CHAPTE R
12
Surface Area and
Volume of Prisms
connectED.mcgraw-hill.com
The
BIG Idea
Investigate
How are threedimensional solids
alike and different?
How do I find the
surface area and
volume of prisms?
Animations
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you organize information
about perimeter, area,
and volume.
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Surfaccee
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Volume
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figure
Polygon polígono a closed
that do
made up of line segments
not cross each other.
Worksheets
Assessment
Key Vocabulary
English
prism
surface area
three-dimensional figure
volume
590
Español
prisma
área total
figura tridimensional
volumen
When Will I Use This?
Your Turn!
You will solve thhiis teerrr.
problem in the chap
Surface Area and Volume of Prisms 591
Are You Ready
for the Chapter?
Text Option
You have two options for checking
Prerequisite Skills for this chapter.
Take the Quick Check below.
Tell whether each pair of figures is congruent or similar.
2.
1.
3.
Describe the lines as parallel, perpendicular, or neither.
5.
4.
6.
Tell whether each shape is a polygon.
8.
7.
Online Option
592
9.
Take the Online Readiness Quiz.
Surface Area and Volume of Prisms
Multi-Part
Lesson
1
PART
Properties of Three-Dimensional Figures
A
Main Idea
I will build nets and
explore properties of
three-dimensional
figures.
B
C
D
E
Build Three-Dimensional
Figures
A net is a two-dimensional pattern of a three-dimensional
figure. You can use a net to build a three-dimensional figure.
A three-dimensional figure has length, width, and height.
Vocabulary
V
net
A vertex is a
point where 3
or more edges
meet.
Materials
colored pencils
grid paper
A face is a
flat side.
An edge is
where two
faces meet.
Step 1
Copy the net shown onto grid paper.
Step 2
Congruent shapes have the same size and
shape. Shade all of the congruent shapes the
same color.
Step 3
Cut out the net. Fold it along the solid black
lines to form the three-dimensional figure. You
have formed a cube.
scissors
Get ConnectED
GLE 0506.4.2
Describe polyhedral solids
and analyze their properties,
including volume and surface
area. SPI 0506.4.3 Identify a
three-dimensional object
from two-dimensional
representations of that
object and vice versa. Also
addresses GLE 0506.1.3.
About It
1. What two-dimensional figure forms the faces of a cube?
How many faces are there? How many are congruent?
2. How many edges and vertices (the plural of vertex)
are there?
Lesson 1A Properties of Three-Dimensional Figures 593
594
Step 1
Copy the net shown onto grid paper.
Step 2
Shade the congruent shapes the same color.
Step 3
Cut out the net. Fold it along
the solid black lines to form the
three-dimensional figure. You
have formed a rectangular prism.
Step 1
Copy the net shown onto grid paper.
Step 2
Shade the congruent shapes the same color.
Step 3
Cut out the net. Fold it along
the solid black lines to form the
three-dimensional figure. You
have formed a triangular prism.
Surface Area and Volume of Prisms
Step 1
Copy the net shown onto grid paper.
Step 2
Shade the congruent shapes
the same color.
Step 3
Cut out the net and fold it along the solid black
lines to form the three-dimensional figure. You
have formed a rectangular pyramid.
About It
3. How many faces, vertices, and edges are there in the figure
in Activity 2?
4. In Activity 3, what two-dimensional figures form the faces of the
triangular prism? How many faces are congruent?
5. How many edges and vertices are there in the figure in Activity 4?
and Apply It
Create a three-dimensional figure from each net. Then describe
the faces, edges, and vertices.
7.
6.
8. Which three-dimensional figure below has the most vertices?
most edges?
A
9.
B
C
D
E WRITE MATH Is there more than one way to create
a net for a cube? Explain.
Lesson 1A Properties of Three-Dimensional Figures 595
Multi-Part
Lesson
1
Properties of Three-Dimensional Figures
PART
A
Main Idea
I will analyze properties
of three-dimensional
figures.
B
C
D
E
Three-Dimensional Figures
Recall that a two-dimensional figure is a plane figure that has
length and width.
Vocabulary
V
tthree-dimensional
figure
polyhedron
A three-dimensional figure has length, width, and height. A
three-dimensional figure with faces that are polygons is called
a polyhedron .
face
edge
vertex
prism
base
pyramid
A vertex is a
point where
3 or more
edges meet.
A face is a
flat side.
An edge is where
two faces meet.
cone
cylinder
Get ConnectED
GLE 0506.4.2
Describe polyhedral solids and
analyze their properties,
including volume and surface
area. SPI 0506.1.1 Given a
series of geometric statements,
draw a conclusion about the
figure described.
Two types of three-dimensional figures are prisms and pyramids.
They are named by the shape of their bases.
Prisms
• A prism has at least three faces that are rectangles.
• The top and bottom faces, called the bases , are congruent
parallel polygons.
Rectangular
prism
A rectangular
prism has six
rectangular faces
and eight vertices.
596
Surface Area and Volume of Prisms
Triangular
prism
A triangular prism
has five faces and
six vertices.
Square prism
or cube
A cube has six
square faces and
eight vertices.
Characteristics of Solids
Describe the faces, edges, and vertices of the
three-dimensional figure. Then identify it.
Parallel
• same distance apart
and never intersects
Congruent
• same size and shape
faces
This figure has 5 faces. The triangular
bases are congruent and parallel. The
other faces are rectangles.
edges
There are 9 edges. The edges that form the vertical
sides of the rectangles are parallel and congruent.
vertices
This figure has 6 vertices.
The figure is a triangular prism.
Pyramids
• A pyramid has at least three faces that are triangles.
• It has only one base, which is a polygon.
Triangular
pyramid
Rectangular
pyramid
C
CAMPING
Describe the faces, edges, and
nd
vertices
of the tent. Then identify the
v
shape of the tent.
faces
The tent has 5 faces. There is
one base that is a rectangle.
The other 4 faces appear to
be triangles.
edges
There are 8 edges. The opposite edges of the base
are parallel and congruent.
vertices
The tent has 5 vertices.
The tent is a rectangular pyramid.
Lesson 1B Properties of Three-Dimensional Figures 597
Some three-dimensional figures have curved surfaces.
Cones and Cylinders
Cones
• A cone has only one base.
• The base is a circle.
• Has one vertex.
Cylinders
• A cylinder has two bases.
• The bases are congruent circles.
• Has no vertices and no edges.
Characteristics
of Solids
SPORTS Describe the faces, edges, and
vertices of the container. Then identify
the shape of the container.
faces
The circular bases are congruent.
They are perpendicular to the curved
surface of the container.
edges
The container has no edges.
vertices
The container has no vertices.
So, the container is a cylinder.
Describe the faces
faces, edges
edges, and vertices of each three
three-dimensional
dimensional
figure. Then identify it. See Examples 1–3
2.
1.
3.
E
TALK MATH Describe the differences between a cylinder and
a rectangular prism.
598
Surface Area and Volume of Prisms
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Begins on page EP2.
Describe
ib the
h faces,
f
edges,
d
and
d vertices
i
off each
h three-dimensional
h
di
i
l
figure. Then identify it. See Examples 1–3
4.
5.
6.
7.
8.
9.
10.
11.
12. What kind of three-dimensional figure is the
tomato soup can at the right?
13. Describe the number of vertices and edges in
an unopened cereal box. Identify the shape of
the box.
14. Vera’s closet is in the shape of a rectangular prism.
Describe the pairs of parallel sides that make up
her closet.
15. WHICH ONE DOESN’T BELONG? Which figure does not belong
with the other three? Explain your reasoning.
16. REASONING Which three-dimensional figure has 4 faces,
4 vertices, and 6 edges?
17. CHALLENGE What figure is formed if only the height of a cube
is increased? Draw the figure to support your answer.
18.
E
WRITE MATH Describe the similarities and differences of
a rectangular prism and a triangular prism.
To assess mastery of SPI 0506.1.1, see your Tennessee Assessment Book.
599
Multi-Part
Lesson
1
PART
Properties of Three-Dimensional Figures
A
Main Idea
I will draw orthogonal
and projective views
of three-dimensional
figures.
B
C
D
E
Views of ThreeDimensional Figures
You can draw different views of three-dimensional figures.
The orthogonal view of a three-dimensional figure is the top,
side, and front views of a figure. A projective view of a threedimensional figure shows the picture view of the figure.
Vocabulary
V
orthogonal view
projective view
Projective View
Get ConnectED
SPI 0506.4.3
Identify a three-dimensional
object from two-dimensional
representations of that object
and vice versa.
Top View
Side View
Front View
Orthogonal View
Draw the Orthogonal View of a Figure
The projective view of a threedimensional figure is shown. Draw
the top, side, and front views of
the figure.
The top view shows 2 rows of 4.
The side view resembles a reverse L.
The front view also shows 2 rows of 4.
600
Surface Area and Volume of Prisms
Draw the Projective View of a Figure
The orthogonal view of a three-dimensional
figure is shown. Draw the projective view of
the figure.
Top View
Front View
top
Use the top view to build the figure’s base.
Use the side and front views to
complete the figure.
Side View
side
front
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Draw the
D
th top,
t
side,
id and
d front
f
t view
i
off each
h figure.
fi
1.
2.
3.
Draw the projective view of each figure.
5.
4.
Top View
Side View Front View
Top View
Side View
Front View
Draw the top, side, and front view of each object.
6.
7.
To assess mastery of SPI 0506.4.3, see your Tennessee Assessment Book.
8.
601
Multi-Part
Lesson
2
PART
Surface Area of Prisms
A
B
C
D
E
Surface Area of Prisms
Main Idea
I will explore using
models to find the
surface area of
rectangular prisms.
Suppose you want to paint all of the
surfaces of the prism. You would need
to find the surface area of this prism.
To find the surface area, you add the
areas of all the faces of the prism.
Materials
grid paper
Create a net to find the surface area of the prism.
scissors
Step 1
Draw and cut out the net below.
Step 2
Fold along the solid black lines.
Tape the edges together to form a prism.
Step 3
Find the area of each of the six faces of the
prism.
Get ConnectED
GLE 0506.4.2
Describe polyhedral solids
and analyze their properties,
including volume and surface
area. SPI 0506.4.4 Solve
problems involving surface
area and volume of
rectangular prisms and
polyhedral solids. Also
addresses GLE 0506.1.4.
top
side
front
side
back
bottom
Face
front and
back
top and
bottom
two sides
30
15
18
Model
Area (units2)
Step 4
Find the sum of the areas.
A = 30 + 30 + 15 + 15 + 18 + 18
A = 126 units2 Surface area has square units
because it measures area.
602
Surface Area and Volume of Prisms
Find the surface area of the rectangular prism.
5 ft
top
7 ft
7 ft
4 ft
back
side
7 ft
5 ft
front
side
4 ft
4 ft
5 ft
7 ft
bottom
7 ft
5 ft
Find the area of each face. Then add.
Face
front and
back
top and
bottom
two sides
28
35
20
Model
Area (ft2)
A = 28 + 28 + 35 + 35 + 20 + 20 or 166 square feet
About It
1. Describe the faces of a cube. Explain how to find the surface area
of a cube.
and Apply It
Make a net to find the surface area of each rectangular prism.
3.
2.
9m
4.
5 ft
4 in.
4m 3m
5.
E
12 ft
2 ft
4 in.
4 in.
WRITE MATH How many pairs of congruent faces are
in a rectangular prism? Describe them.
Lesson 2A Surface Area of Prisms 603
Multi-Part
Lesson
2
Surface Area of Prisms
PART
A
Main Idea
I will find the surface
area of rectangular
prisms.
Vocabulary
V
surface area
B
C
D
E
Surface Area of Prisms
The sum of the areas of all the faces of a prism is called
the surface area of the prism. Each face of a rectangular prism
has a congruent opposite face. So, the following formula can
also be used to find surface area.
Surface Area of a Rectangular Prism
Get ConnectED
GLE 0506.4.2
Describe polyhedral solids
and analyze their properties,
including volume and
surface area. SPI 0506.4.4
Solve problems involving
surface area and volume
of rectangular prisms and
polyhedral solids. Also
addresses GLE 0506.1.6.
Words
To find the surface area of a rectangular prism,
add the areas of all the faces of the prism.
Model
h
h
back
w side bottom side
top
w
Symbols
front
h
S.A. = 2h + 2w + 2hw
Find the Surface Area
GIFTS Find the surface area for the
amount of wrapping paper needed
to cover the gift.
Find the area of each face.
top and bottom:
2w = 2 × 6 × 3 or 36
front and back:
2h = 2 × 6 × 9 or 108
9 in.
6 in.
3 in.
two sides:
2wh = 2 × 3 × 9 or 54
Add to find the surface area.
The surface area is 36 + 108 + 54 or 198 square inches.
604
Surface Area and Volume of Prisms
Area is the number of
square units needed to
cover a region. It is
measured in square
units.
2 in.
C
CAMERAS
Digital cameras
are made small enough
a
to fit in a pocket. This
camera is shaped like
a rectangular prism. Find
the surface area of
the camera.
4 in.
6 in.
Find the area of each face.
top and bottom: 2w = 2 × 6 × 2 or 24
front and back:
2h = 2 × 6 × 4 or 48
two sides:
2wh = 2 × 2 × 4 or 16
Add to find the surface area.
The surface area is 24 + 48 + 16 or 88 square inches.
Find the surface area of each rectangular prism
prism. See Examples 1 and 2
1.
2.
3.
5 ft
9 ft
3 ft
12 mm
2 in.
4 cm
5 cm
5. Find the surface area of a rectangular prism with
a length of 9 meters, a width of 7 meters, and a
height of 4 meters.
E
15 in.
7 mm
4. A box of animal crackers is shaped like
a rectangular prism. What is the surface area
of the box of crackers?
6.
6 in.
11 mm
7 cm
TALK MATH The formula for the surface area of
a rectangular prism is S.A. = 2w + 2h + 2wh.
Explain why there are three 2s in the formula.
Lesson 2B Surface Area of Prisms 605
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Begins on page EP2.
Find
Fi
d th
the surface
f
area off each
h rectangular
t
l prism.
i
See Examples
l 1 andd 2
7.
8.
9.
4 in.
3 in.
14 in.
3 ft
7 ft
12 mm
4 cm
8 cm
10.
12 mm
6 cm
11.
12 mm
12.
8m
2 ft
9 in.
6m
18 m
4 in.
5 in.
13. Alyssa owns a toolbox that is 16 inches by 22 inches by
5 inches. What is the surface area of the toolbox?
14. Michelle put her sister’s birthday present in a box with
a length of 13 mm, a width of 4 mm, and a height of
8 mm. How many square millimeters of wrapping paper
will Michelle need to completely cover the box?
15. A package of three golf balls comes in the box shown.
What is the surface area of the box?
2 in.
7 in.
16. Which has a greater surface area: a box that is 2 inches
by 3 inches by 2 inches or a box that is 1 inch by
2 inches by 4 inches?
17. CHALLENGE What is the possible length, width, and height of a
rectangular prism with the surface area of 110 square centimeters?
18. OPEN ENDED Estimate the surface area of a cereal box. Then
measure and find the actual surface area. Compare to the
estimate.
19.
E
WRITE MATH Explain how to find the surface area of
a cube without using the formula.
606
Surface Area and Volume of Prisms
2 in.
Test Practice
20. What is the surface area of the box of
hot chocolate?
A. 210 in 2
22.
SHORT RESPONSE If the area
of the top of the figure shown is
16 square centimeters, what is the
area of the bottom?
B. 216 in 2
6 in.
C. 325 in 2
D. 340 in 2
10 in.
3 in.
21. For a science project, Madison uses
foil to cover the outside of a can. She
does not cover the top or the bottom
of the can.
What two-dimensional
figure represents the
shape of the piece of
foil that Madison uses?
23. Which statement is true about the
figure?
A. The figure has a triangular base.
B. The figure has exactly 3 pairs of
parallel faces.
F. circle
G. hexagon
H. rectangle
C. The figure has exactly 2 pairs of
parallel faces.
I. triangle
D. The figure has 7 vertices.
Describe the faces, edges, and vertices of each three-dimensional
figure. Then identify it. (Lesson 1B)
24.
25.
26.
27. Cara made the candle shown for her mother. Describe
the faces, edges, and vertices of the candle. Then identify it.
To assess partial mastery of SPI 0506.4.4, see your Tennessee Assessment Book.
607
10
Questions
Three-Dimensional Figures
Get Ready!
Players: 2
Get Set!
Decide who will be Player 1
and Player 2.
GO!
Player 1 will start by
writing the name of a
three-dimensional figure
on a piece of paper.
Player 2 will ask questions
about the figure. For
example: Does this figure
have 5 faces? Does this
figure have less than
6 vertices? Are there
9 edges in this figure?
Player 1 will respond to
Player 2’s questions with
“yes” or “no.”
Player 2 will ask up to
10 questions about the
three-dimensional figure.
608
Surface Area and Volume of Prisms
You Will Need: paper and pencil
If Player 2 guesses the
correct figure within the
10 questions, then he/she
earns 1 point.
Alternate asking and
guessing between the
two players.
The first person with
5 points wins.
Mid-Chapter
Check
Create a three-dimensional figure from
each net. Then describe the faces, edges,
and vertices. (Lesson 1A)
8. MULTIPLE CHOICE Nelly gave her
mother the present shown. Find the
surface area of the present. (Lesson 2B)
1.
5 in.
14 in.
10 in.
2.
F. 140 square inches
G. 260 square inches
H. 520 square inches
I.
Describe the faces, edges, and vertices of
each three-dimensional figure. Then
identify it. (Lesson 1B)
3.
4.
600 square inches
Find the surface area of each rectangular
prism. (Lesson 2B)
9.
10.
3 in.
3m
5.
6.
2m
11.
6 in.
12.
4 ft
7. MULTIPLE CHOICE Which
h
statement is true about
the figure? (Lesson 1B)
A. The figure has 4 vertices.
2 in.
1m
4 ft
4 ft
4 cm
5 cm
15 cm
13. The dimensions of a jewelry box are
7 inches by 4 inches by 5 inches. Find
the surface area of the jewelry box.
B. The figure has 2 circular bases.
C. The figure has a triangular base.
D. The figure has 1 circular base.
14.
E
WRITE MATH Draw and label a cube
with a surface area of 150 square units.
Mid-Chapter Check 609
Multi-Part
Lesson
3
Volume of Prisms
A
PART
B
C
D
E
Problem-Solving Strategy:
Make a Model
Main Idea I will solve problems by making a model.
Nick is helping his younger sister put away her
alphabet blocks. She has already put away one
layer of blocks. To fill up one layer, it takes nine
blocks. If the box is filled with six layers of blocks,
how many blocks would be in the box?
ABC
Understand
What facts do you know?
• The number of blocks in one layer of a box.
• The number of layers in the box.
What do you need to find?
• The number of blocks in the box when there are six layers.
Plan
Solve the problem by making a model.
Solve
Use your plan to solve the problem.
Make a model of one layer of the box
by arranging 9 cubes in a 3 × 3 array.
Continue stacking the cubes until there
are six layers.
There are a total of 54 cubes. So, the
box would have 54 blocks.
Check
Look back. Use logical reasoning and multiplication. There are 6 layers and
each layer has 9 cubes. So, the total number of cubes is 6 × 9 or 54. The
answer is correct. GLE 0506.4.2 Describe polyhedral solids and analyze their properties, including volume and surface area.
SPI 0506.4.4 Solve problems involving surface area and volume of rectangular prisms and polyhedral
solids. Also addresses GLE 0506.1.2.
610
Surface Area and Volume of Prisms
Refer to the problem on the previous page.
1. How many blocks would the box
contain if it had only five layers of
blocks?
3. What are the advantages of the make
a model strategy?
2. List the length, width, and height of
two different boxes that can hold
exactly 54 blocks with none left over.
4. List some objects that you could use
to make a model.
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Solve. Use the make a model strategy.
5. Measurement On an assembly line
that is 150 feet long, there is a work
station every 15 feet. The first station is
at the beginning of the line. How many
work stations are there?
6. A store is stacking cans of food into a
pyramid-shaped display. The bottom
layer has 9 cans. There are 5 layers. If
there are two less cans in each layer,
how many cans are in the display?
9. In the figure below, there are
22 marbles in Box A. To go from
Box A to Box B, four marbles must
pass through the triangular machine
at a time. Five marbles must pass
through the square machine at a time.
Describe how to move all the marbles
from Box A to Box B in the fewest
moves possible.
ä>Qä>ä
QFJB
7. Measurement The distance around
the center ring at the circus is 80 feet.
A clown stands every 10 feet along the
circle. How many clowns are there?
8. Measurement Martino wants to
arrange 18 square tiles into a
rectangular shape with the least
perimeter possible. How many tiles will
be in each row?
ä>Qä>ä
QFJB
10. Drake lined up 15 pennies on his desk.
He replaced every third penny with a
nickel. Then he replaced every fourth
coin with a dime. Finally, he replaced
every fifth coin with a quarter. What is
the value of the remaining 15 coins
on his desk? Explain.
11.
E
WRITE MATH Describe when you
would use the make a model
strategy.
Lesson 3A Volume of Prisms 611
Multi-Part
Lesson
3
Volume of Prisms
PART
A
B
C
D
E
Volume of Prisms
Main Idea
I will use models to
find the volumes of
prisms.
You can use centimeter cubes to build
rectangular prisms like the ones shown
at the right.
Materials
centimeter cubes
Step 1
Use centimeter cubes to build four different
rectangular prisms.
Step 2
For each prism, record the dimensions and the
number of cubes used in a table like the one below.
Get ConnectED
Length
( )
Prism
GLE 0506.4.2
Describe polyhedral solids
and analyze their properties,
including volume and
surface area. SPI 0506.4.4
Solve problems involving
surface area and volume of
rectangular prisms and
polyhedral solids.
Width
(w)
Height
(h)
Number of
Cubes
A
B
C
D
Since volume can be measured using cubes, volume is
measured in cubic units or units 3.
and Apply It
1. Describe the relationship between the dimensions of the
prism and number of cubes.
2. Use , w, and h to write a formula for the volume V of a
rectangular prism.
3. Use the formula you wrote in Exercise 2 to
find the volume of the prism at the right in
appropriate units. Verify your solution by
counting the number of cubes.
612
Surface Area and Volume of Prisms
Multi-Part
Lesson
3
PART
Volume of Prisms
A
Main Idea
I will find the volumes
of rectangular prisms.
Vocabulary
V
vvolume
B
C
D
E
Volume of Prisms
Volume is the amount of space inside a three-dimensional
figure. Volume is measured in cubic units. A cubic unit has
length, width, and height.
1 cubic unit
Get ConnectED
GLE 0506.4.2
Describe polyhedral solids
and analyze their properties,
including volume and surface
area. SPI 0506.4.4 Solve
problems involving surface
area and volume of
rectangular prisms and
polyhedral solids. Also
addresses GLE 0506.1.6,
GLE 0506.1.7.
2 cubic units
4 cubic units
You can find the volume of a rectangular prism by using models.
A cube with 1 unit on an edge is a standard unit for measuring
volume. When cubes are placed in a prism to determine volume,
there are no gaps or overlaps between the cubes.
M
MAIL
Evelyn wants to mail a package to her cousin. What
iis the volume of the package if it is 6 inches long, 4 inches
wide, and 4 inches tall?
Count the number of 1-inch
cubes that will fill the bottom
of the rectangular prism. The
prism is 6 cubes long and
4 cubes wide. There are
24 cubes on the bottom.
4 in.
4 in.
6 in.
There are 4 layers of cubes.
So, there are 4 × 24 or
96 cubes.
4 in.
4 in.
6 in.
Lesson 3C Volume of Prisms 613
Some common units of volume are cubic inch, cubic foot, cubic
yard, cubic centimeter, and cubic meter.
ts
Volume measuremen
ing
us
can be written
abbreviations and an
exponent of 3.
For example:
3
cubic units = units
3
cubic inches = in
The volume of a rectangular prism is related to its dimensions.
You can use a formula to find the volume of a prism.
Volume of a Rectangular Prism
Words
3
cubic feet = ft
3
cubic meters = m
Symbols
To find the volume of a
rectangular prism multiply
the length, width, and
h
height.
Model
w
V = wh
Volume of a Prism
ART Armando makes sand
paintings by filling clear plastic
cases with colored sand. Find
the volume of the plastic case.
7 in.
Estimate 5 × 5 × 5 = 125
V = wh
Formula for volume
V=5×4×7
= 5, w = 4, h = 7
V = 140
Multiply.
4 in.
5 in.
The volume of the prism is 140 cubic inches.
Check for Reasonableness 140 ≈ 125 Volume of a Prism
Find the volume of the prism.
9 cm
Estimate 10 × 10 × 10 = 1,000
V = wh
Formula for volume
V = 12 × 9 × 10
= 12, w = 9, h = 10
V = 1,080
Multiply.
The volume of the prism is 1,080 cm 3.
Check for Reasonableness 1,080 ≈ 1,000 614
Surface Area and Volume of Prisms
10 cm
12 cm
Find the volume of each prism.
prism See Examples 11–33
1.
2.
3.
3m
2 cm
4 in.
9 cm
4m
5 cm
4 in.
6m
4 in.
4. = 21 cm, w = 8 cm, h = 4 cm
5. = 19 ft, w = 9 ft, h = 16 ft
6. Find the cubic feet of air in a room that is 13 feet long, 10 feet
high, and 11 feet wide.
7.
E
TALK MATH Describe which units would be appropriate to
measure the volume of a jewelry box. What other units might be
reasonable to use? Would it be reasonable to use the same units
to measure the volume of a garage? Explain.
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Find
Fi
d the
th volume
l
off each
h prism.
i
See Examples
l 1–3
8.
9.
3 cm
11 ft
3 ft
12 cm
26 cm
30 ft
10.
11. 16 m
23 m
11 in.
11 in.
11 in.
9m
12.
13.
3 in.
9 cm
7 cm
2 in.
4 in.
17 cm
Lesson 3C Volume of Prisms 615
Find the volume of each prism. See Examples 1–3
14. = 5 yd, w = 16 yd, h = 6 yd
15. = 2 m, w = 8 m, h = 10 m
16. = 13 in., w = 3 in., h = 2 in.
17. = 13 cm, w = 8 cm, h = 10 cm
18. Find the volume of a bank vault that is 14 feet by 20 feet by 19 feet.
Use the information to solve the problem.
19. Determine the volume of each pet carrier. Which one should
Emma purchase?
20. OPEN ENDED Estimate the volume of a shoe box. Then measure
the box. Check your estimate by finding the actual volume.
21. NUMBER SENSE Describe the dimensions of two different prisms
that have a volume of 2,400 cubic centimeters.
22. CHALLENGE A store sells lunch boxes that measure 11 inches by
7 inches by 4 inches. How many lunch boxes will fit in a box that
measures 22 inches by 15 inches by 8 inches? Explain.
4 in.
11 in.
7 in.
8 in.
15 in.
22 in.
23.
E
WRITE MATH Write a real-life problem that could be
solved by finding the volume of a prism. Then solve.
616
Surface Area and Volume of Prisms
Test Practice
24. Popcorn tins are stacked in a display
so that there are 12 tins in the
bottom row. There are 10 tins in the
next row, and 8 tins in the row above
that. There are five rows of tins. If the
pattern continues, how many
popcorn tins are there in all?
25.
A. 22
C. 40
B. 30
D. 42
27. Shawn keeps his photos in a box
like the one shown.
What is the volume in cubic inches
of the box?
SHORT RESPONSE Find the
volume in cubic meters of a cube
with the dimension shown.
A. 22
B. 72
C. 288
D. 300
4m
26. What is the volume of the rectangular
prism shown below?
28.
SHORT RESPONSE A
rectangular prism made of 1-inch
cubes is shown below.
5 cm
7 cm
3 cm
F. 88 cm 3
H. 120 cm 3
G. 105 cm 3
I. 142 cm 3
What is the volume of the prism?
29. A box of crayons has a length of 9 inches, a width of 4 inches,
and a height of 6 inches. What is the surface area of the box of
crayons? (Lesson 2B)
30. What kind of three-dimensional shape is shown?
(Lesson 1B)
To assess partial mastery of SPI 0506.4.4, see your Tennessee Assessment Book.
617
Multi-Part
Lesson
3
PART
Volume of Prisms
A
Main Idea
I will select and use
appropriate units and
formulas to measure
surface area and
volume.
B
C
D
E
Select Appropriate
Measurement Formulas
It is important to be able to choose the appropriate
measurement for a given situation.
Get ConnectED
GLE 0506.4.2
Describe polyhedral solids
and analyze their properties,
including volume and surface
area. SPI 0506.4.4 Solve
problems involving surface
area and volume of
rectangular prisms and
polyhedral solids.
Measurement Formulas For Rectangular
Prisms
Measure
Used to Find
Formula
Surface Area
area of all surfaces
Volume
space enclosed by
a figure
S.A. =
2h + 2w + 2hw
V = wh
TRUCKS A family wants to rent the truck that can hold the
most. To determine which truck can hold the most, should
you find the surface area or volume? Solve.
Truck Sizes
Length
Width
Height
Truck A
10 ft
6 ft
6 ft
Truck B
16 ft
7 ft
7 ft
Truck C
24 ft
7 ft
7 ft
You need to determine how much the truck can hold.
So, you need to find the volume.
Truck A
10 × 6 × 6 or 360 cubic feet
Truck B 16 × 7 × 7 or 784 cubic feet
Truck C
24 × 7 × 7 or 1,176 cubic feet
Truck C can hold the most.
618
Surface Area and Volume of Prisms
DESIGN An interior designer needs to
calculate how much wallpaper it will take to
cover 4 decorative columns. The figure
represents one column. Determine whether
she should find the surface area or volume
of the columns. Then solve.
Surface area is given in
square units, and
volume is given in
cubic units.
6m
1m
1m
The interior designer needs to know how
much surface to cover. So, she needs to find the surface area.
S.A. = 2h + 2w + 2hw
Formula for surface area
of a rectangular prism
S.A. = 2(1 × 6) + 2(1 × 1) + 2(6 × 1) = 1, w = 1, and h = 6
S.A. = 12 + 2 + 12
Simplify.
S.A. = 26
Add.
So, 26 square meters of wall paper are needed for one column.
Multiply 26 by 4 to find the surface area of 4 columns.
26 × 4 = 104
The surface area of 4 columns is 104 m2.
Determine whether you need to find the surface area or volume.
volume
Then solve. See Examples 1 and 2
1. Shane is mailing a DVD player as a gift
to his cousin. He wraps the gift in
bubble wrap to help protect it. How much
bubble wrap is needed?
2. The gift described in Exercise 1 will be
mailed in a box that has a length of 30 inches, a width of
15 inches, and a height of 7 inches. How much space will
the box enclose?
3. How much water is needed to fill a pool that is 50 meters long,
25 meters wide, and 3 meters deep?
4. Which units would be most appropriate to measure the volume of
a cake pan: cubic inches, cubic feet, or cubic yards? Explain.
5.
E
TALK MATH Explain how you determine whether to use the
formula for surface area or volume for a given situation.
Lesson 3D Volume of Prisms 619
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D t
Determine
i whether
h th you need
d tto fi
find
d th
the surface
f
area or volume.
l
Then solve. See Examples 1 and 2
6. A path is 3 feet wide and 14 feet long. How much gravel will
Mr. James need if he wants to add 2 inches of gravel over the
entire path?
7. A company packages sticks of butter to be sold
in stores. How much wrapping does the company
need for each stick of butter?
8. Beth is painting her dresser white. Her dresser
is 3 feet high, 3 feet long, and 2 feet wide. Is
Beth painting the surface area or volume?
To help stay fit, some people have swimming pools. Swimming pools
can come in many shapes and sizes. The table shows two examples
of rectangular swimming pools.
Swimming Pool Sizes
Pool
Length
Width
Height
Pool A
15 m
10 m
2m
Pool B
20 m
15 m
2m
9. If you wanted to fill a pool, would you find the
surface area or volume?
10. How many cubic meters of water would fill Pool A?
11. How many cubic meters of water would fill Pool A and Pool B?
12. FIND THE ERROR Tom is
finding the volume of a prism
V
with length 3 meters, height
8 meters, and width 17 meters.
Help find and correct his mistake.
13.
E
=3×8
= 24 cubic meters
WRITE MATH Explain the difference between finding
the surface area and the volume of a rectangular prism.
620
Surface Area and Volume of Prisms
Test Practice
14. Bonita is making a building out of
boxes. She wants to cover a box with
the dimensions shown below with
brown paper.
15. Gabriel wants to determine how
much the box shown can hold.
8 in.
8 in.
14 in.
How much paper will it take to cover
the entire box?
Which formula should he use?
F. A = w
A. 112 square inches
G. S.A. = 2h + 2w + 2hw
B. 576 square inches
H. V = wh
C. 896 cubic inches
1
I. A = _bh
D. 900 cubic inches
2
16. Craig is painting his bedroom. The room has four walls that
are each 9 feet long by 10 feet tall. A gallon of paint covers
120 square feet. How many gallons should he buy to cover
all four walls? (Lesson 3A)
Find the surface area of each rectangular prism. (Lesson 2B)
17.
18.
6 cm
18 in.
17 cm
19.
9 yd
4 cm
25 yd
22 yd
12 in.
9 in.
Describe the faces, edges, and vertices of each three-dimensional
figure. Then identify it. (Lesson 1B)
20.
21.
22.
Lesson 3D Volume of Prisms 621
Multi-Part
Lesson
3
PART
Volume of Prisms
A
B
C
D
E
Problem-Solving Investigation
Main Idea I will choose the best strategy to solve a problem.
JACINDA: I have twelve cubes. There are
many ways to arrange them to form a
rectangular prism. I want to find the
arrangement of cubes that will have the
least surface area.
YOUR MISSION: Find the arrangement of
cubes that has the least surface area.
understand
You need to find the arrangement of cubes that has the least
surface area.
Plan
Solve the problem by making a table.
Solve
Make a table listing possible ways 12 cubes can be arranged to
form a rectangular prism with different surface areas. All prisms
have a volume of 12 cubic units.
Rectangular Prism
Dimensions (units)
Surface Area (units2)
(S.A. = 2h + 2w + 2hw)
= 3, h = 2, w = 2
32
= 6, h = 2, w = 1
40
= 4, h = 1, w = 3
38
= 12, h = 1, w = 1
50
The rectangular prism with the least surface area would have a
length of 3 units, a height of 2 units, and a width of 2 units.
check
622
Reread the problem to see if the answer matches the information
given. Surface Area and Volume of Prisms
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•
•
•
•
•
Guess, check, and revise.
Look for a pattern.
Make a model.
Make a table.
Draw a diagram.
Use any strategy to solve each problem.
1. Geometry Mariana has 24 cubes.
How can she stack the cubes to create
a figure with the greatest possible
surface area?
4. The volume of a rectangular prism is
5,376 cubic inches. The prism is
14 inches long and 16 inches wide.
How tall is the prism?
5. Algebra The table below shows the
number of minutes Danielle spent
practicing the trumpet over the last
7 days. If she continues this pattern of
practicing, in how many days will she
have practiced 340 minutes?
2. Measurement A hexagon that has
each side equal to 1 inch has a
perimeter of 6 inches.
perimeter = 6 in.
Two hexagons placed side by side
have a perimeter of 10 inches.
Three hexagons have a perimeter of
14 inches.
perimeter = 10 in.
perimeter = 14 in.
What would be the perimeter of five
hexagons placed side by side?
Time (min)
1
20
2
20
3
35
4
20
5
20
6
35
7
20
For Exercises 6 and 7, use the following
information.
Marita wants to make a rectangle with a
perimeter of 20 inches.
6. How many rectangles can Marita make
if she only uses whole numbers for the
side lengths? List the dimensions.
7. Which rectangle has the greatest area?
8.
3. Five friends are standing in a circle and
playing a game where they toss a ball
of yarn to one another. If each person
is connected by the yarn to each other
person only once, how many lines of
yarn will connect the group?
Day
E
WRITE MATH One wall of a building
is 80 feet long and 16 feet high. A
one-gallon can of paint covers up to
450 square feet. If each can of paint
costs $22.50, find the total cost of paint
for the wall. Explain the steps you used
to solve the problem.
Lesson 3E Volume of Prisms 623
You probably eat packaged frozen
food every day. Frozen food might
seem like a simple concept, but
there’s more to it than just putting
a container of food in the freezer.
Clarence Birdseye is sometimes
called “the father of frozen food”
because he was the first to
develop a practical way to preserve
food by flash freezing.
Birdseye experimented with
freezing fruits and vegetables, as
well as fish and meat. His method
of freezing food preserved the
food’s taste, texture, and
appearance. He also was the first
to package food in waxed
cardboard packages that could be
sold directly to consumers.
148 patents
were issued that
related to Clarence
Birdseye’s flash-freezing
method, his type of
packaging, and the
packaging materials
he used.
624
Surface Area and Volume of Prisms
Dimensions of Frozen Food Packages in Inches
Item
Length
Width
Height
12
12
2
5
6
2
11
8
2
Fish Sticks
9
5
3
Hamburger Patties
9
10
4
Pizza
Vegetables
Frozen Dinner
Use the information above to solve each problem.
1. What is the volume of a frozen pizza
5. A larger package of frozen
package?
vegetables has the same length and
width but twice the height. What is
the volume of this package?
2. How much more space does a
package of fish sticks occupy than a
package of vegetables?
6. Use a centimeter ruler to measure
the length, width, and height of an
actual frozen food package to the
nearest whole unit. Then find the
surface area of the package.
3. Is 200 cubic inches a reasonable
estimate for the volume of a frozen
dinner package? Explain.
4. A freezer has 2,600 cubic inches of
available space. After seven
packages of hamburger patties are
placed inside, how much available
freezer space is left?
7.
E
WRITE MATH Explain the
differences between surface area
and volume and the units used to
represent them.
Problem Solving in Science 625
Chapter Study
Guide and Review
Be sure the following Big
Ideas are written in your
Foldable.
Vocabulary
cone
cube
cylinder
prism
pyramid
rectangular prism
Three - Dimensional Surfa
ce
Figures
Area
square pyramid
Volu
surface area
me
three-dimensional figure
triangular prism
triangular pyramid
Key Concepts
volume
Three-Dimensional Figures (Lesson 1)
• A three-dimensional figure has length,
width, and height.
A vertex is a
point where
3 or more
edges meet.
A face is a
flat side.
An edge is where
two faces meet.
Surface Areas of Rectangular Prisms
(Lesson 2)
• The surface area S.A. of a rectangular
prism is the sum of the areas of the faces.
S.A. = 2h + 2w + 2hw
Vocabulary Check
State whether each sentence is true
or false. If false, replace the
underlined word or number to make
a true sentence.
1. To find the surface area of a
rectangular prism, multiply the
length by the width w by the
height h.
2. A cube with all sides 4 centimeters
long has a volume of 64 cubic
centimeters.
3. A square pyramid has 6 vertices.
Volumes of Prisms (Lesson 3)
• The volume V of a rectangular prism is
length times width w times height h.
V = wh
626
Surface Area and Volume of Prisms
4. A three-dimensional figure with
faces that are polygons is called a
polyhedron.
5. A cylinder has two parallel
congruent circular bases.
Lesson 1
Properties of Three-Dimensional Figures
Three-Dimensional Figures
(Lesson 1B)
Describe the faces, edges, and vertices
of the three-dimensional figure. Then
identify it.
6.
7.
8. Describe the faces,
edges, and vertices
of the vase. Then
identify the shape
of the vase.
Lesson 2
Describe the faces, edges, and vertices
of the three-dimensional figure. Then
identify it.
Opposite faces are parallel and congruent.
There are 12 edges and 8 vertices.
The figure is a rectangular prism.
Surface Area of Prisms
Surface Area of Prisms
(Lesson 2B)
EXAMPLE 2
Find the surface area of each
rectangular prism.
9.
EXAMPLE 1
Find the surface
area of the
rectangular prism.
10.
2 in.
3 ft
8 in.
S.A. = 2h + 2w + 2hw
front and back:
3 in.
11. A DVD player measures 17 inches by
15 inches by 3 inches. What is the
minimum surface area of a box to
hold the DVD player?
4 cm
5 cm
4 ft
10 ft
8 cm
2h = 2 × 8 × 4 or 64
top and bottom: 2w = 2 × 8 × 5 or 80
two sides:
2hw = 2 × 4 × 5 or 40
The surface area is 64 + 80 + 40 or 184
square centimeters.
Chapter Study Guide and Review 627
Chapter Study Guide and Review
Lesson 3
Volume of Prisms
Problem-Solving Strategy: Make a Model
EXAMPLE 3
Solve by making a model.
12. A box is filled with 48 cubes that
measure 1 inch on each side. The
cubes completely fill the box. What
are possible dimensions of the box?
13. Haley is making a bracelet by placing
beads and charms on a 6-inch chain.
She places a charm at 1 inch from
1
each end, and at every _ inch in
2
between. How many charms does
she use?
14. Destiny has 24 feet of fencing material
to make a pet enclosure. Describe
three different rectangular areas that
can be enclosed by the fencing.
Volume of Prisms
How many centimeter
cubes will fit in the
container at the right?
You can use cubes to
model the situation.
First, arrange 2 rows of
5 cubes.
Next, add three more
layers of cubes.
The total number
of cubes used is 40. So,
40 centimeter cubes will fit
in the container.
(Lesson 3C)
EXAMPLE 4
Find the volume of each prism.
16.
15.
(Lesson 3A)
Find the volume of the prism.
3 in.
6 ft
3 in.
8 ft
3 in.
2 ft
3m
17.
5m
7m
15 m
18. 16 cm
9 cm
22 cm
19. Victoro keeps his pet rabbit in a cage
that is shaped like a rectangular prism.
The cage measures 2 feet by 3 feet
by 2 feet. What is the volume of
the cage?
628
8m
10 m
Surface Area and Volume of Prisms
V = wh
V = 15 × 10 × 8
V = 1,200
Volume of a prism
= 15, w = 10, h = 8
Multiply.
The volume of the prism is 1,200
cubic meters.
Lesson 3
Volume of Prisms
(continued)
Select Appropriate Measurement Formulas
Determine whether you need to find the
surface area or volume. Then solve.
20. Julius wants to purchase the
wastebasket that can hold the
most. Which wastebasket should
he purchase?
(Lesson 3D)
EXAMPLE 5
How much space is inside the box
shown? Determine whether you need to
find the surface area or volume. Solve.
Wastebasket Dimensions
Length Width
Height
Basket
(cm)
(cm)
(cm)
A
10
6
15
B
12
7
12
C
10
10
15
21. A safe is painted red on each side.
How much paint do you need to
cover the safe?
You need to determine how much the box
can hold. So, you need to find the volume.
V = wh
Volume of a prism.
V=4×2×3
= 4, w = 2, and h = 3.
V = 24 cubic feet
4 in.
EXAMPLE 6
8 in.
How much plastic wrap will it take to
cover the entire box? Determine whether
you need to find the surface area or
volume. Solve.
5 in.
22. A box has a length of 6 meters, a
width of 4 meters, and a height of
7 meters. Could you fit 200 cubic
meters of material inside of the
box? Explain.
23. Kristin is wrapping a present. The
present is in a box with a length
of 9 centimeters, a width of
5 centimeters, and a height of
10 centimeters. How much wrapping
paper will she need?
12 in.
16 in.
9 in.
You need to find the surface area.
S.A. = 2(h) + 2(w) + 2(hw)
S.A. = 2(16 × 12) + 2(16 × 9) + 2(12 × 9)
S.A. = 2(192) + 2(144) + 2(108) Multiply.
S.A. = 384 + 288 + 216
Multiply.
S.A. = 888 square inches
Add.
Chapter Study Guide and Review 629
Chapter Study Guide and Review
Lesson 3
Volume of Prisms
(continued)
Problem-Solving Investigation: Choose a Strategy
Use any strategy to solve.
24. Barrett has 18 sports cards. He collects
football and baseball cards. He has
twice as many baseball cards. How
many of each kind does he have?
25. Leon has $5 to buy a bottle of water
that costs $1.49, a granola bar for
$1.09, and the magazine shown
below. Does he have enough money?
Explain.
(Lesson 3E)
EXAMPLE 7
What two whole numbers have a sum of
12 and a product of 32?
Understand
• You know that there are two numbers
that added together equal 12.
• You also know the same two numbers,
when multiplied by each other,
equal 32.
Plan
Use the guess, check, and
revise strategy.
Solve
Guess: 2 and 10
Check: 2 + 10 = 12, 2 × 10 = 20
26. Sarah went bowling with her family.
She had 3 more strikes than her
brother, but 1 less strike than her dad.
If there were a total of 7 strikes, how
many strikes did each person bowl?
Revise: Since 20 ≠ 32, try 2 different
numbers
Guess: 3 and 9
Check: 3 + 9 = 12, 3 × 9 = 27
27. Ian has $21.75 in his pocket after he
purchased a DVD for $38.99. Would
$50, $60, or $70 be a reasonable
estimate for the money in Ian’s pocket
before he bought the DVD?
28. The volume of a rectangular prism is
112 cubic inches. The prism has a
length of 7 inches and a height of
8 inches. What is the width of
the prism?
630
Surface Area and Volume of Prisms
Revise: Since 27 ≠ 32, try 2 different
numbers
Guess: 4 and 8
Check: 4 + 8 = 12, 4 × 8 = 32
The whole numbers are 4 and 8.
Check
Since 4 + 8 = 12 and 4 × 8 = 32
the two numbers chosen are correct. Practice
Chapter Test
Describe the faces, edges, and vertices
of each three-dimensional figure. Then
identify it.
1.
9. MULTIPLE CHOICE Which figure below
has 3 more edges than faces?
F.
H.
G.
I.
2.
3.
4.
Find the volume of each prism.
5. If you place one cube on a table, you
can see 5 faces of the cube. If you place
a second cube on top of the first, you
can see 9 faces. How many faces can
you see in a stack of 6 cubes?
6. MULTIPLE CHOICE For an art project,
Heather needs to cover the box shown
with craft sticks.
What is the surface area of the box?
A. 17 in2
C. 42 in2
B. 28 in2
D. 188 in2
Find the surface area of each prism.
8.
7.
9m
9m
3m
10 m
4m
7 in.
7 in.
7 in.
For each problem, determine whether you
need to find the surface area or volume.
Then solve.
12. A design is printed
on the outside of
a tissue box. To
determine how much
ink you will need for
the design, find the
amount of space
you need to cover.
8 cm
5 cm
6 cm
13. A storage shed is 13 feet long, 8 feet
wide, and 10 feet tall. How much can
the storage shed hold?
5 ft
14.
10 ft
9m
11.
10.
3 ft
E
WRITE MATH Describe the difference
between finding the surface area of a
rectangular prism and finding the
volume of a rectangular prism.
Practice Chapter Test
631
Test Practice
Drew has a fish aquarium. How many cubic inches
of water can the aquarium hold?
14 in..
Before beginning a problem,
you may need to determine
if you should find the surface
area or the volume.
9 in.
20 in.
Read the Test Item
252 0
You need to find the volume of the aquarium.
Solve the Test Item
V = wh
V = 20 × 9 × 14
V = 2,520 in3
The aquarium can hold 2,520 cubic inches of water.
Fill in the grid.
Read each question. Then fill in the correct answer on the answer
sheet provided by your teacher or on a sheet of paper.
1. Which of the following could be a
volume measurement?
A. 250 inches
B. 250 square inches
2. Which of the following
would you use to
find the surface
area of the figure
shown?
C. 250 in2
F. S.A. = × w
D. 250 in3
G. S.A. = 2 + 2w
H. S.A. = wh
I. S.A. = 2h × 2w × 2hw
632
Surface Area and Volume of Prisms
3. How many faces, edges, and vertices
does the figure have?
6.
GRIDDED RESPONSE A
rectangular driveway measures 16 feet
by 12 feet. What is the area of the
driveway in square feet?
7. Paul plays a video game twice. He
receives a score of 159.25 points the
first time and 212.75 points the second
time. How many more points did Paul
score the second time?
A. 5 faces, 8 edges, 5 vertices
B. 5 faces, 6 edges, 8 vertices
C. 5 faces, 8 edges, 6 vertices
D. 6 faces, 10 edges, 6 vertices
4. Look at the pattern of numbers shown
below.
7, ___, 17, 22, 27, 32
Which expression is equal to the
missing number in the pattern?
A. 50
C. 371.75
B. 53.5
D. 372
8. Which statement about the figures
shown below is true?
Q
R
S
T
F. (27 - 18) + 3
G. (7 + 12) - 5
H. (17 - 12) + 5
F. Figures Q and R have opposite sides
that are parallel and congruent.
I. (28 - 23) + 3
G. Figures S and T are squares.
5.
SHORT RESPONSE Derrick is
painting a toy chest with the dimensions
shown. Find the surface area and
volume of the chest.
H. Figures S and R have the same
number of vertices.
I. Figures Q and T each have 5 edges.
9.
11 in.
9 in.
14 in.
SHORT RESPONSE It takes 3 feet
of wood to make 1 birdhouse. If you
have 23 feet of wood, can you make
8 birdhouses? Explain.
NEED EXTRA HELP?
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1
2
3
12–3C 12–2B 12–1B
4
9–1A
5
6
12–2B 11–2C
7
8
9
5–3C
12–1B
3–3B
SPI 4.4 SPI 4.4 GLE 4.2 SPI 3.2 SPI 4.4 GLE 4.1 SPI 2.5 SPI 1.1 SPI 2.3
Test Practice 633