oblique cylinder.
3 cm
4 cm
13 in.
h
8 in.
5 in.
V = πr2h
Volume of a cylinder
2
= π(3) (4)
r = 3, h = 4
≈ 113.1
Simplify.
The volume is about 113.1 cubic
centimeters.
Use the Pythagorean Theorem to find the height of
the cylinder.
h2 + 52 = 132
Pythagorean Theorem
2
h = 144
Simplify.
h = 12
Take the positive square root of each side.
Exercises
Find the volume of each cylinder. Round to the nearest tenth.
1.
2. 2 cm
2 ft
18 cm
1 ft
3.
4.
1.5 ft
12 ft
5.
20 ft
20 ft
6.
10 cm
1 yd
13 cm
Chapter 12
26
4 yd
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
V = πr2h
Volume of a cylinder
2
= π(4) (12)
r = 4, h = 12
≈ 603.2
Simplify.
The Volume is about 603.2 cubic inches.
NAME
DATE
12-4
PERIOD
Skills Practice
Volumes of Prisms and Cylinders
Find the volume of each prism or cylinder. Round to the nearest tenth
if necessary.
1.
2.
8 cm
2 ft
8 ft
16 cm
18 cm
6 ft
3.
4.
34 in.
13 m
5m
16 in.
22 in.
3m
5.
6.
10 yd
15 mm
Find the volume of each oblique prism or cylinder. Round to the nearest
tenth if necessary.
7.
4 cm
8.
18 cm
5 in.
17 cm
Chapter 12
Lesson 12-4
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
23 mm
6 yd
3 in.
27
Glencoe Geometry
NAME
DATE
12-4
PERIOD
Practice
Volumes of Prisms and Cylinders
Find the volume of each prism or cylinder. Round to the nearest tenth if
necessary.
1.
2.
26 m
5 in.
10 m
17 m
5 in.
9 in.
5 in.
3.
4.
16 mm
25 ft
7 ft
17.5 mm
5.
8 cm
6.
10 yd
30 cm
20 yd
13 yd
2
a. What is the volume of the aquarium in cubic feet?
b. If there are 7.48 gallons in a cubic foot, how many gallons of water does the aquarium
hold?
c. If a cubic foot of water weighs about 62.4 pounds, what is the weight of the water in
the aquarium to the nearest five pounds?
Chapter 12
28
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
7. AQUARIUM Mr. Gutierrez purchased a cylindrical aquarium for his office.
1
The aquarium has a height of 25 −
inches and a radius of 21 inches.
NAME
12-4
DATE
PERIOD
Word Problem Practice
Volumes of Prisms and Cylinders
1. TRASH CANS The Meyer family uses a
kitchen trash can shaped like a cylinder.
It has a height of
18 inches and a base
diameter of 12 inches.
What is the volume
18 in.
of the trash can? Round
your answer to the
nearest tenth of a
cubic inch.
4. PENCIL GRIPS A pencil grip is shaped
like a triangular prism with a cylinder
removed from the middle. The base of
the prism is a right isosceles triangle
with leg lengths of 2 centimeters. The
diameter of the base of the removed
cylinder is 1 centimeter. The heights of
the prism and the cylinder are the same,
and equal to 4 centimeters.
12 in.
2. BENCH Inside a lobby, there is a piece
of furniture for sitting. The furniture is
shaped like a simple block with a square
base 6 feet on each side and a height of
3
1−
feet.
5
What is the exact volume of the pencil
grip?
3
6 ft
6 ft
5. TUNNELS Construction workers are
digging a tunnel through a mountain.
The space inside the tunnel is going to
be shaped like a rectangular prism. The
mouth of the tunnel will be a rectangle
20 feet high and 50 feet wide and the
length of the tunnel will be 900 feet.
What is the volume of the seat?
3. FRAMES Margaret makes a square
frame out of four pieces of wood. Each
piece of wood is a
rectangular prism
with a length of
40 centimeters,
a height of
4 centimeters,
and a depth of
6 centimeters.
What is the total
volume of the
wood used in the frame?
Chapter 12
a. What will the volume of the tunnel be?
b. If instead of a rectangular shape, the
tunnel had a semicircular shape with
a 50-foot diameter, what would be its
volume? Round your answer to the
nearest cubic foot.
29
Glencoe Geometry
Lesson 12-4
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
1 5 ft
NAME
12-4
DATE
PERIOD
Enrichment
Visible Surface Area
Use paper, scissors, and tape to make five cubes that have one-inch edges.
Arrange the cubes to form each shape shown. Then find the volume and
the visible surface area. In other words, do not include the area of surface
covered by other cubes or by the table or desk.
1.
2.
volume =
volume =
visible surface area =
visible surface area =
3.
4.
5.
volume =
volume =
visible surface area =
visible surface area =
visible surface area =
3 in.
4 in.
6. Find the volume and the visible surface
area of the figure at the right.
3 in.
volume =
5 in.
visible surface area =
3 in.
3 in.
8 in.
Chapter 12
30
5 in.
4 in.
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
volume =
NAME
DATE
12-5
PERIOD
Study Guide and Intervention
Volumes of Pyramids and Cones
Volumes of Pyramids This figure shows a prism and a pyramid
that have the same base and the same height. It is clear that the volume
of the pyramid is less than the volume of the prism. More specifically,
the volume of the pyramid is one-third of the volume of the prism.
Volume of
a Pyramid
If a pyramid has a volume of V cubic units, a height of h units,
1
and a base with an area of B square units, then V = −
Bh.
Example
Find the volume of the square pyramid.
1
Bh
V=−
3
1
=−
(8)(8)10
3
3
10 ft
Volume of a pyramid
B = (8)(8), h = 10
8 ft
8 ft
≈ 213.3
Multiply.
The volume is about 213.3 cubic feet.
Exercises
Find the volume of each pyramid. Round to the nearest tenth if necessary.
2.
10 ft
15 ft
6 ft
8 ft
12 ft
3.
10 ft
4.
12 cm
18 ft
8 cm
regular
hexagon
4 cm
5.
16 in.
6.
6 yd
8 yd
15 in.
5 yd
15 in.
Chapter 12
6 ft
Lesson 12-5
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
1.
31
Glencoe Geometry
NAME
DATE
12-5
PERIOD
Study Guide and Intervention (continued)
Volumes of Pyramids and Cones
Volumes of Cones For a cone, the volume is one-third the product of the
height and the area of the base. The base of a cone is a circle, so the area of the
base is πr2.
Volume of
a Cone
Example
If a cone has a volume of V cubic units, a height of h units,
1 2
and the bases have a radius of r units, then V = −
πr h.
r
3
Find the volume of the cone.
1 2
πr h
V=−
3
1
=−
π(5)212
3
h
5 cm
Volume of a cone
12 cm
r = 5, h = 12
≈ 314.2
Simplify.
The volume of the cone is about 314.2 cubic centimeters.
Exercises
Find the volume of each cone. Round to the nearest tenth.
1.
10 ft
4.
12 in.
30 in.
20 ft
Chapter 12
18 yd 45°
20 yd
6.
26 ft
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
3.
5.
8 ft
2.
10 cm
6 cm
45°
16 cm
32
Glencoe Geometry
NAME
DATE
12-5
PERIOD
Skills Practice
Volumes of Pyramids and Cones
Find the volume of each pyramid or cone. Round to the nearest tenth if necessary.
1.
2.
8 cm
8 ft
5 ft
3.
12 m
4.
14 in.
25 m
8 in.
10 in.
5.
6.
14 yd
18 mm
66°
25 yd
Find the volume of each oblique pyramid or cone. Round to the nearest tenth
if necessary.
7.
8.
6 cm
12 cm
6 ft
4 ft
4 ft
Lesson 12-5
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
4 cm
7 cm
5 ft
Chapter 12
33
Glencoe Geometry
NAME
DATE
12-5
PERIOD
Practice
Volumes of Pyramids and Cones
Find the volume of each pyramid or cone. Round to the nearest tenth if necessary.
1.
2.
23 cm
13 yd
9.2 yd
9.2 yd
12.5 cm
25 cm
3.
9 ft
4.
19 ft
5.
12 mm
52°
6.
11 ft
6 in.
37 ft
6 in.
7. CONSTRUCTION Mr. Ganty built a conical storage shed. The base of the shed is
4 meters in diameter and the height of the shed is 3.8 meters. What is the volume of
the shed?
8. HISTORY The start of the pyramid age began with King Zoser’s pyramid, erected in the
27th century B.C. In its original state, it stood 62 meters high with a rectangular base
that measured 140 meters by 118 meters. Find the volume of the original pyramid.
Chapter 12
34
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
11 in.
NAME
12-5
DATE
PERIOD
Word Problem Practice
Volumes of Pyramids and Cones
1. ICE CREAM DISHES The part of a dish
designed for ice cream is shaped like an
upside-down cone. The base of the cone
has a radius of 2 inches and the height
is 1.2 inches.
4. SCULPTING A sculptor wants to remove
stone from a cylindrical block 3 feet high
and turn it into a cone. The diameter of
the base of the cone and cylinder is
2 feet.
What is the volume of the cone? Round
your answer to the nearest hundredth.
What is the volume of the stone that the
sculptor must remove? Round your
answer to the nearest hundredth.
5. STAGES A stage has the form of a
square pyramid with the top sliced off
along a plane parallel to the base. The
side length of the top square is 12 feet
and the side length of the bottom square
is 16 feet. The height of the stage is
3 feet.
18 yd
30 yd
What is the volume of the greenhouse?
12 feet
3 feet
16 feet
3. TEEPEE Caitlyn made a teepee for a
class project. Her teepee had a diameter
of 6 feet. The angle the side of the teepee
made with the ground was 65°.
a. What is the volume of the entire
square pyramid that the stage is
part of?
b. What is the volume of the top of the
pyramid that is removed to get the
stage?
65˚
c. What is the volume of the stage?
What was the volume of the teepee?
Round your answer to the nearest
hundredth.
Chapter 12
35
Glencoe Geometry
Lesson 12-5
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
2. GREENHOUSES A greenhouse has the
shape of a square pyramid. The base has
a side length of 30 yards. The height of
the greenhouse is 18 yards.
NAME
12-5
DATE
PERIOD
Enrichment
Frustums
A frustum is a figure formed when a plane intersects a pyramid or
cone so that the plane is parallel to the solid’s base. The frustum is
the part of the solid between the plane and the base. To find the
volume of a frustum, the areas of both bases must be calculated and
used in the formula.
1
V=−
h(B1 + B2 + √
B1B2 ),
3
where h = height (perpendicular distance between the bases),
B1 = area of top base, and B2 = area of bottom base.
Describe the shape of the bases of each frustum. Then find the
volume. Round to the nearest tenth.
13 cm
1.
2.
3 in.
6 cm
7.5 in.
5 cm
19.5 cm
3.
3m
2.25 m
4.5 m
4.
7 ft
8m
5m
6m
12 ft
13 ft
12 m
Chapter 12
36
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
4.5 in.
9 cm
NAME
DATE
12-6
PERIOD
Study Guide and Intervention
Surface Areas of Spheres You can think of the surface area of a sphere
as the total area of all of the nonoverlapping strips it would take to cover
the sphere. If r is the radius of the sphere, then the area of a great circle of
the sphere is πr2. The total surface area of the sphere is four times the area
of a great circle.
Surface Area
of a Sphere
r
If a sphere has a surface area of S square units and a radius of r units, then S = 4πr2.
Example
Find the surface area of a sphere to the nearest tenth
if the radius of the sphere is 6 centimeters.
S = 4πr2
= 4π(6)2
≈ 452.4
Surface area of a sphere
r=6
6 cm
Simplify.
The surface area is 452.4 square centimeters.
Exercises
Find the surface area of each sphere or hemisphere. Round to the nearest tenth.
1.
2.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
5m
7 in
3.
4.
3 ft
9 cm
5. sphere: circumference of great circle = π cm
6. hemisphere: area of great circle ≈ 4π ft2
Chapter 12
37
Glencoe Geometry
Lesson 12-6
Surface Areas and Volumes of Spheres
NAME
DATE
12-6
PERIOD
Study Guide and Intervention (continued)
Surface Areas and Volumes of Spheres
Volumes of Spheres A sphere has one basic measurement, the
length of its radius. If you know the length of the radius of a sphere, you
can calculate its volume.
r
Volume of
a Sphere
4 3
If a sphere has a volume of V cubic units and a radius of r units, then V = −
πr .
Example
Find the volume of a sphere with radius 8 centimeters.
4 3
πr
V=−
3
4
= − π (8)3
3
3
8 cm
Volume of a sphere
r=8
≈ 2144.7
Simplify.
The volume is about 2144.7 cubic centimeters.
Exercises
Find the volume of each sphere or hemisphere. Round to the nearest tenth.
2.
1.
3.
6 in.
16 in.
5 ft
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
4. hemisphere: radius 5 in.
5. sphere: circumference of great circle ≈ 25 ft
6. hemisphere: area of great circle ≈ 50 m2
Chapter 12
38
Glencoe Geometry
NAME
12-6
DATE
PERIOD
Skills Practice
Find the surface area of each sphere or hemisphere. Round to the nearest tenth.
1.
2.
32 m
7 in.
3. hemisphere: radius of great circle = 8 yd
4. sphere: area of great circle ≈ 28.6 in2
Find the volume of each sphere or hemisphere. Round to the nearest tenth.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
5.
6.
94.8 ft
16.2 cm
7. hemisphere: diameter = 48 yd
8. sphere: circumference of a great circle ≈ 26 m
9. sphere: diameter = 10 in.
Chapter 12
39
Glencoe Geometry
Lesson 12-6
Surface Areas and Volumes of Spheres
NAME
DATE
12-6
PERIOD
Practice
Surface Areas and Volumes of Spheres
Find the surface area of each sphere or hemisphere. Round to the nearest tenth.
1.
2.
6.5 cm
89 ft
3. hemisphere: radius of great circle = 8.4 in.
4. sphere: area of great circle ≈ 29.8 m2
Find the volume of each sphere or hemisphere. Round to the nearest tenth.
5.
6.
32 m
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
12.32 ft
7. hemisphere: diameter = 18 mm
8. sphere: circumference ≈ 36 yd
9. sphere: radius = 12.4 in.
Chapter 12
40
Glencoe Geometry
NAME
12-6
DATE
PERIOD
Word Problem Practice
1. ORANGES Mandy cuts a spherical
orange in half along a great circle. If the
radius of the orange is 2 inches, what is
the area of the cross section that Mandy
cut? Round your answer to the nearest
hundredth.
4. THE ATMOSPHERE About 99% of
Earth’s atmosphere is contained in a
31-kilometer thick layer that enwraps
the planet. The Earth itself is almost a
sphere with radius 6378 kilometers.
What is the ratio of the volume of the
atmosphere to the volume of Earth?
Round your answer to the nearest
thousandth.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
2. BILLIARDS A billiard ball set consists
1
inches in
of 16 spheres, each 2 −
4
diameter. What is the total volume of
a complete set of billiard balls? Round
your answer to the nearest thousandth
of a cubic inch.
5. CUBES Marcus builds a sphere inside of
a cube. The sphere fits snugly inside the
cube so that the sphere touches the cube
at one point on each side. The side
length of the cube is 2 inches.
3. MOONS OF SATURN The planet
Saturn has several moons. These can
be modeled accurately by spheres.
Saturn’s largest moon Titan has a
radius of about 2575 kilometers. What is
the approximate surface area of Titan?
Round your answer to the nearest tenth.
a. What is the surface area of the cube?
b. What is the surface area of the
sphere? Round your answers to the
nearest hundredth.
c. What is the ratio of the surface area
of the cube to the surface area of the
sphere? Round your answer to the
nearest hundredth.
Chapter 12
41
Glencoe Geometry
Lesson 12-6
Surface Areas and Volumes of Spheres
NAME
DATE
12-6
PERIOD
Enrichment
Spheres and Density
The density of a metal is a ratio of its mass to its volume. For
example, the mass of aluminum is 2.7 grams per cubic centimeter.
Here is a list of several metals and their densities.
2.7 g/cm3
19.32 g/cm3
11.35 g/cm3
10.50 g/cm3
Aluminum
Gold
Lead
Silver
Copper
Iron
Platinum
8.96 g/cm3
7.874 g/cm3
21.45 g/cm3
To calculate the mass of a piece of metal, multiply volume by density.
Example
Find the mass of a silver ball that is 0.8 cm in diameter.
M=D·V
4
π(0.4)3
= 10.5 · −
3
≈ 10.5(0.27)
≈ 2.81
The mass is about 2.81 grams.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Exercises
Find the mass of each metal ball described. Assume the balls are
spherical. Round your answers to the nearest tenth.
1. a copper ball 1.2 cm in diameter
2. a gold ball 0.6 cm in diameter
3. an aluminum ball with radius 3 cm
4. a platinum ball with radius 0.7 cm
Solve. Assume the balls are spherical. Round your answers
to the nearest tenth.
5. A lead ball weighs 326 grams. Find the radius of the ball to the nearest
tenth of a centimeter.
6. An iron ball weighs 804 grams. Find the diameter of the ball to the
nearest tenth of a centimeter.
7. A silver ball and a copper ball each have a diameter of 3.5 centimeters.
Which weighs more? How much more?
8. An aluminum ball and a lead ball each have a radius of 1.2 centimeters.
Which weighs more? How much more?
Chapter 12
42
Glencoe Geometry
NAME
DATE
12-7
PERIOD
Study Guide and Intervention
Spherical Geometry
Geometry On A Sphere Up to now, we have been studying Euclidean geometry,
where a plane is a flat surface made up of points that extends infinitely in all directions.
In spherical geometry, a plane is the surface of a sphere.
Name each of the following on sphere K.
"
$
a. two lines containing the point F
#
&
and BH
are lines on sphere K that contain the point F
EG
%
b. a line segment containing the point J
−−
ID is a segment on sphere K that contains the point J
'
K
(
)
c. a triangle
*
+
AHI is a triangle on sphere K
Exercises
Name two lines containing point Z, a segment containing point R, and a triangle
in each of the following spheres.
1.
2.
5
#
4
$
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
6
:
;
F
9
;
8
%
"
3
'
7
3
(
M
&
Determine whether figure u on each of the spheres shown is a line in spherical
geometry.
3.
4.
V
V
5. GEOGRAPHY Lines of latitude run horizontally across the surface of Earth. Are there
any lines of latitude that are great circles? Explain.
Chapter 12
43
Glencoe Geometry
Lesson 12-7
Example
NAME
12-7
DATE
PERIOD
Study Guide and Intervention (continued)
Spherical Geometry
Comparing Euclidean and Spherical Geometries Some postulates and
properties of Euclidean geometry are true in spherical geometry. Others are not true or are
true only under certain circumstances.
Example
Tell whether the following postulate or property of plane Euclidean
geometry has a corresponding statement in spherical geometry. If so, write the
corresponding statement. If not, explain your reasoning.
Given any line, there are an infinite number of parallel lines.
On the sphere to the right, if we are given line m we see that it
goes through the poles of the sphere. If we try to make any other
line on the sphere, it would intersect line m at exactly 2 points.
This property is not true in spherical geometry.
A corresponding statement in spherical geometry would be:
“Given any line, there are no parallel lines.”
N
Exercises
Tell whether the following postulate or property of plane Euclidean
geometry has a corresponding statement in spherical geometry. If so,
write the corresponding statement. If not, explain your reasoning.
2. Given a line and a point on the line, there is only one perpendicular line going through
that point.
3. Given two parallel lines and a transversal, alternate interior angles are congruent.
4. If two lines are perpendicular to a third line, they are parallel.
5. Three noncollinear points determine a triangle.
6. A largest angle of a triangle is opposite the largest side.
Chapter 12
44
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
1. If two nonidentical lines intersect at a point, they do not intersect again.
NAME
DATE
12-7
PERIOD
Skills Practice
Spherical Geometry
Name two lines containing point K, a segment containing point T, and a triangle
in each of the following spheres.
2.
'
C
,
)
#
L
(
"
6
5
$
Lesson 12-7
,
1.
*
5
&
4
%
Determine whether figure u on each of the spheres shown is a line in
spherical geometry.
3.
V
4.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
V
basketball
Tell whether the following postulate or property of plane Euclidean geometry has
a corresponding statement in spherical geometry. If so, write the corresponding
statement. If not, explain your reasoning.
5. If two lines form vertical angles, then the angles are equal in measure.
6. If two lines meet a third line at the same angle, those lines are parallel.
7. Two lines meet at two 90° angles or they meet at angles whose sum is 180°.
8. Three non-parallel lines divide the plane into 7 separate parts.
Chapter 12
45
Glencoe Geometry
NAME
DATE
12-7
PERIOD
Practice
Spherical Geometry
Name two lines containing point K, a segment containing point T, and a triangle
in each of the following spheres.
1.
,
,
2.
#
"
%
;
'
5
$
9
L
5
M
&
:
Determine whether figure u on each of the spheres shown is a line in spherical
geometry.
3.
4.
V
V
tennis ball
6. The sum of the angles of a triangle is 180°.
7. Given a line and a point not on the line, there is exactly one line that goes through the
point and is perpendicular to the line.
8. All equilateral triangles are similar.
9. AIRPLANES When flying an airplane from New York to Seattle, what is the shortest
route: flying directly west, or flying north across Canada? Explain.
Chapter 12
46
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Tell whether the following postulate or property of plane Euclidean geometry has
a corresponding statement in spherical geometry. If so, write the corresponding
statement. If not, explain your reasoning.
5. A triangle can have at most one obtuse angle.
NAME
DATE
12-7
PERIOD
Word Problem Practice
1. PAINTING Consider painting
quadrilateral ABCD on the beach ball
with radius 1 ft. What is the surface
area you would need to paint?
5. GEOGRAPHY Latitude and longitude
lines are imaginary lines on Earth. The
lines of latitude are horizontal concentric
circles that help to define the distance a
place is from the equator. Lines of
latitude are measured in degrees. The
equator is 0°. The north pole is 90° north
latitude. The lines of longitude are great
circles that help to define the distance a
place is from the Prime Meridan, which
is located in England and considered the
longitude of 0°.
A
D
B
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
C
2. EARTH The Equator and the Prime
Meridian are perpendicular great circles
that divide Earth into North, South and
East, West hemispheres. If Earth has a
surface area of 197,000,000 square
miles, what is the surface area of the
North-East section of Earth?
a. The mean radius of Earth is 3963
miles. Atlanta, Georgia, has
coordinates (33°N, 84°W) and
Cincinnati, Ohio, has coordinates
(39°N, 84°W). Estimate the distance
between Atlanta and Cincinnati to
the nearest tenth.
Source: NASA
3. OCEAN If the oceans cover 70% of
Earth’s surface, what is the surface area
of the oceans?
Source: NASA
b. Seattle, Washington, has coordinates
(47°N, 122°W) and Portland, Oregon,
has coordinates (45°N, 122°W).
Estimate the distance between
Portland and Seattle to the nearest
tenth.
4. GEOMETRY Three nonidentical lines on
the circle divide it into either
6 sections or 8 triangles. What condition
is needed so that the three lines form
6 sections?
Chapter 12
47
Glencoe Geometry
Lesson 12-7
Spherical Geometry
NAME
12-7
DATE
PERIOD
Enrichment
Spherical Geometry
Projections
180˚ W
90˚ W
When making maps of Earth, cartographers
must show a sphere on a plane. To do this they
have to use projections, a method of converting a
sphere into a plane. But these projections have
their limitations.
The map on the right is a Mercator projection
of Earth. On this map Greenland appears to be
the same size as Africa. But Greenland has a land
area of 2,166,086 square kilometers and Africa
has a land area of 30,365,700 square kilometers.
0˚
90˚ E
180˚ E
60˚ N
40˚ N
20˚ N
0˚
20˚ S
40˚ S
60˚ S
The map on the right is a Lambert projection.
When a pilot draws a straight line between two
points on this map the line shows true bearing, or
relative direction to the North Pole. However, the
bottom area of this map distorts distances.
2. Does each square on the Mercator projection have the same surface area? Explain.
3. Does each square on the Lambert projection have the same surface area? Explain.
4. The Mercator projection uses a cylinder to map Earth, while the Lambert projection uses
a cone to map Earth. What other shapes do you think could be used to map Earth?
Chapter 12
48
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
1. When would it be useful to use a Mercator projection of Earth?
NAME
DATE
12-8
PERIOD
Study Guide and Intervention
Congruent and Similar Solids
Identify Congruent or Similar Solids Similar solids have exactly the same shape
but not necessarily the same size. Two solids are similar if they are the same shape and the
ratios of their corresponding linear measures are equal. All spheres are similar and all
cubes are similar. Congruent solids have exactly the same shape and the same size.
Congruent solids are similar solids with a scale factor of 1:1. Congruent solids have the
following characteristics:
Corresponding angles are congruent
Corresponding edges are congruent
Corresponding faces are congruent
Volumes are equal
Example
Determine whether the pair of solids
is similar, congruent, or neither. If the solids are
similar, state the scale factor.
3
1
ratio of width: −
=−
4
1
ratio of length: −
=−
5
1
=−
ratio of hypotenuse: −
4
1
ratio of height: −
=−
6
2
10
8
2
8
2
6 in.
8 in.
8 in.
4 in.
3 in.
4 in.
2
10 in.
5 in.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
The ratios of the corresponding sides are equal, so the triangular prisms are similar. The
scale factor is 1:2. Since the scale factor is not 1:1, the solids are not congruent.
Exercises
Determine whether the pair of solids is similar, congruent, or neither. If the solids
are similar, state the scale factor.
1.
2.
5 cm
4.2 in.
12.3 in.
12.3 in.
4.2 in.
1 cm
2 cm
10 cm
3.
4.
2m
8 in.
4m
2m
1m
1m
3m
4 in.
Chapter 12
49
Glencoe Geometry
Lesson 12-8
•
•
•
•
NAME
DATE
12-8
PERIOD
Study Guide and Intervention (continued)
Congruent and Similar Solids
Properties of Congruent or Similar Solids When pairs of solids are congruent or
similar, certain properties are known.
If two similar solids have a scale factor of a:b then,
• the ratio of their surface areas is a2:b2.
• the ratio of their volumes is a3:b3.
Example
Two spheres have radii of 2 feet and 6 feet.
What is the ratio of the volume of the small sphere to the
volume of the large sphere?
First, find the scale factor.
radius of the small sphere
radius of the large sphere
6 ft
2
1
−− = −
or −
6
2 ft
3
1
.
The scale factor is −
3
3
(1)3
(3)
a
1
−
= −3 or −
3
b
27
So, the ratio of the volumes is 1:27.
Exercises
2. Two similar cones have heights of 3 feet and 12 feet. What is the ratio of the volume of
the small cone to the volume of the large cone?
3. Two similar triangular prisms have volumes of 27 square meters and 64 square meters.
What is the ratio of the surface area of the small prism to the surface area of the large
prism?
4. COMPUTERS A small rectangular laptop has a width of 10 inches and an area of
80 square inches. A larger and similar laptop has a width of 15 inches. What is the
length of the larger laptop?
5. CONSTRUCTION A building company uses two similar sizes of pipes. The smaller size
has a radius of 1 inch and length of 8 inches. The larger size has a radius of 2.5 inches
What is the volume of the larger pipes?
Chapter 12
50
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
1. Two cubes have side lengths of 3 inches and 8 inches. What is the ratio of the surface
area of the small cube to the surface area of the large cube?
NAME
DATE
12-8
PERIOD
Skills Practice
Congruent and Similar Solids
Determine whether each pair of solids is similar, congruent, or neither. If the
solids are similar, state the scale factor.
1.
2.
9 cm
12 cm
9 cm
3 cm
12 cm
4 cm
6 cm
6 cm
3.
4.
5m
3 ft
10 m
1 ft
3 ft
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
8 cm
1 ft
3 ft
9 ft
5. Two similar pyramids have heights of 4 inches and 7 inches What is the ratio of the
volume of the small pyramid to the volume of the large pyramid?
6. Two similar cylinders have surface areas of 40π square feet and 90π square feet. What
is the ratio of the height of the large cylinder to the height of the small cylinder?
7. COOKING Two stockpots are similar cylinders. The smaller stockpot has a height of
10 inches and a radius of 2.5 inches. The larger stockpot has a height of 16 inches. What
is the volume of the larger stockpot? Round to the nearest tenth.
Chapter 12
51
Glencoe Geometry
Lesson 12-8
2 cm
NAME
12-8
DATE
PERIOD
Practice
Congruent and Similar Solids
Determine whether the pair of solids is similar, congruent, or neither. If the solids
are similar, state the scale factor.
1.
2.
10 cm
18 cm
5 cm
24 cm
6 cm
24 cm
12 cm
8 cm
3. 1 m
5m
4m
1m
5m
4.
10 cm
5 cm
4m
10 cm
5 cm
3m
3m
2 cm
1.5 cm
6. Two similar ice cream cones are made of a half sphere on top and a cone on bottom.
They have radii of 1 inch and 1.75 inches respectively. What is the ratio of the volume
of the small ice cream cone to the volume of the large ice cream cone? Round to the
nearest tenth.
7. ARCITHECTURE Architects make scale models of buildings to present their ideas to
clients. If an architect wants to make a 1:50 scale model of a 4000 square foot house,
how many square feet will the model have?
Chapter 12
52
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
5. Two cubes have surface areas of 72 square feet and 98 square feet. What is the ratio of
the volume of the small cube to the volume of the large cube?
NAME
DATE
12-8
PERIOD
Word Problem Practice
Congruent and Similar Solids
1. COOKING A cylindrical pot is 4.5 inches
tall and has a radius of 4 inches. How
tall would a similar pot be if its radius is
6 inches?
4. PLANETS Earth has a surface area of
about 196,937,500 square miles. Mars
has a surface area of about 89,500,000
square miles. What is the ratio of the
radius of Earth to the radius of Mars?
Round to the nearest tenth.
Source: NASA
2. MANUFACTURING Boxes, Inc. wants
to make the two boxes below. How long
does the second box need to be so that
they are similar?
25 cm
5. BASEBALL Major League Baseball or
MLB, rules state that baseballs must
have a circumference of 9 inches. The
National Softball Association, or NSA,
rules state that softballs must have a
circumference not exceeding 12 inches.
15 cm
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
15 cm
25 cm
Source: MLB, NSA
a. Find the ratio of the circumference of
MLB baseballs to the circumference
of NSA softballs.
3. FARMING A farmer has two similar
cylindrical grain silos. The smaller silo
is 25 feet tall and the larger silo is
40 feet tall. If the smaller silo can hold
1500 cubic feet of grain, how much can
the larger silo hold?
Chapter 12
b. Find the ratio of the volume of MLB
baseballs to the volume of NSA
softballs. Round to the nearest tenth.
53
Glencoe Geometry
Lesson 12-8
24 cm
NAME
12-8
DATE
PERIOD
Enrichment
Doubling Sizes
Consider what happens to surface area when the sides of a figure are doubled.
The sides of the large cube are twice the size of the sides of the
small cube.
5 in.
1. How long are the edges of the large cube?
2. What is the surface area of the small cube?
3. What is the surface area of the large cube?
4. The surface area of the large cube is how many times greater
than that of the small cube?
The radius of the large sphere at the right is twice the radius of
the small sphere.
3m
5. What is the surface area of the small sphere?
6. What is the surface area of the large sphere?
7. The surface area of the large sphere is how many times
greater than the surface area of the small sphere?
Now consider how doubling the dimensions affects the volume of a cube.
The sides of the large cube are twice the size of the sides of the small cube.
9. How long are the edges of the large cube?
5 in.
10. What is the volume of the small cube?
11. What is the volume of the large cube?
12. The volume of the large cube is how many times greater than
that of the small cube?
The large sphere at the right has twice the radius of the small sphere.
3m
13. What is the volume of the small sphere?
14. What is the volume of the large sphere?
15. The volume of the large sphere is how many times greater
than the volume of the small sphere?
16. It appears that if the dimensions of a solid are doubled, the
volume is multiplied by
.
Chapter 12
54
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
8. It appears that if the dimensions of a solid are doubled, the
surface area is multiplied by
.
NAME
DATE
12
PERIOD
Student Recording Sheet
Assessment
Use this recording sheet with pages 894– 895 of the Student Edition.
Multiple Choice
Read each question. Then fill in the correct answer.
1. A
B
C
D
3. A
B
C
D
5. A
B
C
D
2. F
G
H
J
4. F
G
H
J
6. F
G
H
J
Short Response/Gridded Response
Record your answer in the blank.
For gridded response questions, also enter your answer in the grid by writing
each number or symbol in a box. Then fill in the corresponding circle for that
number or symbol.
7. ————————
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
8. ————————
9. ———————— (grid in)
10. ————————
11. ———————— (grid in)
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12. ————————
Extended Response
Record your answers for Question 13 on the back of this paper.
Chapter 12
55
Glencoe Geometry
NAME
DATE
12
PERIOD
Rubric for Scoring Extended-Response
General Scoring Guidelines
• If a student gives only a correct numerical answer to a problem but does not show
how he or she arrived at the answer, the student will be awarded only 1 credit.
All extended response questions require the student to show work.
• A fully correct answer for a multiple-part question requires correct responses for
all parts of the question. For example, if a question has three parts, the correct
response to one or two parts of the question that required work to be shown is not
considered a fully correct response.
• Students who use trial and error to solve a problem must show their method.
Merely showing that the answer checks or is correct is not considered a complete
response for full credit.
Exercise 13 Rubric
Specific Criteria
4
A correct solution that is supported by well-developed, accurate
explanations. The scale factor of the prisms, and the correct volumes of
the prisms are provided. The student displays an understanding of how
volume changes based upon changing dimensions by correctly answering
parts c and d.
3
A generally correct solution, but may contain minor flaws in reasoning
or computation.
2
A partially correct interpretation and/or solution to the problem.
1
A correct solution with no evidence or explanation.
0
An incorrect solution indicating no mathematical understanding of the
concept or task, or no solution is given.
Chapter 12
56
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Score
NAME
12
DATE
PERIOD
Chapter 12 Quiz 1
SCORE
1. Given the corner view of a figure,
draw the left view.
1.
2. A cylinder has a lateral area of 120π square meters, and a
height of 7 meters. Find the radius. Round to the nearest
tenth.
2.
3. Find the lateral area of the
hexagonal prism.
3.
10
Assessment
(Lessons 12-1 and 12-2)
2
4. Find the surface area of a rectangular prism with a length and
width of 6 centimeters and a height of 12 centimeters.
4.
3.5
5. MULTIPLE CHOICE Find the surface
area of the prism to the nearest hundredth.
A 30.50
B 54.00
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
C 49.45
3
5.
D 52.44
NAME
12
5
DATE
PERIOD
Chapter 12 Quiz 2
SCORE
(Lessons 12-3 and 12-4)
1. Find the surface area of the solid figure
at the right to the nearest tenth.
6 in.
9 in.
1.
9 in.
For Questions 2 and 3, use a right circular cone with a
radius of 5 feet and a slant height of 12 feet. Round to the
nearest tenth.
2. Find the lateral area.
2.
3. Find the surface area.
3.
4. A rectangular prism has a length of 16 feet, a width of 9 feet,
and a height of 8 feet. Find the volume of the prism.
4.
5. A cylinder has a diameter of 20 inches and a height of 9 inches.
Find the volume of the cylinder, round to the nearest tenth.
5.
Chapter 12
57
Glencoe Geometry
NAME
12
DATE
PERIOD
Chapter 12 Quiz 3
SCORE
(Lessons 12-5 and 12-6)
1. A pyramid has a height of 18 centimeters and a base with an
area of 26 square centimeters. Find the volume.
1.
2. Find the volume of the cone.
Round to the nearest tenth.
2.
16 cm
17 cm
3. A hemisphere has a base with an area that is 25π square
centimeters. Find the volume of the hemisphere. Round to the
nearest tenth.
3.
4. A sphere has a great circle with a circumference of 8π meters.
What is the surface area of the sphere?
4.
5. MULTIPLE CHOICE A sphere has a radius that is 15.6 inches
long. Find the volume of the sphere. Round to the nearest
tenth.
A 1019.4 in3
C 15,902 in3
3
B 7951.2 in
D 47,702.2 in3
5.
NAME
12
DATE
PERIOD
Chapter 12 Quiz 4
SCORE
1. Name two lines containing point Z,
a segment containing point R, and
a triangle in the sphere F.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
(Lessons 12-7 and 12-8)
5
6
4
8
;
1.
:
3
9
F
7
2. Do all lines have an infinite number of points in spherical
geometry? If not, explain your reasoning.
2.
3. Determine whether the pair of
solids is similar, congruent, or
neither. If the solids are similar,
state the scale factor.
3.
1m
0.5 m
4. Two similar prisms have heights of 12 feet and 20 feet. What
is the ratio of the volume of the small prism to the volume of
the large prism?
4.
5. Two cubes have surface areas of 81 square inches and
144 square inches. What is the ratio of the volume of the
small cube to the volume of the large cube?
5.
Chapter 12
58
Glencoe Geometry
NAME
12
DATE
PERIOD
Chapter 12 Mid-Chapter Test
SCORE
Part I Write the letter for the correct answer in the blank at the right of each question.
1. A cylinder is standing on one of its bases. It is sliced by a plane horizontally.
What is the shape of the cross section?
A triangle
C circle
B square
D rectangle
1.
2. Choose the correct formula for the surface area of a cone.
1
F S = Ph + 2B
H S=−
P + B
2
G S = πr + πr2
2.
J S = πr + 2πr
3. The surface area of a prism is 120 square centimeters and the area of each
base is 32 square centimeters. Find the lateral area of the prism.
A 184 cm2
B 152 cm2
C 86 cm2
D 56 cm2
3.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
For Questions 4 and 5, refer to the solid figure. Round to the nearest
tenth.
46.2 ft
4. Find the lateral area.
F 9289.1 ft2
G 9434.2 ft2
H 10,965.4 ft2
J 12,641.8 ft2
5. Find the surface area.
A 9289.1 ft2
B 9434.2 ft2
C 10,965.4 ft2
64 ft
4.
D 12,641.8 ft2
5.
Part II
6. Draw the top view of this orthogonal
drawing.
6.
For Questions 7 and 8, refer to the regular hexagonal
prism.
7. Sketch the cross section from a vertical
slice of the figure.
2
7.
6
8. Find the surface area. Round to the
nearest tenth.
8.
9. A barrel in the shape of a right cylinder
has a diameter of 18 inches and a height of
42 inches. Find the surface area of the barrel.
9.
10. Find the lateral area of the solid.
Round to the nearest tenth.
4 in.
10.
6 in.
Chapter 12
59
Glencoe Geometry
Assessment
(Lessons 12-1 through 12-4)
NAME
DATE
12
PERIOD
Chapter 12 Vocabulary Test
altitude
axis
base edges
composite solid
congruent solid
cross section
Euclidean geometry
great circle
SCORE
regular pyramidright cone
right cylinder
right prism
similar solids
slant height
spherical geometry
topographical map
isometric view
lateral area
lateral edge
lateral face
non-Euclidean geometry
oblique cone
oblique cylinder
oblique prism
Choose from the terms above to complete each sentence.
1. The height of each lateral face of a regular pyramid is called
?
a(n)
.
?
2.
have the same shape but not the same size.
3. If the axis of a cylindrical is also the altitude, then the
?
cylinder is called a(n)
.
4.
?
is the measure of the amount of space that a
figure encloses.
1.
2.
3.
4.
Choose the correct term to complete each sentence.
5.
6. Two solids that have the same shape and the same size are
called (congruent solids, composite solids).
6.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
5. The segment whose endpoints are the centers of the circular
bases of a cylinder is the (axis, hemisphere)
State whether each sentence is true or false. If false,
replace the underlined word or phrase to make a true
sentence.
7. A polyhedron that has all but one face intersecting at one
point is a prism.
7.
8. A cross section is the intersection of a plane and a solid figure.
8.
9. A hexagonal prism has a six lateral faces.
9.
10. A cone with an axis that is not an altitude is a right cone.
10.
Define each term in your own words.
11. lateral area
11.
12. right prism
12.
Chapter 12
60
Glencoe Geometry
12
DATE
PERIOD
Chapter 12 Test, Form 1
SCORE
Write the letter for the correct answer in the blank at the right of each question.
1. Which of these is part of an orthographic drawing?
A a perspective view
C a two-dimensional top view
B a corner view
D a three-dimensional view
For Questions 2–4, refer to the figure.
E
2. Identify this solid figure.
F square pyramid
G square prism
H
J
triangular pyramid
triangular prism
3. Name the base.
A ABE
C
CDE
B ABCD
4. The shape of a vertical cross section of a cone is a
F circle
G triangle
H square
5. Find the surface area of the cube.
A 9 in2
B 27 in2
C
D
1.
C
B
A
D
2.
3.
D E
?
J
.
trapezoid
4.
3 in.
36 in2
54 in2
3 in.
5.
3 in.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
For Questions 6 and 7, refer to the figure.
6. Find the lateral area. Round to the nearest tenth.
F 75.4 ft2
H 50.3 ft2
G 62.8 ft2
J 25.1 ft2
2 ft
4 ft
7. Find the surface area. Round to the nearest tenth.
A 75.4 ft2
B 62.8 ft2
C 50.3 ft2
6.
7.
D 25.1 ft2
For Questions 8 and 9, refer to the figure.
8. Find the lateral area.
F 108 cm2
G 144 cm2
9. Find the surface area.
A 108 cm2
B 144 cm2
H
J
9 cm
162 cm2
324 cm2
8.
6 cm
6 cm
C
162 cm2
D 324 cm2
9.
50
10. Find the surface area to the nearest tenth.
F 546.6 units2
H 1017.9 units2
2
G 989.6 units
J 1046.2 units2
5
6
10.
11. The radius of a cone is 17 inches long and the slant height is 20 inches. Find
the surface area to the nearest tenth.
A 18,158.4 in2
B 1976.1 in2 C 1068.1 in2
D 340 in2
Chapter 12
61
11.
Glencoe Geometry
Assessment
NAME
NAME
12
DATE
PERIOD
Chapter 12 Test, Form 1 (continued)
12. The area of the base of a prism is 96 square centimeters and the
height is 9 centimeters. Find the volume of the prism.
F 288 cm3
G 864 cm3
H 932 cm3
J 7776 cm3
12.
13. The volume of a cylinder is 62.8 cubic meters and the radius is
2 meters. Find the height of the cylinder. Round to the nearest
meter.
A 20 m
B 10 m
C 8m
D 5m
13.
14. A pyramid has a height of 10 inches and a base with an area of
21 square inches. Find the volume of the pyramid.
F 210 in3
G 105 in3
H 70 in3
J 35 in3
14.
6 in.
15. Find the volume of the oblique cone.
Round to the nearest tenth.
A 1206.4 in3
C 301.6 in3
B 402.1 in3
D 100.5 in3
10 in.
15.
16.
17. A sphere has a volume that is 36π cubic meters. Find the radius of
the sphere.
A 2m
B 3m
C 6m
D 12 m
17.
18. Which of the following postulates or properties of spherical
geometry are true?
F A line has an infinite number of lines parallel to it.
G No two lines are parallel.
H Alternate interior angles formed by two parallel lines and a
transversal are equal in measure.
J Four noncollinear points form two parallel lines.
18.
19. Which of the following describes the two spheres?
A congruent
C both congruent and similar
B similar
D neither congruent nor similar
9 ft
6 ft
19.
20. The ratio of the side lengths of two cubes is 3:7. Find the ratio of
their volumes.
F 3:7
G 9:21
H 9:49
J 27:343
8 ft
Bonus Find the amount of glass needed
to cover the sides of the greenhouse
shown. The bottom, front, and back
are not glass.
Chapter 12
20.
8 ft
9 ft
9 ft
30 ft
B:
15 ft
62
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
16. The diameter of a sphere is 42 centimeters. Find the surface area to
the nearest tenth.
F 5541.8 cm2
G 2770.9 cm2
H 2167.1 cm2 J 527.8 cm2
DATE
12
Chapter 12 Test, Form 2A
PERIOD
SCORE
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Write the letter for the correct answer in the blank at the right of each question.
1. What do the dark segments represent in an orthographic drawing?
A changes in color
C designs on the surface
B where paper should be folded
D different heights in the surface
1.
For Questions 2 and 3, refer to the figure.
2. Identify the figure.
F pyramid
H cone
G prism
J cylinder
2.
X
Y
3. Identify the shape of a horizontal cross section of the figure.
A triangle
B ellipse
C rectangle
D circle
3.
4. The lateral area of a cube is 36 square inches. How long is each edge?
in.
F √6
G 3 in.
H 6 in.
J 9 in.
4.
5. Find the surface area of the outside of the open box.
A 1920 in2
C 752 in2
B 998 in2
D 400 in2
5.
8 in.
12 in.
20 in.
For Questions 6 and 7, use a right cylinder with a radius of 3 inches
and a height of 17 inches. Round to the nearest tenth.
6. Find the lateral area.
F 320.4 in2
G 348.7 in2
H 377.0 in2
J 537.2 in2
6.
7. Find the surface area.
A 320.4 in2
B 348.7 in2
7.
C 377.0 in2
D 537.2 in2
For Questions 8 and 9, refer to the figure.
8. Find the lateral area.
F 144 cm2
H 196 cm2
G 144 + 24 √
3 cm2
J 288 cm2
12 cm
8.
4 cm
9. Find the surface area.
A 144 cm2
B 144 + 24 √
3 cm2 C 196 cm2
For Questions 10 and 11, refer to the figure.
Round to the nearest tenth.
D 288 cm2
12 in.
9.
2 in.
10. Find the lateral area.
G 75.4 in2
F 44.0 in2
H 88.0 in2
J 100.5 in2
10.
11. Find the surface area.
A 44.0 in2
B 75.4 in2
C 88.0 in2
D 100.5 in2
11.
Chapter 12
63
Glencoe Geometry
Assessment
NAME
NAME
12
DATE
PERIOD
Chapter 12 Test, Form 2A (continued)
12. The surface area of a cube is 96 square feet. Find the volume of
the cube.
F 4 ft3
G 16 ft3
H 64 ft3
J 256 ft3
12.
13. A cylinder whose height is 5 meters has a volume of 320π cubic
meters. Find the radius of the cylinder.
A 8m
B 12.8 m
C 64 m
D 201 m
13.
14. A square pyramid has a height that is 8 centimeters long and a
base with sides that are each 9 centimeters long. Find the volume
of the pyramid.
F 648 cm3
G 324 cm3
H 216 cm3
J 162 cm3
14.
15. Find the volume to the nearest tenth.
A 3619.1 m3
C 14,476.5 m3
B 4825.5 m3
D 43,429.4 m3
15.
24 m
60°
16. Find the surface area to the nearest tenth.
F 4536.5 m2
H 477.5 m2
2
G 2268.2 m
J 238.8 m2
19 m
D 9 in.
17.
18. The shortest distance between any two points in spherical
geometry is
F a straight line.
G any circle.
H a great circle.
J a line through the sphere.
18.
19. Two square pyramids are similar. The sides of the bases are 4 inches
and 12 inches. The height of the smaller pyramid is 6 inches.
Find the height of the larger pyramid.
A 24 in.
B 18 in.
C 16 in.
D 14 in.
19.
20. The ratio of the radii of two similar cylinders is 3:5. The volume of
the smaller cylinder is 54π cubic centimeters. Find the volume of the
larger cylinder.
F 90π cm3
G 150π cm3
H 250π cm3
J 540π cm3
20.
1 ft
Bonus Find the surface area of the figure
to the nearest tenth.
B:
12 ft
4 ft
Chapter 12
64
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
17. A sphere has a volume of 972π cubic inches.
Find the radius of the sphere.
A 2 in.
B 3 in.
C 6 in.
16.
12
DATE
PERIOD
Chapter 12 Test, Form 2B
SCORE
Write the letter for the correct answer in the blank at the right of each question.
1. Given the corner view of a figure, which is the top view?
A
B
C
D
1.
For Questions 2 and 3, refer to the figure.
2. Identify the figure.
F pyramid
H cone
G prism
J cylinder
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
3. Identify the shape of a vertical cross section of the figure.
A rectangle
B circle
C triangle
M
2.
N
D parabola
3.
4. Find the lateral area of an equilateral triangular prism if the area of each
lateral face is 10 square centimeters.
F 10 √
3 cm2
G 30 cm2
H 50 cm2
J 100 cm2
4.
5. The surface area of a rectangular prism is 190 square inches, the length is
10 inches, and the width 3 inches. Find the height.
A 30 in.
B 20 in.
C 10 in.
D 5 in.
5.
For Questions 6–9, use a right cylinder with a radius of
5 centimeters and a height of 22 centimeters. Round to the nearest
tenth.
6. Find the lateral area.
F 848.2 cm2
G 769.7 cm2
H 691.2 cm2
J 345.6 cm2
6.
7. Find the surface area.
A 848.2 cm2
B 769.7 cm2
C
691.2 cm2
D 345.6 cm2
7.
8. Find the volume.
F 345.6 cm3
H 1727.9 cm3
J 2290.2 cm3
8.
G 691.2 cm3
For Questions 9–11, refer to the figure. Round to the nearest tenth.
9. Find the lateral area.
8 cm
3 cm
A 75.4 cm2
C 131.9 cm2
B 103.7 cm2
D 150.8 cm2
9.
10. Find the surface area.
G 75.4 cm2
G 103.7 cm2
H 131.9 cm2
J 150.8 cm2
10.
11. Find the volume.
A 50.3 cm3
C
209.7 cm3
J 226.2 cm3
11.
Chapter 12
B 69.9 cm3
65
Glencoe Geometry
Assessment
NAME
NAME
12
DATE
PERIOD
Chapter 12 Test, Form 2B (continued)
12. The lateral area of a cube is 324 square centimeters. Find the volume of
the cube.
G 81 cm3
H 729 cm3
J 972 cm3
F 9 cm3
12.
13. Find the volume of the solid. Round
to the nearest tenth.
A 31.4 in3
C 125.7 in3
B 41.9 in3
D 502.7 in3
13.
4 in.
10 in.
14. A right triangular pyramid has a 12-meter height and a base with legs
that are 3 meters and 4 meters long. Find the volume of the triangular
pyramid.
F 144 m3
G 72 m3
H 48 m3
J 24 m3
14.
15. Find the volume of the cone. Round to the nearest tenth.
A 41,224.0 m3
C 10,306.0 m3
B 20,612.0 m3
D 763.4 m3
27 m
15.
45°
16. The surface area of a sphere is 64π square centimeters. Find the radius.
F 16 cm
G 8 cm
H 4 cm
J 2 cm
16.
18. In spherical geometry, two lines must meet at least how many times?
F zero times
H two times
G one time
J an infinite number of times
18.
19. Find the scale factor between the two
similar cones.
A 3:8
C 1:2
B 1:3
D 1:4
19.
8 ft
16 ft
3 ft
6 ft
20. The ratio of the heights of two similar solids is 6:11. Find the ratio of
their surface areas.
F 6:11
G 36:121
H 216:1331
J 24:44
Bonus Find the surface area of the frustum
of a square pyramid.
2 ft
4 ft
20.
B:
3 ft
Chapter 12
66
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
17. A sphere has a 48-centimeter diameter. Find the volume of the sphere.
Round to the nearest tenth.
A 463,246.7 cm3
B 57,905.8 cm3
C 28,952.9 cm3
D 7238.2 cm3 17.
12
DATE
PERIOD
Chapter 12 Test, Form 2C
SCORE
1. Given the corner view of a figure,
sketch the front view.
Assessment
NAME
1.
2. Name the faces of the solid.
2.
T
R
Q
P
S
3. Sketch the shape of a horizonal cross
section of the solid.
3.
4. Find the lateral area of a triangular prism with a height of
8 centimeters, and with bases having sides that measure
4 centimeters, 5 centimeters, and 6 centimeters.
4.
5. Find the surface area of the solid.
1 in.
5.
1 in.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
1 in.
6. Find the lateral area of a right cylinder with a diameter of
8.6 yards and a height of 19.4 yards. Round to the nearest tenth.
7. The surface area of a cylinder is 180π square inches and the
height is 9 inches. Find the radius.
6.
7.
For Questions 8 and 9, use a regular hexagonal pyramid
with base edges of 10 inches and a slant height of 9 inches.
8. Find the lateral area.
8.
9. Find the surface area.
9.
For Questions 10 and 11, use a right circular cone with
a radius of 4 feet and a height of 3 feet. Round to the
nearest tenth.
10. Find the lateral area.
10.
11. Find the surface area.
11.
Chapter 12
67
Glencoe Geometry
NAME
12
DATE
PERIOD
Chapter 12 Test, Form 2C (continued)
12. The volume of a rectangular prism is 120 cubic feet and the
area of the base is 60 square feet. Find the length of a lateral
edge of the prism.
12.
13. A cylinder has a 12-foot radius and a 17-foot height. Find the
volume of the cylinder. Round to the nearest tenth.
13.
14. A regular hexagonal pyramid has a height that is 15 feet and a
base 6 feet on each side. Find the volume of the pyramid. Round
to the nearest tenth.
14.
8 ft
15. Find the volume of the oblique cone.
Round to the nearest tenth.
16. Find the surface area of this
hemisphere to the nearest tenth.
5 ft
30°
15.
11 in.
16.
17. A sphere has a diameter of 7.36 inches long. Find the volume of
the sphere. Round to the nearest tenth.
17.
18. What best describes a line in spherical geometry?
18.
4 in.
10 in.
3 in.
9 in.
19.
20. The ratio of the heights of two similar prisms is 2:7. The surface
area of the smaller prism is 50 square meters. Find the surface
area of the larger prism.
20.
Bonus The length of each side of a cube is 6 inches long. Find the
surface area of a sphere inscribed in the cube. Round to the
nearest tenth.
B:
Chapter 12
68
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
19. Determine whether these two
cylinders are congruent, similar,
or neither.
Glencoe Geometry
DATE
12
Chapter 12 Test, Form 2D
1. Given the corner view of a figure,
sketch the back view.
PERIOD
SCORE
Assessment
NAME
1.
J
2. Name the edges of the solid.
2.
H
G
I
3. Sketch the shape of a horizonal
cross section of the solid.
3.
4. Find the lateral area of a regular pentagonal prism if the
perimeter of the base is 50 inches and the height is 15 inches.
4.
5. Find the surface area of the prism.
5.
6 ft
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
9 ft
5 ft
6. A right cylinder has a diameter of 23.6 meters and a height of
11.4 meters. Find the lateral area of the cylinder. Round to
the nearest tenth.
6.
7. The surface area of a right cylinder is 252π square feet and
the height is 11 feet. Find the radius of the cylinder.
7.
For Questions 8 and 9, use a regular octagonal pyramid
with base edges 9 feet long, slant height 15 feet, and a base
with an apothem of 10.86 feet.
8. Find the lateral area.
8.
9. Find the surface area to the nearest tenth.
9.
For Questions 10 and 11, use a cone with a radius of
5 centimeters and a height of 12 centimeters. Round to the
nearest tenth.
10. Find the lateral area.
10.
11. Find the surface area.
11.
Chapter 12
69
Glencoe Geometry
NAME
12
DATE
PERIOD
Chapter 12 Test, Form 2D (continued)
12. An aquarium is 18 inches long, 8 inches wide, and 14 inches
high. The water in it is 4 inches deep. Find the volume of
the water.
12.
13. Find the volume of the cylinder.
Round to the nearest tenth.
13.
25 cm
7 cm
14. A square pyramid has a height that is 51 inches and a base
with sides that are each 11 inches long. Find the volume of
the pyramid.
14.
4 cm
15. Find the volume of the oblique cone.
Round to the nearest tenth.
6 cm
16. Find the surface area of the hemisphere.
Round to the nearest tenth.
13 in.
17. A sphere has a radius that is 2.94 centimeters long. Find the
volume of the sphere. Round to the nearest tenth.
15.
16.
17.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
18. Why do parallel lines not exist in spherical geometry?
19. Determine whether these
cubes are congruent, similar,
or neither.
18.
6 cm
8 cm
19.
20. The ratio of the heights of two similar pyramids is 2:5 and the
volume of the smaller pyramid is 100 cubic feet. Find the
volume of the larger pyramid.
20.
Bonus The length of each side of a cube is 8 inches long.
Find the surface area of a sphere inscribed in the
cube. Round to the nearest tenth.
B:
Chapter 12
70
Glencoe Geometry
12
DATE
PERIOD
Chapter 12 Test, Form 3
1. Draw the back view of a
figure given its orthographic
drawing.
top view
left view
SCORE
front view
right view
Assessment
NAME
1.
2.
2. Describe the cross section.
3. Find the surface area of the solid.
Show the exact solution.
√21
3
3.
3
4
8
4. Find the lateral area of a triangular prism with a right
triangular base with legs that measure 2 feet and 3 feet and
a height of 7 feet. Show the exact solution.
4.
5. Find the surface area of the prism.
5.
10
20
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
6
For Questions 6 and 7, use a right cylinder with a diameter
of 96.4 feet and a height of 58.9 feet. Round to the nearest
tenth.
6. Find the lateral area.
6.
7. Find the surface area.
7.
For Questions 8 and 9, refer to the solid.
Round to the nearest tenth if necessary.
13 in.
8. Find the lateral area.
8.
12 in.
9. Find the surface area to the nearest tenth.
9.
For Questions 10 and 11, use a right circular cone with a
radius of 7 inches and a height of 8 inches. Round to the
nearest tenth.
10. Find the lateral area.
10.
11. Find the surface area.
11.
Chapter 12
71
Glencoe Geometry
NAME
12
DATE
PERIOD
Chapter 12 Test, Form 3 (continued)
10 cm
12. Find the volume of the solid.
12.
8 cm
12 cm
3 cm
2 cm
13. The volume of a cylinder is 96π cubic meters and the height is
13.
6 meters. Find the length of the diameter of this cylinder.
14. Sam is filling a rectangular pan
with liquid from a cylindrical can.
2 in.
The can is three-fourths full
8 in.
of water. Determine whether all
of the water will fit in the pan. Explain.
7 in.
3 in.
6 in.
14.
8 in.
2 in.
15. Find the volume of the solid.
15.
13 in.
16.
17. A cone is 9 centimeters deep and 4 centimeters across the top.
A single scoop of ice cream, 4 centimeters in diameter, is
placed on top of the cone. If the ice cream melts into the cone,
determine whether the melted ice cream will fit in the cone.
Explain.
17.
18. Is the angle sum of a triangle in spherical geometry always
180°? If not give an example.
18.
19. Determine whether these
two pyramids are
congruent, similar, or
neither.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
16. Write a formula for the surface area of a hemisphere in terms
of π and the radius r.
4m
5m
5m
6.4 m
8m
8m
5m
19.
8m
20. The ratio of the volumes of two similar solids is 1:2. Find the
ratio of their surface areas.
20.
6 ft
Bonus Find the surface area of the solid
to the nearest square foot. Do not
include the area of the base.
B:
10 ft
8 ft
Chapter 12
72
Glencoe Geometry
12
DATE
PERIOD
Chapter 12 Extended-Response Test
SCORE
Demonstrate your knowledge by giving a clear, concise solution to each
problem. Be sure to include all relevant drawings and justify your answers.
You may show your solution in more than one way or investigate beyond the
requirements of the problem.
1. Explain the difference between the lateral area and the
surface area of a prism.
2. Draw an oblique cylinder and a right cylinder.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
3. Write a practical application problem involving the surface area or lateral area of a
solid figure studied in this chapter.
4. Give the dimensions of two cylinders in which the first has a greater volume than the
second, but the second has greater surface area than the first.
5. Draw and label the dimensions of a prism and a pyramid that have the same
volume.
6. Write a formula for the volume of this solid in terms of the radius r. Explain.
45°
r
2r
r
Chapter 12
73
Glencoe Geometry
Assessment
NAME
NAME
DATE
PERIOD
12
Standardized Test Practice
SCORE
(Chapters 1–12)
Part 1: Multiple Choice
Instructions: Fill in the appropriate circle for the best answer.
1. Which method could you use to prove
B
E
−−− −−
F
BE AC if AF = BF? (Lesson 4-5)
A Show that ABE BAC by SSS,
−−− −−
C
A
then BE AC by CPCTC.
−−− −−
B Show that ABE BAC by ASA, then BE AC by CPCTC.
−−− −−
C Show that BFE AFC by SAS, then BE AC by CPCTC.
−−− −−
D Show that ABE BAC by AAS, then BE AC by CPCTC.
1. A
B
C
D
2. F
G
H
J
3. A
B
C
D
4. A square has side length 18 centimeters. Find the area of
the square. (Lesson 11-1)
F 36 cm2
G 40 cm2
H 81 cm2
J 324 cm2
4.
F
G
H
J
5. What can you assume from the
figure? (Lesson 10-3)
A ABC is isosceles.
B ABC is equilateral.
C DF = EG
D radius of O = x + y
5. A
B
C
D
6.
F
G
H
J
7. A
B
C
D
8. F
G
H
J
2. Find a. (Lesson 7-3)
F 28.5
G 6.3
9.5
40°
H 12.6
J 14 12.6
4.2
40°
3. Find r. (Lesson 8-6)
A about 34.0
B about 8.9
a
P
C about 11.8
D about 6.6
49°
r
Q
97°
8.9
R
E
D
x
O x
F
A
C
y
G
= 210. If K is the
6. Points D, E, and F are on a circle so that mDEF
center of the circle, what is m ∠DKF? (Lesson 10-2)
F 210
G 105
H 70
J 35
7. Which net could be folded into a triangular prism? (Lesson 12-1)
A
B
C
D
8. Find the surface area of a square pyramid with a height of
9 centimeters and base with a side measuring 24 centimeters.
(Lesson 12-3)
F 1296 cm2
Chapter 12
G 1806 cm2
H 2016 cm2
74
J 8640 cm2
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
B
NAME
PERIOD
Standardized Test Practice (continued)
9. Find y to the nearest centimeter. (Lesson 8-6)
A 19 cm
C 34 cm
B 28 cm
D 37 cm
X
y
Z
9. A
25 cm
B
C
D
F
G
H
J
C
11. A
B
C
D
3 cm
12. F
G
H
73°
32 cm
Y
10. A plane figure is the locus of all points in a plane equidistant from
point B. What is the shape of this figure? (Lesson 10-1)
F square
G cylinder
H rhombus
J circle
10.
E
11. Find m∠C. (Lesson 10-6)
A 18º
B 25º
C 28º
D 60º
D
88°
32°
B
A
12. Find the area of the figure. (Lesson 11-1)
F 76 cm2
H 88 cm2
G 80 cm2
J 92 cm2
4 cm
7 cm
2 cm
J
4 cm
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
6 cm
Part 2: Gridded Response
Instructions: Enter your answer by writing each digit of the answer in a column box
and then shading in the appropriate circle that corresponds to that entry.
13. Quadrilateral PQSR P
is a rectangle.
Find a. (Lesson 6-4)
R
14. Find x. Assume that
segments that appear
tangent are tangent.
13.
Q
(6a - 21)°
(2a - 1)°
S
M
14.
0
0
0
0
1
1
1
1
1
2
2
2
2
2
3
3
3
3
3
4
4
4
4
4
5
5
5
5
5
6
6
6
6
6
7
7
7
7
7
8
8
8
8
8
9
9
9
9
9
0
0
0
0
0
1
1
1
1
1
2
2
2
2
2
3
3
3
3
3
4
4
4
4
4
5
5
5
5
5
6
6
6
6
6
7
7
7
7
7
8
8
8
8
8
9
9
9
9
9
N
5x - 7
(Lesson 10-5)
2x + 8
Chapter 12
0
75
Glencoe Geometry
Assessment
12
DATE
NAME
12
DATE
PERIOD
Standardized Test Practice (continued)
Part 3: Short Response
Instructions: Write your answer in the space provided.
to the nearest
15. Find the length of SR
tenth. (Lesson 10-2)
15.
S
50° T
3m
R
16. Find x. (Lesson 10-7)
15
18
17. Find the area of the shaded region
to the nearest tenth. (Lesson 11-3)
16.
x
30
17.
15
cm
60°
60°
5√3 cm
60°
18.
19. A right circular cone has a slant height of 15 inches and a
radius that is 25 inches long. Find the surface area of the
cone. Round to the nearest tenth. (Lesson 12-3)
19.
20. A ball has a diameter of 26.5 centimeters. Find the surface
area of the ball. Round to the nearest tenth. (Lesson 12-6)
20.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
18. Identify the solid. (Lesson 12-1)
21. Find the following measurements for a sphere with a diameter
of 66 meters. Round to the nearest tenth. (Lesson 12-6)
21a.
a. surface area
b. circumference of the great circle
b.
c. area of the great circle
c.
d. surface area of the hemisphere
d.
Chapter 12
76
Glencoe Geometry
Chapter 12
A1
Glencoe Geometry
D
A
D
D
A
2. The lateral area of a prism is equal to the sum of the
areas of each face.
3. The axis of an oblique cylinder is different than the
4. The slant height and height of a regular pyramid are
the same.
5. The lateral area of a cone equals the product of π, the
radius, and the height of the cone.
6. The volume of a right cylinder with radius r and
height h is πr2h.
Answers
3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter Resources
PERIOD
Representations of Three-Dimensional Figures
Study Guide and Intervention
DATE
E
B
top view
Chapter 12
3.
2. rectangular prism 1 unit high, 5 units
long, and 4 units wide
left view
front view
right view
5
4.
top view
left view
front view
Lesson 12 -1
4/10/08 9:12:58 PM
Glencoe Geometry
right view
F
C
right view
object
front view
D
A
Use isometric dot paper and each orthographic drawing to sketch a solid.
1. cube with 4 units on each side
Sketch each solid using isometric dot paper.
Exercises
Connect the dots on the isometric dot paper to represent the edges of the
solid. Shade the tops of each column.
Example 2
Use isometric dot paper and the
orthographic drawing to sketch a solid.
• The top view indicates two columns.
top view
left view
• The right and left views indicate that the height of figure is
three blocks.
• The front view indicates that the columns have heights 2 and 3 blocks.
Example 1
Use isometric dot paper to sketch a triangular
prism 3 units high, with two sides of the base that are 3 units
long and 4 units long.
−−
−−
Step 1 Draw AB at 3 units and draw AC at 4 units.
−−− −−−
−−
Step 2 Draw AD, BE, and CF, each at 3 units.
−−−
Step 3 Draw BC and DEF.
Draw Isometric Views Isometric dot paper can be used to draw isometric views, or
corner views, of a three-dimensional object on two-dimensional paper.
12-1
NAME
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001_025_GEOCRMC12_890521.indd
PM
5
Glencoe Geometry
For those statements that you mark with a D, use a piece of paper to write an
example of why you disagree.
Chapter 12
Did any of your opinions about the statements change from the first column?
•
A
10. All spheres and all cubes are similar solids.
•
D
9. All postulates and properties of Euclidean geometry are true in
spherical geometry.
After you complete Chapter 12
Step 2
Reread each statement and complete the last column by entering an A or a D.
A
8. To find the surface area of a sphere with radius r,
multiply πr2 by 4.
•
D
7. The volume of a pyramid or a cone is found by multiplying the
area of the base by the height.
height of the cylinder.
D
STEP 2
A or D
1. The shape of a horizontal cross section of a square pyramid
is a triangle.
Statement
•
STEP 1
A, D, or NS
Decide whether you Agree (A) or Disagree (D) with the statement.
Write A or D in the first column OR if you are not sure whether you agree or
disagree, write NS (Not Sure).
•
PERIOD
Read each statement.
Before you begin Chapter 12
Extending Surface Area and Volume
Anticipating Guide
DATE
•
Step 1
12
NAME
001_025_GEOCRMC12_890521.indd 3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Answers (Anticipation Guide and Lesson 12-1)
PERIOD
Representations of Three-Dimensional Figures
Study Guide and Intervention (continued)
DATE
A2
Glencoe Geometry
circle
001_025_GEOCRMC12_890521.indd 6
Chapter 12
1.
Describe each cross section.
Exercises
2.
6
ellipse
3.
If the plane cuts across the entire cone, then
the resulting cross section will be an ellipse.
Angled cross
section
Vertical cross
section
Horizontal
cross section
Glencoe Geometry
rectangle
If the plane cuts through the cone
perpendicular to the base and through the
center of the cone, then the resulting
cross section will be a triangle.
b.
c.
If the plane is parallel to the base of the cone,
then the resulting cross section will be a
circle.
a.
There are several interesting shapes that are cross sections of a cone. Determine the shape
resulting from each cross section of the cone.
Example
solid. The shape of a cross section depends upon the angle of the plane.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
DATE
PERIOD
Representations of Three-Dimensional Figures
Skills Practice
2. rectangular prism 2 units high,
5 units long, and 2 units wide
top view
left view
front view
right view
Chapter 12
7.
5.
triangle
square
Describe each cross section.
3.
7
8.
6.
4.
top view
front view
rectangle
Lesson 12-1
6/14/08 2:45:42 PM
Glencoe Geometry
right view
rectangle
left view
Use isometric dot paper and each orthographic drawing to sketch a solid.
1. cube 2 units on each edge
Use isometric dot paper to sketch each prism.
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9:13:03 PM
7
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 12
Cross Sections The intersection of a solid and a plane is called a cross section of the
12-1
NAME
Answers (Lesson 12-1)
Chapter 12
DATE
PERIOD
Representations of Three-Dimensional Figures
Practice
2. triangular prism 3 units high, whose bases
are right triangles with legs 2 units and
4 units long
top view
left view
front view
right view
4.
top view
left view
front view
A3
circle
6.
trapezoid
right view
Glencoe Geometry
Answers
8
Glencoe Geometry
a cut through
diagonally opposite
top and bottom edges
to get a rectangle
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
regular hexagon
3. CUBES Nathan marks the midpoints of
three edges of a
cube as shown.
He then slices the
cube along a plane
that contains these
three points.
Describe the
resulting cross section.
Draw the right view of the sculpture.
2. BLOCKS Margot’s three-year-old son
made the magnetic block sculpture
shown below in corner view.
Chapter 12
DATE
PERIOD
9
c. Draw the right view.
b. Draw the front view.
a. Draw the top view.
Lesson 12-1
4/10/08 9:13:33 PM
Glencoe Geometry
5. DESK SUPPORTS The figure shows the
support for a desk.
Sample answer: A cylinder with
its height equal to its diameter.
4. ENGINEERING Stephanie needs an
object whose top view is a circle and
whose left and front views are squares.
Describe an object that will satisfy these
conditions.
Representations of Three-Dimensional Figures
Word Problem Practice
1. LABELS Jamal removes the label from a
cylindrical soup can to earn points for
his school. Sketch the shape of the label.
12-1
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9
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
001_025_GEOCRMC12_890521.indd 8
Chapter 12
a cut parallel to
the bases to get
a square
8. MINERALS Pyrite, also known as fool’s gold, can form crystals that are perfect cubes.
Suppose a gemologist wants to cut a cube of pyrite to get a square and a rectanglar face.
What cuts should be made to get each of the shapes? Illustrate your answers.
All spherical cross sections are circles.
7. SPHERES Consider the sphere in Exercise 5. Based on the cross section resulting from
a horizontal and a vertical slice of the sphere, make a conjecture about all spherical
cross sections.
5.
Sketch the cross section from a vertical slice of each figure.
3.
Use isometric dot paper and each orthographic drawing to sketch a solid.
1. rectangular prism 3 units high,
3 units long, and 2 units wide
Use isometric dot paper to sketch each prism.
12-1
NAME
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Answers (Lesson 12-1)
Enrichment
DATE
A4
6.
5.
Glencoe Geometry
001_025_GEOCRMC12_890521.indd 10
10
4.
3.
Chapter 12
2.
1.
For each solid shown, draw another solid whose dimensions
are twice as large.
Isometric dot paper is helpful for drawing solids. Remember to use
dashed lines for hidden edges.
PERIOD
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Perspective Drawings
Graphing Calculator Activity
DATE
PERIOD
H
E
C
B
G
F
Chapter 12
11
Lesson 12-1
6/14/08 2:45:57 PM
Glencoe Geometry
A'(-7, -4), B'(1, -4), C'(8, -1), D'(0, 3); for sketches, see students’ work.
2. The points A(10, 2, 0), B(10, 10, 0), C(2, 10, 0), and D(3, 3, 4) are vertices of a pyramid.
Find the projection coordinates, using a = 25. Round coordinates to the nearest integer.
−−− −−− −−− −−−
Then graph the pyramid on a graphing calculator by drawing AB, BC, CD, DA,
−−−
and DB. Make a sketch of the display.
A(__, __) B(__, __) C(__, __) D(__, __)
A'(-4, 1), B'(1, 1), C'(1, -4), D'(-4, -4), E'(0, 5), F '(5, 5), G'(5, 0) H'(0,0);
For sketches, see students’ work.
1. The drawing with the coordinates given below is a cube.
A(5, 0, 5), B(5, 5, 5), C(5, 5, 0), D(5, 0, 0),
E(0, 0, 5), F(0, 5, 5), G(0, 5, 0), H(0, 0, 0)
Use the formulas above to find the projection coordinates of each A
point, using a = 45. Round projection coordinates to the nearest
integer. Graph the cube on a graphing calculator. Make a sketch
of the display.
A'(__, __) B'(__, __) C(__, __) D(__, __) E'(__, __) F'(__, __)
D
G(__, __) H(__, __)
The formulas below will draw one type of projection in which the y-axis is
drawn horizontally, the z-axis vertically, and the x-axis at an angle of a˚
with the y-axis. If the three-dimensional coordinates of a point are (x, y, z),
then the projection coordinates (X, Y) are given by
X = x(-cos a) + y and Y = x(-sin a) + z.
Although this type of projection gives a fairly good perspective drawing, it
does distort some lengths.
Today, computers are often used to make perspective drawings, particularly
elaborate graphics used in television and movies. The three-dimensional
coordinates of objects are figured. Then algebra is used to transform these
into two-dimensional coordinates. The graph of these new coordinates is
called a projection.
The science of perspective drawing studies how to draw a threedimensional object on a two-dimensional page. This science became highly
refined during the Renaissance with the work of artists such as Albrecht
Dürer and Leonardo da Vinci.
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11
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 12
Drawing Solids on Isometric Dot Paper
12-1
NAME
Answers (Lesson 12-1)
Chapter 12
If a prism has a surface area of S square units, a lateral area of L
square units, and each base has an area of B square units, then
S = L + 2B or S = Ph + 2B
Surface Area
of a Prism
pentagonal prism
Multiply.
P = 75, h = 10
Lateral area of a prism
≈ 1524.2
7.5
= 750 + −
(75)
(2 )
( tan 36° )
S = L + 2B
1
= 750 + 2 −
aP
36°
7.5
a=−
tan 36°
7.5
tan 36° = −
a
15 cm
a
A5
10 m
18 in.
12 in.
6.
4.
2.
20 cm
8 cm
12 cm
4m
16 m
L = 588 cm2; S = 828 cm2
9 cm
10 cm
10 cm
L = 460 in
(8 in. × 15 in. base) or
2
8 in. L = 400 in
15 in.
(10 in. × 15 in. base) or
L = 540 in2 (10 in. × 8 in. base);
S = 700 in2
10 in.
Glencoe Geometry
Answers
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
PERIOD
Surface Areas of Prisms and Cylinders
If a cylinder has a surface area of S square units, a height of h units, and a
base has a radius of r units, then S = L + 2B or 2πrh + 2πr2.
Surface Area
of a Cylinder
radius of base
height
base
14 cm
6 cm
L ≈ 150.8 m2; S ≈ 377.0 m2
4m
12 m
L ≈ 113.1 cm2; S ≈ 169.6 cm2
3 cm
3 cm
Chapter 12
5.
3.
12 cm
L ≈ 301.6 cm2; S ≈ 402.1 cm2
4 cm
13
6.
4.
6 in.
20 cm
1m
Lesson 12-2
5/30/09 3:07:00 PM
Glencoe Geometry
L ≈ 12.6 m2; S ≈ 37.7 m2
2m
L ≈ 502.7 cm2; S ≈ 603.2 cm2
8 cm
L ≈ 377.0 in2; S ≈ 603.2 in2
10 in.
Find the lateral area and surface area of each cylinder. Round to the nearest
tenth.
1.
2.
Exercises
S = 2πrh + 2πr2
Surface area of a cylinder
≈ 527.8 + 2π(6)2
2πrh ≈ 527.8, r = 6
≈ 754.0
Use a calculator.
The lateral area is about 527.8 square centimeters and the surface area is about
754.0 square centimeters.
Example
Find the lateral and surface area of the cylinder. Round to the
nearest tenth.
If d = 12 cm, then r = 6 cm.
L = 2πrh
Lateral area of a cylinder
= 2π(6)(14)
r = 6, h = 14
12 cm
≈ 527.8
Use a calculator.
If a cylinder has a lateral area of L square units, a height of h units, and a base
has a radius of r units, then L = 2πrh.
Lateral Area
of a Cylinder
axis
base
Study Guide and Intervention (continued)
DATE
Lateral and Surface Areas of Cylinders A cylinder is a
solid with bases that are congruent circles lying in parallel planes.
The axis of a cylinder is the segment with endpoints at the centers
of these circles. For a right cylinder, the axis is also the altitude
of the cylinder.
12-2
NAME
001_025_GEOCRMC12_890521.indd 13
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001_025_GEOCRMC12_890521.indd 12
12
L = 128 in2 (rectangular base) or
L = 384 m2; S = 467.1 m2
L = 192 in2 (square base); S = 224 in2
4 in.
4 in.
L = 540 in2; S ≈ 663.9 in2
6 in.
L = 120 m2; S = 132 m2
4m
3m
Chapter 12
5.
3.
1.
2
Find the lateral area and surface area of each prism. Round to the nearest tenth
if necessary.
Exercises
The lateral area is 750 square centimeters and the surface area is about 1524.2 square
centimeters.
L = Ph
= 75(10)
= 750
lateral
face
altitude
Example
Find the lateral and surface area of the regular pentagonal
prism above if each base has a perimeter of 75 centimeters and the height is
10 centimeters.
If a prism has a lateral area of L square units, a height of h units,
and each base has a perimeter of P units, then L = Ph.
Lateral Area
of a Prism
faces that are not bases are lateral faces. The lateral area is
the sum of the area of the lateral faces. The surface area is
the sum of the lateral area and the area of the bases.
PERIOD
lateral
edge
Surface Areas of Prisms and Cylinders
Study Guide and Intervention
DATE
Lateral and Surface Areas of Prisms In a solid figure,
12-2
NAME
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Answers (Lesson 12-2)
Surface Areas of Prisms and Cylinders
Skills Practice
DATE
PERIOD
A6
12 yd
10 in.
L = 120 in2
S = 168 in2
8 in.
5 in.
6 in.
L = 480 yd2 (square base)
L = 528 yd2 (rectangular base)
S = 768 yd2
10 yd
12 yd
4.
2.
8m
9 cm
12 cm
9 cm
L = 324 cm2
S = 394.2 cm2
9 cm
7.8 cm
L = 240 m2 (8 × 12 base)
L = 288 m2 (12 × 6 base)
L = 336 m2 (8 × 6 base)
S = 432 m2
12 m
6m
12 in.
10 in.
2
Glencoe Geometry
001_025_GEOCRMC12_890521.indd 14
Chapter 12
L ≈ 37.7 yd
S ≈ 94.2 yd2
2 yd
L ≈ 377.0 in2
S ≈ 603.2 in2
7. 3 yd
5.
14
8.
6.
8 in.
L ≈ 603.2 in
S ≈ 1005.3 in2
12 in.
2
2m
L ≈ 25.1 m2
S ≈ 50.3 m2
2m
Glencoe Geometry
Find the lateral area and surface area of each cylinder. Round to the nearest
tenth.
3.
1.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Surface Areas of Prisms and Cylinders
Practice
DATE
PERIOD
11 m
L = 132 m2; S ≈ 152.8 m2
2m
4.
8 ft
5 yd
9.5 yd
4 yd
5 ft
L = 123.5 yd2; S ≈ 139.1 yd2
4 yd
L = 224.3 ft2;
S = 264.3 ft2
10 ft
7 ft
5 ft
2
L ≈ 1014.7 in ;
S ≈ 1581.8 in2
17 in.
19 in.
L ≈ 219.9 ft2;
S ≈ 377.0 ft2
Chapter 12
7.
5.
15
8.
6.
8.5 m
30 m
L ≈ 2261.9 m2;
S ≈ 3166.7 m2
12 m
L ≈ 106.8 m2;
S ≈ 131.9 m2
4m
Lesson 12-2
3/5/13 10:08:32 PM
Glencoe Geometry
Find the lateral area and surface area of each cylinder. Round to the nearest
tenth.
3.
32 cm
L = 1920 cm2 (square base) or
L = 1410 cm2 (rectangular base);
S = 2370 cm2
15 cm
15 cm
Find the lateral and surface area of each prism. Round to the nearest tenth if
necessary.
2.
1.
12-2
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5/30/09001_025_GEOCRMC12_890521.indd
3:07:04 PM
15
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 12
Find the lateral area and surface area of each prism. Round to the nearest tenth
if necessary.
12-2
NAME
Answers (Lesson 12-2)
Chapter 12
A7
15"
15 ft
20 ft
16 ft
What is the lateral surface area of the
stairwell?
9 ft
2. STAIRWELLS Management decides to
enclose stairs connecting the first and
second floors of a parking garage in a
stairwell shaped like an oblique
rectangular prism.
5850 in2
What is the lateral surface area of
this “Z”?
Glencoe Geometry
14,250.26 in2
30,536.28 in2
Glencoe Geometry
b. Another tower is constructed by
placing the original tower on top of
another cylinder with a height of
18 inches and a radius of 54 inches.
What is the total surface area of the
new tower? Round your answer to the
nearest hundredth.
Answers
16
18 in.
a. What is the total surface area of the
tower? Round your answer to the
nearest hundredth.
18 in.
5. TOWERS A circular tower is made by
placing one cylinder on top of another.
Both cylinders have a height of 18
inches. The top cylinder has a radius of
18 inches and the bottom cylinder has a
radius of 36 inches.
628.32 in2
4. EXHAUST PIPES An exhaust pipe is
shaped like a cylinder with a height of
50 inches and a radius of 2 inches. What
is the lateral surface area of the exhaust
pipe? Round your answer to the nearest
hundredth.
PERIOD
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Enrichment
DATE
PERIOD
πr
40
h=−
2
r
h
Chapter 12
1 cent
1 cent
4 cents
6 cents
1 cent
3 cents
1 cent
1 cent
2 cents
5 cents
Cost
Cylinder
Cost Top
& Bottom
17
12.24
10.84
9.34
7.71
5.88
h
Minimum
Lesson 12-2
6/2/09 4:28:37 PM
Glencoe Geometry
The height increases as the cost of the top and bottom go up.
6. Compute the table for the cost value given. What happens to the height of the can as the
cost of the top and bottom increases?
See students’ work. Sample answer: The manufacturer might make the
cans taller and narrower.
5. What would you expect to happen as the cost of the top and bottom increases?
80
80
2 cents: C = 4πr2 + −
= 5.88 in. ; 4 cents: C = 8πr2 + −
= 9.34 in.
r
r
4. Repeat the procedure using 2 cents per square inch for the top and bottom and
4 cents per square inch for the top and bottom.
The minimum height is 7.71 inches, which gives a minimum cost of 93.
3. Use a graphing calculator to graph the formula, letting Y1 represent the cost and X
represent r. Use the graph to estimate the point at which the cost is minimized.
πr
40
80
C = 3(2πr2) + 1(2πr −
) or 6πr2 + −
2
r
2. Write a formula for the cost in terms of r.
1. Write the value of h in terms of r, given v = πr 2 h .
Suppose that a manufacturer wants to make a can that has a volume of
40 cubic inches. The cost to make the can is 3 cents per square inch for
the top and bottom and 1 cent per square inch for the side.
Minimizing Cost in Manufacturing
12-2
NAME
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Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
001_025_GEOCRMC12_890521.indd 16
Chapter 12
396 in2
3. CAKES A cake is a rectangular prism
with height 4 inches and base 12 inches
by 15 inches. Wallace wants to apply
frosting to the sides and the top of the
cake. What is the surface area of the
part of the cake that will have frosting?
840 ft2
DATE
Surface Areas of Prisms and Cylinders
Word Problem Practice
1. LOGOS The Z company specializes in
caring for zebras. They want to make a
3-dimensional “Z” to put in front of their
company headquarters. The “Z” is
15 inches thick and the perimeter of the
base is 390 inches.
12-2
NAME
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Answers (Lesson 12-2)
Surface Areas of Pyramids and Cones
Study Guide and Intervention
DATE
where ℓ is the slant height, P is the perimeter of the base,
and B is the area of the base.
1
The surface area S of a regular pyramid is S = −
Pℓ + B,
2
is the slant height and P is the perimeter of the base.
lateral edge
height
base
A8
Simplify.
P = 4 12 or 48, ℓ = 10
Lateral area of a regular pyramid
= 384
= 240 + 144
1
S=−
Pℓ + B
2
2
1
−
Pℓ = 240, B = 12 · 12 or 144
Surface area of a regular pyramid
15 cm
Glencoe Geometry
001_025_GEOCRMC12_890521.indd 18
Chapter 12
L ≈ 266.7 cm2; S ≈ 400.0 cm2
60°
L ≈ 450 cm2; S ≈ 547.4 cm2
20 cm
3. 10 cm
1.
18
4.
2.
45°
15 in.
Glencoe Geometry
L ≈ 326.3 in2; S ≈ 456.8 in2
8.7 in.
6 in.
L ≈ 362.0 ft2; S ≈ 618.0 ft2
8 ft
Find the lateral area and surface area of each regular pyramid. Round to the
nearest tenth if necessary.
Exercises
The lateral area is 240 square centimeters, and the surface area is 384 square centimeters.
= 240
1
L=−
Pℓ
2
1
= − (48)(10)
2
Find the slant height.
ℓ2 = 62 + 82
Pythagorean Theorem
ℓ2 = 100
Simplify.
ℓ = 10
Take the positive square root of each side.
Example
For the regular square pyramid above, find the lateral area and
surface area if the length of a side of the base is 12 centimeters and the height is
8 centimeters. Round to the nearest tenth if necessary.
Surface Area of
a Regular Pyramid
Lateral Area of
a Regular Pyramid
1
The lateral area L of a regular pyramid is L = −
Pℓ, where ℓ
2
A pyramid is a
solid with a polygon base. The lateral faces intersect in a common
slant height
point known as the vertex. The altitude is the segment from the
vertex that is perpendicular to the base. For a regular pyramid,
the base is a regular polygon and the altitude has an endpoint at
the center of the base. All the lateral edges are congruent and all
the lateral faces are congruent isosceles triangles. The height of each
lateral face is called the slant height.
PERIOD
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
PERIOD
V
base
slant height
axis
right cone
altitude
Simplify.
r = 6, ℓ = 10
Lateral area of a right cone
S = πrℓ + πr2
≈ 188.5 + π(62)
≈ 301.6
Simplify.
πrℓ ≈ 188.5, r = 6
Surface area of a right cone
9 cm
12 cm
2
L ≈ 204.2 cm ;
S ≈ 282.7 cm2
13 cm
12 cm
L ≈ 424.1 cm2;
S ≈ 678.6 cm2
Chapter 12
3.
1.
19
4.
2.
L ≈ 71.1 in2;
S ≈ 121.4 in2
4 in.
45°
L ≈ 157.1 ft2;
S ≈ 235.6 ft2
30°
5 ft
Lesson 12-3
3/27/10 10:45:38 PM
Glencoe Geometry
Find the lateral area and surface area of each cone. Round to the nearest tenth if
necessary.
Exercises
The lateral area is about 188.5 square centimeters and the surface area is about
301.6 square centimeters.
L = πrℓ
= π(6)(10)
≈ 188.5
Find the slant height.
ℓ2 = 6 2 + 8 2
Pythagorean Theorem
ℓ2 = 100
Simplify.
ℓ = 10
Take the positive square root of each side.
Example
For the right cone above, find the lateral area and surface area if
the radius is 6 centimeters and the height is 8 centimeters. Round to the nearest
tenth if necessary.
The surface area S of a right cone is S = πr + πr2, where r is
the radius and is the slant height.
Surface Area of
a Cone
oblique cone
The lateral area L of a right circular cone is L = πr, where r is
the radius and is the slant height.
base
Lateral Area of
a Cone
a circular base and a vertex. The axis of the cone is the
segment with endpoints at the vertex and the center of
the base. If the axis is also the altitude, then the cone is a
right cone. If the axis is not the altitude, then the cone
is an oblique cone.
V
Surface Areas of Pyramids and Cones
Study Guide and Intervention (continued)
DATE
Lateral and Surface Areas of Cones A cone has
12-3
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19
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 12
Lateral and Surface Areas of Pyramids
12-3
NAME
Answers (Lesson 12-3)
Chapter 12
Surface Areas of Pyramids and Cones
Skills Practice
DATE
PERIOD
4 cm
L ≈ 283.2 m2
S ≈ 455.3 m2
10 m
9m
L = 56 cm2
S = 72 cm2
7 cm
4.
2.
L ≈ 389.0 ft2
S ≈ 585.0 ft2
12 ft
L = 480 in2
S = 646.3 in2
8 in.
20 in.
14 ft
A9
5m
L ≈ 527.8 in2
S ≈ 728.8 in2
8 in.
21 in.
L ≈ 219.9 m2
S ≈ 298.5 m2
14 m
Glencoe Geometry
25 ft
L ≈ 480.7 mm2
S ≈ 735.1 mm2
17 mm
9 mm
L ≈ 845.9 ft2
S ≈ 1160.1 ft2
10 ft
Answers
20
8.
6.
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Surface Areas of Pyramids and Cones
Practice
DATE
PERIOD
9 yd
L = 162.5 ft2; S ≈ 205.5 ft2
5 ft
13 ft
L = 180 yd2; S = 261 yd2
10 yd
4.
2.
7m
2.5 cm
L ≈ 60 cm2; S ≈ 76.2 cm2
8 cm
L = 126 m2; S ≈ 147.2 m2
12 m
L ≈ 80.5 m2; S ≈ 130.7 cm2
5m
4m
6.
21 cm
468.8 cm2 ; S ≈ 640.7 cm2
7 cm
Chapter 12
about 2513.3 cm2
21
Lesson 12-3
5/30/09 3:07:34 PM
Glencoe Geometry
10. HATS Cuong bought a conical hat on a recent trip to central Vietnam. The basic frame
of the hat is 16 hoops of bamboo that gradually diminish in size. The hat is covered in
palm leaves. If the hat has a diameter of 50 centimeters and a slant height of
32 centimeters, what is the lateral area of the conical hat?
3.8 m2
9. GAZEBOS The roof of a gazebo is a regular octagonal pyramid. If the base of the
pyramid has sides of 0.5 meter and the slant height of the roof is 1.9 meters, find the
area of the roof.
1338.6 in2
8. Find the surface area of a cone if the height is 12 inches and the diameter is 27 inches.
669.3 cm2
7. Find the surface area of a cone if the height is 14 centimeters and the slant height is
16.4 centimeters.
5.
Find the lateral area and surface area of each cone. Round to the nearest tenth if
necessary.
3.
1.
Find the lateral area and surface area of each regular pyramid. Round to the
nearest tenth if necessary.
12-3
NAME
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2:46:29 PM
21
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
001_025_GEOCRMC12_890521.indd 20
Chapter 12
7.
5.
Find the lateral area and surface area of each cone. Round to the nearest tenth.
3.
1.
Find the lateral area and surface area of each regular pyramid. Round to the
nearest tenth if necessary.
12-3
NAME
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Answers (Lesson 12-3)
A10
Glencoe Geometry
001_025_GEOCRMC12_890521.indd 22
Chapter 12
6.54 in2
What is the lateral surface area of this
pyramid? Round your answers to the
nearest hundredth.
3. PAPERWEIGHTS Daphne uses a
paperweight shaped like a pyramid with
a regular hexagon for a base. The side
length of the regular hexagon is 1 inch.
The altitude of the pyramid is 2 inches.
68 in2
2. TETRAHEDRON Sung Li builds a paper
model of a regular tetrahedron, a
pyramid with an equilateral triangle for
the base and three equilateral triangles
for the lateral faces. One of the faces of
the tetrahedron has an area of 17 square
inches. What is the total surface area of
the tetrahedron?
600 cm2
22
PERIOD
3814.89 in2
Glencoe Geometry
c. What is the lateral surface area of the
megaphone?
158.95 in2
b. What is the lateral surface area of the
tip that is removed?
3973.84 in2
a. What is the lateral surface area of the
original cone?
5. MEGAPHONES A megaphone is
formed by taking a cone with a radius
of 20 centimeters and an altitude of
60 centimeters and cutting off the tip.
The cut is made along a plane that is
perpendicular to the axis of the cone and
intersects the axis 12 centimeters from
the vertex. Round your answers to the
nearest hundredth.
78.54 in2
4. SPRAY PAINT A can of spray paint
shoots out paint in a cone shaped mist.
The lateral surface area of the cone is
65π square inches when the can is held
12 inches from a canvas. What is the
area of the part of the canvas that gets
sprayed with paint? Round your answer
to the nearest hundredth.
Surface Areas of Pyramids and Cones
Word Problem Practice
DATE
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Enrichment
DATE
PERIOD
120°
height of cone = 4.5 cm
(to nearest tenth of a centimeter)
lateral area = 24π cm
2
diameter of base = 8 cm
6 cm
Chapter 12
12.
23
13.
Lesson 12-3
5/30/09 4:13:23 PM
Glencoe Geometry
height of cone = 8.7 cm
(to nearest tenth of a centimeter)
lateral area = 50π cm2
diameter of base = 10 cm
20 cm
Make a paper pattern for each cone with the given measurements.
Then cut the pattern out and make the cone. Find the measurements.
11. Find the total surface area. 5.25 cm2
10. Find the lateral area. 3π cm2
9. Use the Pythagorean Theorem to calculate the height of the cone.
Use a decimal approximation. Check your calculation by measuring
the height with a metric ruler. 1.32 cm
8. What is the slant height of the cone? 2 cm
7. What is the circumference of the base of the cone? 3π cm
6. Measure the diameter of the circular base of the cone. 3 cm
5. Cut out the pattern and tape it together to
form a cone. See students’ work.
4. How long is the circular arc that is the
outside of the pattern? 3π cm
3. What is the circumference of the complete
circle? 4π cm
4
2. The pattern is what fraction of the
3
complete circle? −
1. Measure the radius of the circle to the
nearest centimeter. 2 cm
The pattern at the right is made from a
circle. It can be folded to make a cone.
Cone Patterns
12-3
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23
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 12
1. PAPER MODELS Patrick is making a
paper model
of a castle.
Part of the
model
involves
20 cm
20 cm
15 cm
cutting out
the net shown
and folding it
into a
pyramid. The
pyramid has a square base. What is
the lateral surface area of the
resulting pyramid?
12-3
NAME
Answers (Lesson 12-3)
Chapter 12
Surface Areas of Cones
Spreadsheet Activity
DATE
PERIOD
A11
8. r = 1.5 mm, = 4.5 mm 28.3 mm2
7. r = 10 mm, = 20 mm 942.5 mm2
Glencoe Geometry
Answers
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Volumes of Prisms and Cylinders
Study Guide and Intervention
DATE
3 cm
8 ft
4 cm
2 cm
27 cm3
1.5 cm
30°
8 ft
467.7 ft3
15 ft
12 ft
512 ft3
8 ft
Chapter 12
5.
3.
1.
6 cm
Find the volume of each prism.
Exercises
V = Bh
Volume of a prism
= (7)(3)(4)
B = (7)(3), h = 4
= 84
Multiply.
The volume of the prism is 84 cubic
centimeters.
7 cm
Find the volume
3.5 ft
base
6.
4.
15 ft
84 yd3
7 yd
1800 ft3
10 ft
9 cm3
3 cm
1.5 cm
4 yd
12 ft
4 cm
3 yd
Lesson 12-4
4/10/08 9:14:47 PM
Glencoe Geometry
V = Bh
Volume of a prism
= (6.3)(3.5)
B = 6.3, h = 3.5
= 22.05
Multiply.
The volume is 22.05 cubic feet.
2.
25
cubic foot
cubic yard
27 cubic feet = 1 cubic yard
PERIOD
Example 2
Find the volume of the
prism if the area of each base is 6.3
square feet.
If a prism has a volume of V cubic units, a height of h units,
and each base has an area of B square units, then V = Bh.
4 cm
Example 1
of the prism.
Volume
of a Prism
Volumes of Prisms The measure of the amount of space
that a three-dimensional figure encloses is the volume of the
figure. Volume is measured in units such as cubic feet, cubic
yards, or cubic meters. One cubic unit is the volume of a cube
that measures one unit on each edge.
12-4
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25
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
001_025_GEOCRMC12_890521.indd 24
24
Chapter 12
Glencoe Geometry
12. r = 11 m, = 13 m 829.4 m2
11. r = 10 m, = 2 m 377.0 m2
10. r = 10 cm, = 15 cm 785.4 cm2
6. r = 3 ft, = 1.5 ft 42.4 ft2
5. r = 1 ft, = 3 ft 12.6 ft2
9. r = 6.2 cm, = 1.2 cm 144.1 cm
4. r = 5 in., = 11 in. 251.3 in2
3. r = 3 in., = 7 in. 94.2 in2
2
2. r = 6 m, = 2 m 150.8 m2
1. r = 12 m, = 2.3 m 539.1 m2
Use a spreadsheet to find the surface area of each cone with the given
dimensions. Round to the nearest tenth.
Exercises
The surface area of the cone is 60.5 cm2 to the nearest tenth.
Click on the bottom right corner of cell C1 and drag it to C2. This returns the
surface area of the cone.
C
Step 2
Sheet 1
B
Use cell A2 for the radius of the cone and
cell B2 for the slant height.
1
2
Step 1
Example 2
Use a spreadsheet to determine
the surface area of a cone that has a radius of
2.5 centimeters and a slant height of 5.2 centimeters.
Round to the nearest tenth.
A
In cell C1, enter an equals sign followed by PI()*A1*B1 + PI()*A1^2. Then press
ENTER. This will return the surface area of the cone.
Step 2
The surface area of the conical box is 84.8 in2 to the nearest tenth.
Use cell A1 for the radius of the cone and cell B1 for the height.
Step 1
Example 1
Lucy wants to wrap a Mother’s Day gift. The gift she has bought for
her mother is in a conical box that has a slant height of 6 inches and has a radius
of 3 inches. She must determine the surface area of the box to determine how
much wrapping paper to buy. Use a spreadsheet to determine the surface area of
the box. Round to the nearest tenth.
You can use a spreadsheet to determine the surface area of a cone.
12-3
NAME
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Answers (Lesson 12-3 and Lesson 12-4)
PERIOD
Volumes of Prisms and Cylinders
Study Guide and Intervention (continued)
DATE
A12
5 in.
8 in.
10 cm
12 ft
2 ft
652.4 cm3
13 cm
84.8 ft3
1.5 ft
12.6 ft3
Glencoe Geometry
026_044_GEOCRMC12_890521.indd 26
Chapter 12
5.
3.
1.
1 ft
h
26
6.
4.
18 cm
4 yd
12.6 yd3
1 yd
6283.2 ft3
20 ft
20 ft
226.2 cm3
2. 2 cm
Glencoe Geometry
V = πr2h
Volume of a cylinder
= π(4)2(12) r = 4, h = 12
≈ 603.2
Simplify.
The Volume is about 603.2 cubic inches.
Use the Pythagorean Theorem to find the height of
the cylinder.
h2 + 52 = 132
Pythagorean Theorem
h2 = 144
Simplify.
h = 12
Take the square root of each side.
h
13 in.
Find the volume of each cylinder. Round to the nearest tenth.
Exercises
V = πr h
Volume of a cylinder
= π(3)2(4)
r = 3, h = 4
≈ 113.1
Simplify.
The volume is about 113.1 cubic
centimeters.
2
4 cm
3 cm
Find the volume of the
oblique cylinder.
Example 2
If a cylinder has a volume of V cubic units, a height of h units,
and the bases have a radius of r units, then V = πr 2h.
Find the volume
of the cylinder.
Example 1
Volume of
a Cylinder
height and the area of the base. When a solid is not a right solid, use
Cavalieri’s Priniciple to find the volume. The principle states that if two
solids have the same height and the same cross sectional area at every
level, then they have the same volume.
r
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Volumes of Prisms and Cylinders
Skills Practice
DATE
PERIOD
16 cm
15 mm
16,257.7 mm3
23 mm
90 m3
3m
13 m
2304 cm3
18 cm
8 cm
5m
6.
4.
2.
226.2 yd3
10 yd
6 yd
5280 in3
16 in.
96 ft3
6 ft
34 in.
22 in.
8 ft
2 ft
1224 cm3
17 cm
Chapter 12
7.
18 cm
4 cm
27
8.
141.4 in3
3 in.
5 in.
Lesson 12-4
6/14/08 2:47:17 PM
Glencoe Geometry
Find the volume of each oblique prism or cylinder. Round to the nearest
tenth if necessary.
5.
3.
1.
Find the volume of each prism or cylinder. Round to the nearest tenth
if necessary.
12-4
NAME
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9:21:18 PM
27
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 12
Volumes of Cylinders The volume of a cylinder is the product of the
12-4
NAME
Answers (Lesson 12-4)
Chapter 12
Volumes of Prisms and Cylinders
Practice
DATE
PERIOD
A13
17.5 mm
2600 yd3
13 yd
20 yd
3518.6 mm3
16 mm
2040 m3
17 m
26 m
10 yd
10 m
2
6.
4.
2.
5 in.
8 cm
25 ft
5 in.
9 in.
6031.9 cm3
30 cm
923.6 ft3
7 ft
97.4 in3
5 in.
Glencoe Geometry
Answers
28
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
6 ft
6 ft
Chapter 12
3840 cm3
3. FRAMES Margaret makes a square
frame out of four pieces of wood. Each
piece of wood is a
rectangular prism
with a length of
40 centimeters,
a height of
4 centimeters,
and a depth of
6 centimeters.
What is the total
volume of the
wood used in the frame?
57.6 ft3
What is the volume of the seat?
3
1 5 ft
2. BENCH Inside a lobby, there is a piece
of furniture for sitting. The furniture is
shaped like a simple block with a square
base 6 feet on each side and a height of
3
1−
feet.
12 in.
29
PERIOD
883,573 ft3
Lesson 12-4
4/10/08 9:11:41 PM
Glencoe Geometry
b. If instead of a rectangular shape, the
tunnel had a semicircular shape with
a 50-foot diameter, what would be its
volume? Round your answer to the
nearest cubic foot.
900,000 ft3
a. What will the volume of the tunnel be?
5. TUNNELS Construction workers are
digging a tunnel through a mountain.
The space inside the tunnel is going to
be shaped like a rectangular prism. The
mouth of the tunnel will be a rectangle
20 feet high and 50 feet wide and the
length of the tunnel will be 900 feet.
8 - π cm3
What is the exact volume of the pencil
grip?
4. PENCIL GRIPS A pencil grip is shaped
like a triangular prism with a cylinder
removed from the middle. The base of
the prism is a right isosceles triangle
with leg lengths of 2 centimeters. The
diameter of the base of the removed
cylinder is 1 centimeter. The heights of
the prism and the cylinder are the same,
and equal to 4 centimeters.
Volumes of Prisms and Cylinders
2035.8 in3
5
DATE
Word Problem Practice
1. TRASH CANS The Meyer family uses a
kitchen trash can shaped like a cylinder.
It has a height of
18 inches and a base
diameter of 12 inches.
What is the volume
18 in.
of the trash can? Round
your answer to the
nearest tenth of a
cubic inch.
12-4
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29
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
026_044_GEOCRMC12_890521.indd 28
Chapter 12
1275 lb
c. If a cubic foot of water weighs about 62.4 pounds, what is the weight of the water in
the aquarium to the nearest five pounds?
152.9 gal
b. If there are 7.48 gallons in a cubic foot, how many gallons of water does the aquarium
hold?
20.4 ft3
a. What is the volume of the aquarium in cubic feet?
7. AQUARIUM Mr. Gutierrez purchased a cylindrical aquarium for his office.
1
The aquarium has a height of 25 −
inches and a radius of 21 inches.
5.
3.
1.
Find the volume of each prism or cylinder. Round to the nearest tenth if
necessary.
12-4
NAME
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Answers (Lesson 12-4)
Enrichment
PERIOD
A14
2
Glencoe Geometry
026_044_GEOCRMC12_890521.indd 30
Chapter 12
visible surface area = 164 in
volume = 136 in3
30
visible surface area = 19 in2
visible surface area = 17 in2
5 in.
volume = 5 in3
volume = 5 in3
3 in.
3 in.
Glencoe Geometry
4 in.
5 in.
3 in.
visible surface area = 19 in2
volume = 5 in3
8 in.
3 in.
4 in.
5.
visible surface area = 15 in2
visible surface area = 14 in2
4.
volume = 4 in3
2.
volume = 4 in3
6. Find the volume and the visible surface
area of the figure at the right.
3.
1.
Use paper, scissors, and tape to make five cubes that have one-inch edges.
Arrange the cubes to form each shape shown. Then find the volume and
the visible surface area. In other words, do not include the area of surface
covered by other cubes or by the table or desk.
DATE
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Volumes of Pyramids and Cones
Study Guide and Intervention
DATE
8 ft
10 ft
1200 in3
15 in.
110.9 cm3
8 cm
12 cm
320 ft3
12 ft
10 ft
Chapter 12
5.
3.
1.
15 in.
16 in.
4 cm
8 ft
31
6.
4.
2.
10 ft
64 yd3
5 yd
561.2 ft3
regular
hexagon
120 ft3
15 ft
6 yd
6 ft
8 yd
6 ft
18 ft
Lesson 12-5
4/10/08 9:11:47 PM
Glencoe Geometry
Find the volume of each pyramid. Round to the nearest tenth if necessary.
Exercises
≈ 213.3
Multiply.
The volume is about 213.3 cubic feet.
B = (8)(8), h = 10
Volume of a pyramid
Find the volume of the square pyramid.
3
1
=−
(8)(8)10
3
1
Bh
V=−
Example
3
If a pyramid has a volume of V cubic units, a height of h units,
1
and a base with an area of B square units, then V = −
Bh.
Volume of
a Pyramid
8 ft
PERIOD
that have the same base and the same height. It is clear that the volume
of the pyramid is less than the volume of the prism. More specifically,
the volume of the pyramid is one-third of the volume of the prism.
Volumes of Pyramids This figure shows a prism and a pyramid
12-5
NAME
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6:34:37 AM
31
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 12
Visible Surface Area
12-4
NAME
Answers (Lesson 12-4 and Lesson 12-5)
Chapter 12
PERIOD
Volumes of Pyramids and Cones
Study Guide and Intervention (continued)
DATE
r = 5, h = 12
Volume of a cone
Find the volume of the cone.
A15
3
2513.3 ft3
26 ft
1131.0 in3
12 in.
Glencoe Geometry
379.1 cm3
16 cm
45°
1332.9 yd3
20 yd
18 yd 45°
3
10 ft
670.2 ft
Answers
32
6.
4.
2.
8 ft
5 cm
12 cm
r
Glencoe Geometry
h
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Volumes of Pyramids and Cones
Skills Practice
DATE
PERIOD
10 in.
1231.5 yd3
14 yd
357.8 in3
14 in.
25 yd
8 in.
5 ft
25 m
12 m
6.
1210.6 mm3
18 mm
3769.9 m3
4.
7 cm
74.7 cm3
8 cm
66°
4 cm
31 ft3
4 ft
Chapter 12
7.
4 ft
6 ft
33
8.
452.4 cm3
12 cm
6 cm
Lesson 12-5
6/14/08 2:47:35 PM
Glencoe Geometry
Find the volume of each oblique pyramid or cone. Round to the nearest tenth
if necessary.
5.
3.
66.7 ft3
5 ft
8 ft
Find the volume of each pyramid or cone. Round to the nearest tenth if necessary.
1.
2.
12-5
NAME
4/10/08026_044_GEOCRMC12_890521.indd
9:11:52 PM
33
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
026_044_GEOCRMC12_890521.indd 32
20 ft
30 in.
10 cm
6 cm
301.6 cm
Chapter 12
5.
3.
1.
Find the volume of each cone. Round to the nearest tenth.
Exercises
3
If a cone has a volume of V cubic units, a height of h units,
1 2
and the bases have a radius of r units, then V = −
πr h.
≈ 314.2
Simplify.
The volume of the cone is about 314.2 cubic centimeters.
3
1
=−
π(5)212
3
1 2
πr h
V=−
Example
Volume of
a Cone
Volumes of Cones For a cone, the volume is one-third the product of the
height and the area of the base. The base of a cone is a circle, so the area of the
base is πr2.
12-5
NAME
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Answers (Lesson 12-5)
Volumes of Pyramids and Cones
Practice
PERIOD
A16
3
132 in
6 in.
3
9.2 yd
11 in.
9 ft
6 in.
1419.4 ft3
19 ft
317.5 yd
9.2 yd
13 yd
6.
4.
2.
3
3
4688.3 ft
11 ft
1104.6 mm3
12 mm
2395.8 cm
25 cm
37 ft
52°
12.5 cm
23 cm
Glencoe Geometry
026_044_GEOCRMC12_890521.indd 34
Chapter 12
about 341,413.3 m3
34
Glencoe Geometry
8. HISTORY The start of the pyramid age began with King Zoser’s pyramid, erected in the
27th century B.C. In its original state, it stood 62 meters high with a rectangular base
that measured 140 meters by 118 meters. Find the volume of the original pyramid.
about 15.9 m3
7. CONSTRUCTION Mr. Ganty built a conical storage shed. The base of the shed is
4 meters in diameter and the height of the shed is 3.8 meters. What is the volume of
the shed?
5.
3.
1.
Find the volume of each pyramid or cone. Round to the nearest tenth if necessary.
12-5
DATE
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 12
60.63 ft3
What was the volume of the teepee?
Round your answer to the nearest
hundredth.
65˚
3. TEEPEE Caitlyn made a teepee for a
class project. Her teepee had a diameter
of 6 feet. The angle the side of the teepee
made with the ground was 65°.
5400 yd3
What is the volume of the greenhouse?
30 yd
2. GREENHOUSES A greenhouse has the
shape of a square pyramid. The base has
a side length of 30 yards. The height of
the greenhouse is 18 yards.
35
PERIOD
3 feet
592 ft3
Lesson 12-5
5/30/09 3:08:51 PM
Glencoe Geometry
c. What is the volume of the stage?
432 ft3
b. What is the volume of the top of the
pyramid that is removed to get the
stage?
1024 ft3
a. What is the volume of the entire
square pyramid that the stage is
part of?
16 feet
12 feet
5. STAGES A stage has the form of a
square pyramid with the top sliced off
along a plane parallel to the base. The
side length of the top square is 12 feet
and the side length of the bottom square
is 16 feet. The height of the stage is
3 feet.
6.28 ft3
What is the volume of the stone that the
sculptor must remove? Round your
answer to the nearest hundredth.
4. SCULPTING A sculptor wants to remove
stone from a cylindrical block 3 feet high
and turn it into a cone. The diameter of
the base of the cone and cylinder is
2 feet.
Volumes of Pyramids and Cones
What is the volume of the cone? Round
your answer to the nearest hundredth.
5.03 in3
DATE
Word Problem Practice
1. ICE CREAM DISHES The part of a dish
designed for ice cream is shaped like an
upside-down cone. The base of the cone
has a radius of 2 inches and the height
is 1.2 inches.
12-5
NAME
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9:12:00 PM
35
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 12
18 yd
NAME
Answers (Lesson 12-5)
Chapter 12
Enrichment
DATE
)
A17
19.5 cm
5 cm
6 cm
13 cm
9 cm
6m
12 m
8m
trapezoids; 151.6 m3
5m
3m
2.25 m
4.5 m
rectangles; 617.5 cm3
Glencoe Geometry
4.5 in.
7.5 in.
3 in.
13 ft
circles; 3480.9 ft3
12 ft
circles; 335.8 in3
Answers
36
4.
2.
Glencoe Geometry
7 ft
PERIOD
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Surface Areas and Volumes of Spheres
Study Guide and Intervention
DATE
PERIOD
If a sphere has a surface area of S square units and a radius of r units, then S = 4πr2.
Simplify.
r=6
Surface area of a sphere
6 cm
r
3 ft
5m
84.8 ft2
314.2 m2
4.
2.
Chapter 12
37
6. hemisphere: area of great circle ≈ 4π ft2 37.7 ft2
7 in
9 cm
5. sphere: circumference of great circle = π cm 3.1 cm2
3.
1.
Lesson 12-6
5/30/09 3:09:01 PM
Glencoe Geometry
190.9 cm2
153.9 in2
Find the surface area of each sphere or hemisphere. Round to the nearest tenth.
Exercises
The surface area is 452.4 square centimeters.
S = 4πr2
= 4π(6)2
≈ 452.4
Example
Find the surface area of a sphere to the nearest tenth
if the radius of the sphere is 6 centimeters.
Surface Area
of a Sphere
as the total area of all of the nonoverlapping strips it would take to cover
the sphere. If r is the radius of the sphere, then the area of a great circle of
the sphere is πr2. The total surface area of the sphere is four times the area
of a great circle.
Surface Areas of Spheres You can think of the surface area of a sphere
12-6
NAME
6/1/09 026_044_GEOCRMC12_890521.indd
10:19:21 AM
37
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
026_044_GEOCRMC12_890521.indd 36
Chapter 12
3.
1.
Describe the shape of the bases of each frustum. Then find the
volume. Round to the nearest tenth.
where h = height (perpendicular distance between the bases),
B1 = area of top base, and B2 = area of bottom base.
(
1
V=−
h B1 + B2 + √
B1B2 ,
3
A frustum is a figure formed when a plane intersects a pyramid or
cone so that the plane is parallel to the solid’s base. The frustum is
the part of the solid between the plane and the base. To find the
volume of a frustum, the areas of both bases must be calculated and
used in the formula.
Frustums
12-5
NAME
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Answers (Lesson 12-5 and Lesson 12-6)
PERIOD
Surface Areas and Volumes of Spheres
A18
523.6 ft3
5 ft
2.
6 in.
452.4 in3
Glencoe Geometry
026_044_GEOCRMC12_890521.indd 38
Chapter 12
38
6. hemisphere: area of great circle ≈ 50 m2 133.0 m3
5. sphere: circumference of great circle ≈ 25 ft 263.9 ft3
4. hemisphere: radius 5 in. 261.8 in3
1.
3.
8 cm
r
Glencoe Geometry
8578.6 in3
16 in.
Find the volume of each sphere or hemisphere. Round to the nearest tenth.
Exercises
≈ 2144.7
Simplify.
The volume is about 2144.7 cubic centimeters.
r=8
Volume of a sphere
Find the volume of a sphere with radius 8 centimeters.
Example
4 3
πr
V=−
3
4
=−
π (8)3
3
4 3
If a sphere has a volume of V cubic units and a radius of r units, then V = −
πr .
Volume of
a Sphere
length of its radius. If you know the length of the radius of a sphere, you
can calculate its volume.
3
Study Guide and Intervention (continued)
DATE
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
PERIOD
Surface Areas and Volumes of Spheres
Skills Practice
DATE
615.8 in2
7 in.
2.
3217.0 m2
32 m
2226.1 cm3
16.2 cm
6.
94.8 ft
446,091.2 ft3
Chapter 12
9. sphere: diameter = 10 in. 523.6 in3
39
8. sphere: circumference of a great circle ≈ 26 m 296.8 m3
7. hemisphere: diameter = 48 yd 28,952.9 yd3
5.
Lesson 12-6
5/30/09 3:09:16 PM
Glencoe Geometry
Find the volume of each sphere or hemisphere. Round to the nearest tenth.
4. sphere: area of great circle ≈ 28.6 in2 114.4 in2
3. hemisphere: radius of great circle = 8 yd 603.2 yd2
1.
Find the surface area of each sphere or hemisphere. Round to the nearest tenth.
12-6
NAME
3/27/10 026_044_GEOCRMC12_890521.indd
10:47:18 PM
39
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 12
Volumes of Spheres A sphere has one basic measurement, the
12-6
NAME
Answers (Lesson 12-6)
Chapter 12
PERIOD
Surface Areas and Volumes of Spheres
Practice
DATE
530.9 cm2
6.5 cm
2.
A19
7832.9 ft3
12.32 ft
6.
Glencoe Geometry
Answers
40
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
PERIOD
83,322,891.2 km2
3. MOONS OF SATURN The planet
Saturn has several moons. These can
be modeled accurately by spheres.
Saturn’s largest moon Titan has a
radius of about 2575 kilometers. What is
the approximate surface area of Titan?
Round your answer to the nearest tenth.
95.426 in3
2. BILLIARDS A billiard ball set consists
1
of 16 spheres, each 2 −
inches in
4
diameter. What is the total volume of
a complete set of billiard balls? Round
your answer to the nearest thousandth
of a cubic inch.
41
1.91
Lesson 12-6
5/30/09 3:09:26 PM
Glencoe Geometry
c. What is the ratio of the surface area
of the cube to the surface area of the
sphere? Round your answer to the
nearest hundredth.
12.57 in2
b. What is the surface area of the
sphere? Round your answers to the
nearest hundredth.
24 in2
a. What is the surface area of the cube?
5. CUBES Marcus builds a sphere inside of
a cube. The sphere fits snugly inside the
cube so that the sphere touches the cube
at one point on each side. The side
length of the cube is 2 inches.
0.015
4. THE ATMOSPHERE About 99% of
Earth’s atmosphere is contained in a
31-kilometer thick layer that enwraps
the planet. The Earth itself is almost a
sphere with radius 6378 kilometers.
What is the ratio of the volume of the
atmosphere to the volume of Earth?
Round your answer to the nearest
thousandth.
Surface Areas and Volumes of Spheres
12.57 in2
Chapter 12
DATE
Word Problem Practice
1. ORANGES Mandy cuts a spherical
orange in half along a great circle. If the
radius of the orange is 2 inches, what is
the area of the cross section that Mandy
cut? Round your answer to the nearest
hundredth.
12-6
NAME
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3:09:21 PM
41
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
026_044_GEOCRMC12_890521.indd 40
Chapter 12
9. sphere: radius = 12.4 in. 7986.4 in3
8. sphere: circumference ≈ 36 yd 787.9 yd3
32 m
8578.6 m3
7. hemisphere: diameter = 18 mm 1526.8 mm3
5.
Find the volume of each sphere or hemisphere. Round to the nearest tenth.
4. sphere: area of great circle ≈ 29.8 m2 119.2 m2
2
24,884.6 ft2
89 ft
3. hemisphere: radius of great circle = 8.4 in. 665.0 in
1.
Find the surface area of each sphere or hemisphere. Round to the nearest tenth.
12-6
NAME
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Answers (Lesson 12-6)
Enrichment
2.7 g/cm
19.32 g/cm3
11.35 g/cm3
10.50 g/cm3
Copper
Iron
Platinum
8.96 g/cm
7.874 g/cm3
21.45 g/cm3
3
3
A20
Glencoe Geometry
026_044_GEOCRMC12_890521.indd 42
Chapter 12
42
8. An aluminum ball and a lead ball each have a radius of 1.2 centimeters.
Which weighs more? How much more? lead; 62.6 g
7. A silver ball and a copper ball each have a diameter of 3.5 centimeters.
Which weighs more? How much more? silver; 34.6 g
6. An iron ball weighs 804 grams. Find the diameter of the ball to the
nearest tenth of a centimeter. 5.8 cm
5. A lead ball weighs 326 grams. Find the radius of the ball to the nearest
tenth of a centimeter. 1.9 cm
Solve. Assume the balls are spherical. Round your answers
to the nearest tenth.
4. a platinum ball with radius 0.7 cm 30.8 g
3. an aluminum ball with radius 3 cm 305.4 g
2. a gold ball 0.6 cm in diameter 2.2 g
1. a copper ball 1.2 cm in diameter 8.1 g
Find the mass of each metal ball described. Assume the balls are
spherical. Round your answers to the nearest tenth.
Exercises
The mass is about 2.81 grams.
≈ 10.5(0.27)
≈ 2.81
PERIOD
Glencoe Geometry
Find the mass of a silver ball that is 0.8 cm in diameter.
4
π(0.4)3
= 10.5 · −
M=D·V
Example
To calculate the mass of a piece of metal, multiply volume by density.
Aluminum
Gold
Lead
Silver
3
The density of a metal is a ratio of its mass to its volume. For
example, the mass of aluminum is 2.7 grams per cubic centimeter.
Here is a list of several metals and their densities.
DATE
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Spherical Geometry
Study Guide and Intervention
DATE
PERIOD
Name each of the following on sphere K.
)
(
&
*
"
+
$
K
'
%
#
8
;
5
7
3
F
6
−−
⎯, XU and SRT
⎯ and SV
WT
9
:
4
2.
'
;
(
&
3
M
$
%
−−
⎯ and EB
⎯, GC and GRZ
AD
"
#
V
No
4.
V
No
Chapter 12
43
Lesson 12-7
5/30/09 3:09:36 PM
Glencoe Geometry
Yes, the equator is a great circle. No other line of latitude is a great
circle.
5. GEOGRAPHY Lines of latitude run horizontally across the surface of Earth. Are there
any lines of latitude that are great circles? Explain.
3.
Determine whether figure u on each of the spheres shown is a line in spherical
geometry.
1.
Name two lines containing point Z, a segment containing point R, and a triangle
in each of the following spheres.
Exercises
AHI is a triangle on sphere K
c. a triangle
b. a line segment containing the point J
−−
ID is a segment on sphere K that contains the point J
and BH
are lines on sphere K that contain the point F
EG
a. two lines containing the point F
Example
where a plane is a flat surface made up of points that extends infinitely in all directions.
In spherical geometry, a plane is the surface of a sphere.
Geometry On A Sphere Up to now, we have been studying Euclidean geometry,
12-7
NAME
5/30/09026_044_GEOCRMC12_890521.indd
3:09:31 PM
43
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 12
Spheres and Density
12-6
NAME
Answers (Lesson 12-6 and Lesson 12-7)
Chapter 12
PERIOD
Spherical Geometry
Study Guide and Intervention (continued)
DATE
A21
N
Glencoe Geometry
Answers
44
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Spherical Geometry
Skills Practice
DATE
PERIOD
6
*
)
−−
⎯ and ⎯, SG, and STH
SF
IH
4
5
C
(
'
,
2.
"
&
L
$
#
%
5
−−
⎯ and ⎯, CA , and ATE
BD
ET
,
No
V
4.
Yes
basketball
V
Chapter 12
45
No. Three lines divide the plane into 6 or 7 separate parts.
8. Three non-parallel lines divide the plane into 7 separate parts.
Yes. The same statement works in spherical geometry.
Lesson 12-7
6/2/09 4:48:36 PM
Glencoe Geometry
7. Two lines meet at two 90° angles or they meet at angles whose sum is 180°.
No. There are no parallel lines in spherical geometry.
6. If two lines meet a third line at the same angle, those lines are parallel.
Yes. The same statement works in spherical geometry.
5. If two lines form vertical angles, then the angles are equal in measure.
Tell whether the following postulate or property of plane Euclidean geometry has
a corresponding statement in spherical geometry. If so, write the corresponding
statement. If not, explain your reasoning.
3.
Determine whether figure u on each of the spheres shown is a line in
spherical geometry.
1.
Name two lines containing point K, a segment containing point T, and a triangle
in each of the following spheres.
12-7
NAME
045_055_GEOCRMC12_890521.indd 45
4/10/08 9:12:33 PM
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
026_044_GEOCRMC12_890521.indd 44
Chapter 12
Yes. The same statement works in spherical geometry.
6. A largest angle of a triangle is opposite the largest side.
Yes. The same statement works in spherical geometry.
5. Three noncollinear points determine a triangle.
No. There are no parallel lines in spherical geometry.
4. If two lines are perpendicular to a third line, they are parallel.
No. There are no parallel lines in spherical geometry.
3. Given two parallel lines and a transversal, alternate interior angles are congruent.
Yes. The same statement works in spherical geometry.
2. Given a line and a point on the line, there is only one perpendicular line going through
that point.
No. If two nonidentical lines intersect at a point, they intersect again
on the opposite side of the sphere.
1. If two nonidentical lines intersect at a point, they do not intersect again.
Tell whether the following postulate or property of plane Euclidean
geometry has a corresponding statement in spherical geometry. If so,
write the corresponding statement. If not, explain your reasoning.
Exercises
On the sphere to the right, if we are given line m we see that it
goes through the poles of the sphere. If we try to make any other
line on the sphere, it would intersect line m at exactly 2 points.
This property is not true in spherical geometry.
A corresponding statement in spherical geometry would be:
“Given any line, there are no parallel lines.”
Given any line, there are an infinite number of parallel lines.
Example
Tell whether the following postulate or property of plane Euclidean
geometry has a corresponding statement in spherical geometry. If so, write the
corresponding statement. If not, explain your reasoning.
properties of Euclidean geometry are true in spherical geometry. Others are not true or are
true only under certain circumstances.
Comparing Euclidean and Spherical Geometries Some postulates and
12-7
NAME
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Answers (Lesson 12-7)
Spherical Geometry
Practice
DATE
PERIOD
&
,
L
%
#
'
⎯ and AE
⎯, FA
⎯, ADE
CB
$
5
"
2.
5
:
M
9
⎯ and ⎯, ZX
⎯ and TZK
ZX
TY
;
A22
No
tennis ball
V
4.
Yes
V
Glencoe Geometry
045_055_GEOCRMC12_890521.indd 46
Chapter 12
46
Glencoe Geometry
The shortest route is to fly along the great circle connecting New York
and Seattle, which crosses Canada.
9. AIRPLANES When flying an airplane from New York to Seattle, what is the shortest
route: flying directly west, or flying north across Canada? Explain.
Yes. The same statement works in spherical geometry.
8. All equilateral triangles are similar.
Yes. The same statement works in spherical geometry.
7. Given a line and a point not on the line, there is exactly one line that goes through the
point and is perpendicular to the line.
No. The sum of the angles of a triangle is more than 180°.
6. The sum of the angles of a triangle is 180°.
No. A triangle can have at most three obtuse angles.
Tell whether the following postulate or property of plane Euclidean geometry has
a corresponding statement in spherical geometry. If so, write the corresponding
statement. If not, explain your reasoning.
5. A triangle can have at most one obtuse angle.
3.
Determine whether figure u on each of the spheres shown is a line in spherical
geometry.
1.
,
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Spherical Geometry
7
C
B
Chapter 12
All three lines must intersect at
two points.
4. GEOMETRY Three nonidentical lines on
the circle divide it into either
6 sections or 8 triangles. What condition
is needed so that the three lines form
6 sections?
137,900,000 square miles
Source: NASA
3. OCEAN If the oceans cover 70% of
Earth’s surface, what is the surface area
of the oceans?
24,625,000 square miles
Source: NASA
2. EARTH The Equator and the Prime
Meridian are perpendicular great circles
that divide Earth into North, South and
East, West hemispheres. If Earth has a
surface area of 197,000,000 square
miles, what is the surface area of the
North-East section of Earth?
6π
−
square feet
D
A
47
DATE
PERIOD
138.3 miles
Lesson 12-7
5/30/09 3:10:18 PM
Glencoe Geometry
b. Seattle, Washington, has coordinates
(47°N, 122°W) and Portland, Oregon,
has coordinates (45°N, 122°W).
Estimate the distance between
Portland and Seattle to the nearest
tenth.
415.4 miles
a. The mean radius of Earth is 3963
miles. Atlanta, Georgia, has
coordinates (33°N, 84°W) and
Cincinnati, Ohio, has coordinates
(39°N, 84°W). Estimate the distance
between Atlanta and Cincinnati to
the nearest tenth.
5. GEOGRAPHY Latitude and longitude
lines are imaginary lines on Earth. The
lines of latitude are horizontal concentric
circles that help to define the distance a
place is from the equator. Lines of
latitude are measured in degrees. The
equator is 0°. The north pole is 90° north
latitude. The lines of longitude are great
circles that help to define the distance a
place is from the Prime Meridan, which
is located in England and considered the
longitude of 0°.
Word Problem Practice
1. PAINTING Consider painting
quadrilateral ABCD on the beach ball
with radius 1 ft. What is the surface
area you would need to paint?
12-7
NAME
045_055_GEOCRMC12_890521.indd 47
6/2/09 4:46:23 PM
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 12
Name two lines containing point K, a segment containing point T, and a triangle
in each of the following spheres.
12-7
NAME
Answers (Lesson 12-7)
Chapter 12
Enrichment
180˚ W
DATE
90˚ W
A23
0˚
90˚ E
PERIOD
60˚ N
40˚ N
20˚ N
0˚
20˚ S
40˚ S
60˚ S
180˚ E
Glencoe Geometry
Answers
48
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Congruent and Similar Solids
Study Guide and Intervention
DATE
PERIOD
Corresponding angles are congruent
Corresponding edges are congruent
Corresponding faces are congruent
Volumes are equal
8
2
4 in.
5 in.
3 in.
4 in.
8 in.
10 in.
6 in.
8 in.
2 cm
Similar, 1:2
4 in.
Similar, 1:5
1 cm
Chapter 12
3.
1.
5 cm
10 cm
8 in.
49
4.
2.
2m
Neither
2m
Congruent
4.2 in.
4m
12.3 in.
1m
3m
Lesson 12-8
5/30/09 3:10:33 PM
Glencoe Geometry
1m
4.2 in.
12.3 in.
Determine whether the pair of solids is similar, congruent, or neither. If the solids
are similar, state the scale factor.
Exercises
The ratios of the corresponding sides are equal, so the triangular prisms are similar. The
scale factor is 1:2. Since the scale factor is not 1:1, the solids are not congruent.
10
4
1
ratio of height: −
=−
2
2
5
1
=−
ratio of hypotenuse: −
8
4
1
ratio of length: −
=−
2
3
1
ratio of width: −
=−
6
Example
Determine whether the pair of solids
is similar, congruent, or neither. If the solids are
similar, state the scale factor.
•
•
•
•
Identify Congruent or Similar Solids Similar solids have exactly the same shape
but not necessarily the same size. Two solids are similar if they are the same shape and the
ratios of their corresponding linear measures are equal. All spheres are similar and all
cubes are similar. Congruent solids have exactly the same shape and the same size.
Congruent solids are similar solids with a scale factor of 1:1. Congruent solids have the
following characteristics:
12-8
NAME
4/12/08045_055_GEOCRMC12_890521.indd
9:02:10 PM
49
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
045_055_GEOCRMC12_890521.indd 48
Chapter 12
Triangles, hexagons, prisms.
4. The Mercator projection uses a cylinder to map Earth, while the Lambert projection uses
a cone to map Earth. What other shapes do you think could be used to map Earth?
No. Lines of longitude become shorter as you travel towards the poles.
3. Does each square on the Lambert projection have the same surface area? Explain.
No. Near the top, the width across is much smaller than the equator.
2. Does each square on the Mercator projection have the same surface area? Explain.
To measure distances close to the equator.
1. When would it be useful to use a Mercator projection of Earth?
The map on the right is a Lambert projection.
When a pilot draws a straight line between two
points on this map the line shows true bearing, or
relative direction to the North Pole. However, the
bottom area of this map distorts distances.
The map on the right is a Mercator projection
of Earth. On this map Greenland appears to be
the same size as Africa. But Greenland has a land
area of 2,166,086 square kilometers and Africa
has a land area of 30,365,700 square kilometers.
When making maps of Earth, cartographers
must show a sphere on a plane. To do this they
have to use projections, a method of converting a
sphere into a plane. But these projections have
their limitations.
Spherical Geometry
Projections
12-7
NAME
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Answers (Lesson 12-7 and Lesson 12-8)
PERIOD
Congruent and Similar Solids
Study Guide and Intervention (continued)
DATE
6
3
3
A24
2 ft
6 ft
Glencoe Geometry
045_055_GEOCRMC12_890521.indd 50
Chapter 12
125π cubic in.
50
Glencoe Geometry
5. CONSTRUCTION A building company uses two similar sizes of pipes. The smaller size
has a radius of 1 inch and length of 8 inches. The larger size has a radius of 2.5 inches
What is the volume of the larger pipes?
12 in.
4. COMPUTERS A small rectangular laptop has a width of 10 inches and an area of
80 square inches. A larger and similar laptop has a width of 15 inches. What is the
length of the larger laptop?
9:16
3. Two similar triangular prisms have volumes of 27 square meters and 64 square meters.
What is the ratio of the surface area of the small prism to the surface area of the large
prism?
1:64
2. Two similar cones have heights of 3 feet and 12 feet. What is the ratio of the volume of
the small cone to the volume of the large cone?
9:64
1. Two cubes have side lengths of 3 inches and 8 inches. What is the ratio of the surface
area of the small cube to the surface area of the large cube?
Exercises
So, the ratio of the volumes is 1:27.
(1)3
a3
1
−
= −3 or −
27
b3
(3)
1
.
The scale factor is −
2
1
−− = −
or −
radius of the small sphere
radius of the large sphere
First, find the scale factor.
Example
Two spheres have radii of 2 feet and 6 feet.
What is the ratio of the volume of the small sphere to the
volume of the large sphere?
If two similar solids have a scale factor of a:b then,
• the ratio of their surface areas is a2:b2.
• the ratio of their volumes is a3:b3.
similar, certain properties are known.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Congruent and Similar Solids
Skills Practice
DATE
PERIOD
2 cm
congruent
5m
12 cm
similar; 1:3
4 cm
3 cm
6 cm
9 cm
10 m
4.
2.
6 cm
3 ft
1 ft
similar; 1:3
1 ft
similar; 3:4
9 cm
3 ft
12 cm
9 ft
8 cm
3 ft
Chapter 12
51
Lesson 12-8
6/1/09 10:25:58 AM
Glencoe Geometry
7. COOKING Two stockpots are similar cylinders. The smaller stockpot has a height of
10 inches and a radius of 2.5 inches. The larger stockpot has a height of 16 inches. What
is the volume of the larger stockpot? Round to the nearest tenth. 804.2 in3
3:2
6. Two similar cylinders have surface areas of 40π square feet and 90π square feet. What
is the ratio of the height of the large cylinder to the height of the small cylinder?
64:343
5. Two similar pyramids have heights of 4 inches and 7 inches What is the ratio of the
volume of the small pyramid to the volume of the large pyramid?
3.
1.
Determine whether each pair of solids is similar, congruent, or neither. If the
solids are similar, state the scale factor.
12-8
NAME
6/3/09 045_055_GEOCRMC12_890521.indd
11:22:33 PM
51
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 12
Properties of Congruent or Similar Solids When pairs of solids are congruent or
12-8
NAME
Answers (Lesson 12-8)
Chapter 12
Congruent and Similar Solids
Practice
DATE
PERIOD
6 cm
A25
4m
1m
24 cm
3m
5m
18 cm
4.
2.
neither
2 cm
5 cm
neither
5 cm
5 cm
12 cm
24 cm
1.5 cm
10 cm
10 cm
10 cm
Glencoe Geometry
Answers
52
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
15 cm
25 cm
25 cm
Chapter 12
6144 cubic feet of grain
3. FARMING A farmer has two similar
cylindrical grain silos. The smaller silo
is 25 feet tall and the larger silo is
40 feet tall. If the smaller silo can hold
1500 cubic feet of grain, how much can
the larger silo hold?
40 cm
15 cm
24 cm
2. MANUFACTURING Boxes, Inc. wants
to make the two boxes below. How long
does the second box need to be so that
they are similar?
6.75 in.
DATE
53
PERIOD
27:64
Lesson 12-8
5/30/09 3:10:59 PM
Glencoe Geometry
b. Find the ratio of the volume of MLB
baseballs to the volume of NSA
softballs. Round to the nearest tenth.
3:4
a. Find the ratio of the circumference of
MLB baseballs to the circumference
of NSA softballs.
Source: MLB, NSA
5. BASEBALL Major League Baseball or
MLB, rules state that baseballs must
have a circumference of 9 inches. The
National Softball Association, or NSA,
rules state that softballs must have a
circumference not exceeding 12 inches.
1.5:1
Source: NASA
4. PLANETS Earth has a surface area of
about 196,937,500 square miles. Mars
has a surface area of about 89,500,000
square miles. What is the ratio of the
radius of Earth to the radius of Mars?
Round to the nearest tenth.
Congruent and Similar Solids
Word Problem Practice
1. COOKING A cylindrical pot is 4.5 inches
tall and has a radius of 4 inches. How
tall would a similar pot be if its radius is
6 inches?
12-8
NAME
5/30/09045_055_GEOCRMC12_890521.indd
3:10:53 PM
53
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
045_055_GEOCRMC12_890521.indd 52
Chapter 12
1.6 square feet
7. ARCITHECTURE Architects make scale models of buildings to present their ideas to
clients. If an architect wants to make a 1:50 scale model of a 4000 square foot house,
how many square feet will the model have?
1:5.4
6. Two similar ice cream cones are made of a half sphere on top and a cone on bottom.
They have radii of 1 inch and 1.75 inches respectively. What is the ratio of the volume
of the small ice cream cone to the volume of the large ice cream cone? Round to the
nearest tenth.
216:343
5. Two cubes have surface areas of 72 square feet and 98 square feet. What is the ratio of
the volume of the small cube to the volume of the large cube?
congruent
3m
4m
5m
similar, 1:3
8 cm
3. 1 m
1.
Determine whether the pair of solids is similar, congruent, or neither. If the solids
are similar, state the scale factor.
12-8
NAME
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Answers (Lesson 12-8)
Enrichment
PERIOD
2
A26
Glencoe Geometry
045_055_GEOCRMC12_890521.indd 54
Chapter 12
54
16. It appears that if the dimensions of a solid are doubled, the
volume is multiplied by
. 8
15. The volume of the large sphere is how many times greater
than the volume of the small sphere? 8 times
14. What is the volume of the large sphere? 288π m3
13. What is the volume of the small sphere? 36π m3
The large sphere at the right has twice the radius of the small sphere.
12. The volume of the large cube is how many times greater than
that of the small cube? 8 times
11. What is the volume of the large cube? 1000 in3
10. What is the volume of the small cube? 125 in
3
9. How long are the edges of the large cube? 10 in.
The sides of the large cube are twice the size of the sides of the small cube.
Now consider how doubling the dimensions affects the volume of a cube.
8. It appears that if the dimensions of a solid are doubled, the
surface area is multiplied by
.
4
7. The surface area of the large sphere is how many times
greater than the surface area of the small sphere? 4 times
6. What is the surface area of the large sphere? 144π m2
5. What is the surface area of the small sphere? 36π m2
The radius of the large sphere at the right is twice the radius of
the small sphere.
4. The surface area of the large cube is how many times greater
than that of the small cube? 4 times
3. What is the surface area of the large cube? 600 in2
2. What is the surface area of the small cube? 150 in
1. How long are the edges of the large cube? 10 in.
The sides of the large cube are twice the size of the sides of the
small cube.
3m
3m
Glencoe Geometry
5 in.
5 in.
Consider what happens to surface area when the sides of a figure are doubled.
DATE
4/10/08 9:11:09 PM
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 12
Doubling Sizes
12-8
NAME
Answers (Lesson 12-8)
Chapter 12 Assessment Answer Key
Quiz 1 (Lessons 12-1 and 12-2)
Page 57
1.
4.
1.
156 cm3
2.
1005.3 cm3
8.6 m
2.
3.
Quiz 3 (Lessons 12-4 and 12-5)
Page 58
3.
120 units2
4.
1.
C
2.
G
3.
D
4.
F
5.
D
261.8 cm3
201.1 m2
360 cm2
5.
C
B
5.
Quiz 4 (Lessons 12-7 and 12-8)
Page 58
Quiz 2 (Lessons 12-3 and 12-4)
Page 57
−−
⎯⎯ and VU
⎯, ZX,
WX
and RSZ
1.
6.
1.
2
116.1 in
2.
3.
188.5 ft
2.
similar, 2:1
2
7.
8.
92.8 units2
267.0 ft2
3.
1152 ft3
4.
5.
yes
2827.4 in3
Chapter 12
4.
5.
27:125
9.
2884.0 in2
10.
150.8 in2
27:64
A27
Glencoe Geometry
Answers
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Mid-Chapter Test
Page 59
Chapter 12 Assessment Answer Key
Vocabulary Test
Page 60
Form 1
Page 61
1.
2.
1.
slant height
2. similar solids
right cylinder
3.
4.
volume
5.
axis
7.
congruent solids
true
10.
11.
12.
false; oblique cone
the sum of the
areas of the lateral
faces
a prism whose
lateral edges are
also altitudes
Chapter 12
13.
D
14.
H
F
3.
B
4.
G
15.
D
5.
D
16.
F
17.
B
18.
G
19.
B
6.
H
7.
A
8.
F
true
9.
G
C
false; pyramid
8.
12.
9.
10.
11.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
6.
Page 62
B
20.
H
B:
B
A28
J
1020 ft2
Glencoe Geometry
Chapter 12 Assessment Answer Key
Form 2A
Page 63
Page 64
12.
D
1.
13.
2.
3.
D
4.
G
C
15.
16.
17.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
6.
9.
J
3.
A
14.
J
4.
G
15.
B
5.
D
16.
G
17.
B
18.
H
19.
C
20.
G
B:
53 ft2
B
F
D
F
C
F
B
19.
20.
7.
A
8.
H
B
H
10.
A
G
G
B:
11.
C
2.
9.
10.
13.
H
18. F
8.
H
B
6. H
7.
12.
A
H
14.
5.
H
Page 66
391.6 ft2
11.
B
C
Chapter 12
A29
Glencoe Geometry
Answers
1.
Form 2B
Page 65
Chapter 12 Assessment Answer Key
Form 2C
Page 67
Page 68
1.
2.
PQRS, PQT,
QTR, RTS, PTS
12.
2 ft
13.
7690.6 ft3
14.
467.7 ft3
3.
15.
103.2 ft3
4.
120 cm2
16.
1140.4 cm2
5.
6 in2
17.
208.8 in3
18.
524.1 yd2
19.
neither
20.
612.5 m2
B:
113.1 in2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
6.
great circle
6 in.
7.
270 in2
8.
9.
270 + 150 √
3 in2
10.
62.8 ft2
11.
113.1 ft2
Chapter 12
A30
Glencoe Geometry
Chapter 12 Assessment Answer Key
Form 2D
Page 69
Page 70
1.
2.
12.
−− −− − −− −− −−
GJ, HJ, IJ, GH, HI, GI
13.
14.
576 in3
923.6 cm3
2057 in3
3.
15.
750 in2
4.
258 ft2
5.
16.
17.
1592.8 in2
106.4 cm3
Lines in spherical
845.2 m2
geometry are great
circles. Great circles
always intersect,
7 ft
7.
therefore they cannot be
18.
540 ft2
8.
19.
20.
11.
similar
931.0 ft2
9.
10.
parallel.
Answers
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
6.
100.5 cm3
1562.5 ft3
204.2 cm2
282.7 cm2
Chapter 12
B:
A31
201.1 in2
Glencoe Geometry
Chapter 12 Assessment Answer Key
Form 3
Page 71
Page 72
12.
184 cm3
13.
8m
1.
2.
3.
It is a rectangle.
21
128 + 4 √
2
units
No, it will overflow.
4.
5.
35 + 7 √
13 ft2
528 units2
14.
15.
16.
17,837.8 ft2
7.
32,435.2 ft2
468 in2
8.
9.
121.5 in3
3πr2
17.
Yes; 33.5 cm3
< 37.7 cm3
19.
similar
20.
1: √
4
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
6.
148.4 in3 > 96 in3
No. Any three points
on a sphere determine
a triangle.
18.
842.1 in2
3
10.
233.8 in2
11.
387.7 in2
B:
Chapter 12
A32
700 ft2
Glencoe Geometry
Chapter 12 Assessment Answer Key
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Score
General Description
Specific Criteria
4
Superior
A correct solution that
is supported by well-developed,
accurate explanations
• Shows thorough understanding of the concepts of
pyramids, prisms, cylinders, cones, spheres, surface area,
lateral area, volume, and properties of solid figures.
• Uses appropriate strategies to solve problems.
• Computations are correct.
• Written explanations are exemplary.
• Figures and drawings are accurate and appropriate.
• Goes beyond requirements of some or all problems.
3
Satisfactory
A generally correct solution, but may
contain minor flaws in reasoning or
computation
• Shows an understanding of the concepts of
pyramids, prisms, cylinders, cones, spheres, surface area,
lateral area, volume, and properties of solid figures.
• Uses appropriate strategies to solve problems.
• Computations are mostly correct.
• Written explanations are effective.
• Figures and drawings are mostly accurate and appropriate.
• Satisfies all requirements of problems.
2
Nearly Satisfactory
A partially correct interpretation and/or
solution to the problem
• Shows an understanding of most of the concepts of
pyramids, prisms, cylinders, cones, spheres, surface area,
lateral area, volume, and properties of solid figures.
• May not use appropriate strategies to solve problems.
• Computations are mostly correct.
• Written explanations are satisfactory.
• Figures and drawings are mostly accurate.
• Satisfies the requirements of most of the problems.
1
Nearly Unsatisfactory
A correct solution with no supporting
evidence or explanation
• Final computation is correct.
• No written explanations or work shown to substantiate the
final computation.
• Figures and drawings may be accurate but lack detail or
explanation.
• Satisfies minimal requirements of some of the problems.
0
Unsatisfactory
An incorrect solution indicating no
mathematical understanding of the
concept or task, or no solution is given
• Shows little or no understanding of most of the concepts of
pyramids, prisms, cylinders, cones, spheres, surface area,
lateral area, volume, and properties of solid figures.
• Does not use appropriate strategies to solve problems.
• Computations are incorrect.
• Written explanations are unsatisfactory.
• Figures and drawings are inaccurate or inappropriate.
• Does not satisfy requirements of problems.
• No answer given.
Chapter 12
A33
Glencoe Geometry
Answers
Extended-Response Test, Page 73
Scoring Rubric
Chapter 12 Assessment Answer Key
Extended-Response Test, Page 73
Sample Answers
In addition to the scoring rubric found on page A33, the following sample answers
may be used as guidance in evaluating open-ended assessment items.
1. The lateral area is the area of the lateral faces. The surface area includes
the area of the lateral faces plus the areas of the two bases.
2.
Oblique
Right
3. Sample answer: Sam is painting the walls of a room. The room is 12 feet
long, 10 feet wide, and 8 feet high. A gallon of paint covers 400 square feet
and costs $16 per gallon. Find the cost of the paint needed to paint the
room.
4. The first cylinder could have a radius of 3, a height of 4, a volume of 36π
cubic units, and a surface area of 42π square units. The second cylinder
could have a radius of 1, a height of 30, a volume of 30π cubic units, and a
surface area of 62π square units.
5.
12 in.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
4 in.
4 in.
4 in.
4 in.
4 in.
16 • 12
The volume of the pyramid is −
or 64 cubic units and
3
the volume of the prism is 4 • 4 • 4 or 64 cubic units.
6. The volume of the cylinder is πr2 • 2r.
3
2πr
The volume of the hemisphere is −
.
3
r
The volume of the cone is πr • −
3
2
Therefore, the total volume is
3
3
2πr
πr
2πr3 + −
+−
or 3πr3 cubic units.
3
Chapter 12
3
A34
Glencoe Geometry
Chapter 12 Assessment Answer Key
Standardized Test Practice
Page 74
Page 75
9. A
B
C
D
F
G
H
J
11. A
B
C
D
12. F
G
H
10.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
2. F
B
G
C
H
D
J
3. A
B
C
D
4.
G
H
J
F
J
13.
5.
A
6. F
B
G
C
H
D
J
1 4
0
0
0
0
0
1
1
1
1
1
2
2
2
2
2
3
3
3
3
3
4
4
4
4
4
5
5
5
5
5
6
6
6
6
6
7
7
7
7
7
8
8
8
8
8
9
9
9
9
9
14.
7. A
8.
F
B
G
Chapter 12
C
H
Answers
1. A
5
D
J
A35
0
0
0
0
0
1
1
1
1
1
2
2
2
2
2
3
3
3
3
3
4
4
4
4
4
5
5
5
5
5
6
6
6
6
6
7
7
7
7
7
8
8
8
8
8
9
9
9
9
9
Glencoe Geometry
Chapter 12 Assessment Answer Key
Standardized Test Practice
Page 76
2.6 m
15.
25
16.
17.
18.
20.
21a.
pentagonal
pyramid
3141.6 in2
2206.2 cm2
13,684.8 m2
207.3 m
b.
c.
d.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
19.
38.5 cm2
3421.2 m2
10,263.6 m2
Chapter 12
A36
Glencoe Geometry