Solids
Polyhedrons – solids in which all of the faces are polygons.
 Vertex – a point of intersection of 3 or more edges
 Face – each flat surface or planar region
 Edge – a line segment formed by the intersection of two faces.
vertex
face
edge
Polyhedrons include prisms, pyramids and platonic solids (regular polyhedrons).
Prism
 a polyhedron with two congruent parallel faces called bases joined by lateral
faces that are parallelograms
 prisms are named by the shape of their bases
 regular prisms have bases that are regular polygons
 cube is a prism in which all the faces are squares
triangular prism
cube
pentagonal prism
Right Rectangular Prism
Oblique Rectangular
Prism
(Lateral edges are perpendicular
perpendicular
to the bases)
(Lateral edges are not
to the bases)
Lateral Edge – edge
of a lateral face
Lateral Face – not a
base, a parallelogram
Lateral Face – not a
Base
– one
of the
base,
a parallelogram
congruent parallel faces
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Altitude – perpendicular to the bases
Platonic Solids (Regular Polyhedron)
 a polyhedron in which all faces are congruent regular polygons.
 there are only five regular polyhedron (yet, there is an infinite number of
regular polygons)
The Five Convex Regular Polyhedra (Platonic solids)
Tetrahedron
Hexahedron
or Cube
Octahedron
Dodecahedron
Icosahedron
4 faces
6 faces
8 faces
12 faces
20 faces
Pyramid
 a polyhedron with at least three triangular faces that meet at a point and the
other face (the base) is any type of polygon.
 pyramids are named by the shape of their bases
Vertex or Apex – point at
which triangular faces meet
Lateral edge – joins the vertex of the
pyramid to the vertex of the base.
Slant Height – height
of any lateral face.
(pyramids that are
vertex
not regular do not
have slant heights.
Lateral face – a triangle
Altitude – segment from
perpendicular to base.
Base – face that need not be a
triangle.
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Leonhard Euler (Oil er), the Swiss mathematician lived from 1707 to 1783. He
proved the following theorem known as Euler’s Theorem for Polyhedrons
F+V=E+2
where F = number of faces, V = number or vertices, E = number of edges.
Using the rectangle prism pictured above, count the faces, vertices, and edges then
verify Euler’s formula.
Type of
Polyhedron
Drawing of
Polyhedron
Number Number of Number Test Euler’s
of Faces Vertices
of Edges Formula
Square
Pyramid
Tetrahedron
Rectangular
Prism
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Non-polyhedron Solids include cylinders, cones and spheres.
Cylinder
 a solid formed from two congruent and parallel circular regions and a curved
surface joining the two
Right Cylinder
Oblique Cylinder
Base edge
Altitude
Lateral surface – is
the curved surface
Cone
 a solid formed from one circular base and a curved region.
Right Cone
Oblique Cone
Vertex
Slant height – segment
from the vertex to the
base edge. Oblique
cones
Altitude
Lateral surface – is
the curved surface
Base
Sphere
 all points in space equidistant from a given point, called its center.
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Perspective Views of Solids
Orthographic drawings provide front, side and top views.
Isometric drawings provide corner views.
Cross Section
A cross section is the intersection of a plane and a solid. A cross section is parallel to the base.
http://www.learner.org/channel/courses/learningmath/geometry/session9/part_c/index.html
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Geometry
Lesson Plan 1
Unit Title: Three-Dimensional Objects
Time Estimate: 1 period
Lesson Title: Nets
Behavioral Objectives

The student will demonstrate synthesis of three-dimensional objects by analyzing their nets.

The student will demonstrate evaluation of three-dimensional objects by determining if a net
will produce a specific object.
Materials



Regular hexagon templates
Scissors
Paper clips
Lesson Procedures
Motivation:
Display the following:
Which of the following will fold into a tetrahedron? (There is more than 1 correct choice).
Activities:
1. Distribute classwork. Working in groups of two or three the students complete the work.
2. Go to the following website: http://mathematics.hellam.net/nets.html Ask students to draw a
net of a square prism. Display the net. Ask students to draw a net of a triangular pyramid.
Display the net.
3. Students work in groups of three. Distribute hexagon template and scissors. Cut large
regular hexagons from heavy paper. Fold each on its three main diagonals and cut along one
to the center. Slide one triangular region behind another to create a pentagonal pyramid. Use
a paper clip to secure. The other
members of the group slide 2 and
3 triangular regions to create
square and triangular pyramids.
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4. Using the three pyramids the students discover the relationship between the number of
vertices in the base and the total number of vertices and of edges. Write the following on the
board:
A pyramid with an n-gon as a base has ___________ vertices.
A pyramid with an n-gon as a base has ___________ edges.
5. Draw a 3 by 4 by 5 unit rectangular prism on the board. Students draw prism and the net of
the prism in their notebook. Students label each rectangular face with correct dimensions.
The teacher models using nets to find surface area of solids.
6. Distribute exit question.
7. Assign homework
Closure:
Assign exit ticket and collect.
Assessments:
 Class participation
 Completion of Activities
 Exit Ticket
 Homework
7
Name:
Date:
Geometry: Three-Dimensional Objects: Nets: Classwork
Nets
A net is a two-dimensional pattern that can be folded to form a solid. A net shows all the faces
of a solid and their positions in relation to each other. Some solids, such as a sphere, can’t be
built from a net. Other solids may have more than one net.
http://gwydir.demon.co.uk/jo/solid/index.htm
Find the 11 that fold into a cube
OVER
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Find the 11 nets that fold into a(n) ______________________
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Geometry: Three-Dimensional Objects: Nets: Regular Hexagon
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Name:
Date:
Geometry: Three-Dimensional Objects: Nets: Exit Question
Name the solid shown. _________________
Draw a net of the solid. Label the dimensions.
Find the surface area of the solid.
10 units
4 units
4 units
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Geometry
Lesson Plan 2
Unit Title: Three-Dimensional Objects
Time Estimate: 1 period
Lesson Title: Pyramids & Cones: Lateral & Surface Area
Behavioral Objectives

The student will demonstrate application of three-dimensional objects by learning and
applying formulas to find the lateral area and total surface area of pyramids and cones.
Materials


Rulers
Copier Paper
A
O
B
Motivation:
Find the area of sector AOB given the diameter of
circle O is 16 and the mAOB = 50º
Activities:
1. Display overhead showing the first drawing activity. Instruct students to carefully follow the
steps for drawing a cone (three-dimensional shape) on a piece of paper. While drawing
students reflect on the two-dimensional figures that make up the solid.
2. After the drawing activity review pervious learning by discussing the types of surfaces in
each solid.
3. Display notes, unveiling line-by-line. Refer back to the motivation as the formula for the
lateral surface and surface area of a cone is derived.
4. Assign homework.
Closure:
Assign exit ticket and collect.
Assessments:
 Class participation
 Completion of Activity Worksheet
 Exit Ticket
 HSPA Open-Ended (Homework)
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Name:
Date:
Geometry: Three-Dimensional Objects: Pyramids & Cones: Lateral & Surface Area: Do Now
On a plain piece of paper follow the step-by-step instructions and draw the solid described.
Note: Dotted lines represent segments in the solids that are hidden from view.
STEP 1:
DRAWING A CONE
Draw two lateral edges
STEP 2:
STEP 3:
Complete the base circle
STEP 4:
Draw the front half of the base circle
Draw the altitude and radius
DRAWING A TRIANGULAR PYRAMID
STEP 1: Draw the front triangular face
STEP 2: Draw the remaining base edges
STEP 3: Draw the remaining lateral edge STEP 4: Draw the altitude
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DRAWING A PENTAGONAL PYRAMID
STEP 1: Draw the front triangular face
STEP 2: Draw the two visible base edges
STEP 3: Draw the remaining base edges
STEP 4: Draw the three remaining lateral edges
STEP 5: Draw the slant height
STEP 6: Draw the altitude and apothem
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Geometry: Three-Dimensional Objects: Pyramids & Cones: Lateral & Surface Area: Notes
A lateral face is any face or surface that is not a base.
l
r
The derivation of the formula for lateral surface of a cone:
Area Sector
Length arc

Area Circle
Circumfere nce
2 r

2 l
2 r  l 2
Area Sector 
2 l
Area Sector
 l2
Area Sector   r l
http://www.schoolmath3d.org/e/teacher/image/unit01/14/f01.jpg
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Name:
Date:
Geometry: Three-Dimensional Objects: Pyramids & Cones: Lateral & Surface Area: Exit
1. Name the solid shown:
_____________________
2. How many lateral faces are there? ______________
3. The base is a regular polygon. Using the most
specifically name possible describe the lateral
faces. ________________________
4. Compute the lateral area and the surface area of a cone if the slant height is 13 feet and the
diameter of the base is 10 feet. Include a sketch.
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Geometry
Lesson Plan 3
Unit Title: Three-Dimensional Objects
Time Estimate: 1 period
Lesson Title: Spheres
Behavioral Objectives

The student will demonstrate application of three-dimensional objects by learning and
applying formulas to find the surface area and volume of a sphere.
Materials


Ball
Paper cut into a rectangle that is the diameter of the ball by the circumference of the ball.
Lesson Procedures
Motivation:
From the book Entertaining Mathematical Puzzles by Martin Gardner:
Display the following problem:
Under the Band
Imagine that you are on a perfectly smooth sphere as big as the sun. A steel band is stretched
tightly around the equator.
One yard of steel is added to this band so that it is raised off the surface of the sphere by the
same distance so that you can:
A. Slip a playing card under it?
B. Slip your hand under it?
C. Slip a baseball under it?
Tally the student responses. Discuss methods for finding the answer. Would your answer be
different if it were a basketball rather than sphere the size of the sun?
Answer: The height the band is raised is the same regardless of how large the sphere may be. It
will be exactly 5.7+ inches whether the sphere is as large as the sun or as small as an orange.
Students may still not be convinced. Have 1/3 of the class start with
C1 = 1,000,000 ft and find d1; C2 = 1,000,003 ft and find d2 then find d2 - d1
Another third of the class solves
C1 = 500 ft and find d1; C2 = 503 ft and find d2 then find d2 - d1
Another third of the class solves
C1 = 10 ft and find d1; C2 = 13 ft and find d2 then find d2 - d1
Activities:
1. Distribute and display fill-in-the-blank. Have students record their responses. Students
compare their answers.
2. Using the illustration as a guide, ask the students to define great circle and hemisphere. The
correct definitions are written on the board for students to copy into their notes.
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3. Display the true/false questions. Left hand – false; right hand – true. Read first question,
pause, ask students to lift left or right hand. Continue activity.
4. Display ball and rectangular piece of paper. Wrap the paper around the ball. Students should
already understand that the surface area of an object can be represented by how much wrapping
paper it would take to cover it. If you wrap the ball with the paper, you see that it would cover
the entire sphere if it weren't for all the overlaps (which would fit into the gaps if you cut them
out). Using the illustration below, have students derive the formula.
Circumference of the sphere
Diameter of sphere
The formula for the surface area of the paper is:
Surface area = area of rectangle
=lw
=Cd
= 2πr 2r
= 4πr2
5. Ask students if the formula looks familiar. Make the connection to the lateral area of a
cylinder. Display overhead.
6. Write the formula for the volume of a sphere on the board, asking students to copy into their
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notebooks. V =  r 3 .
3
7. Have students practice using the formula by trying the following:
1. Find the volume of a sphere with diameter 4 inches.
2. Find the volume of a hemisphere with radius 3 cm.
8. Assign and collect the following problem. Find the surface area and volume of the sphere.
=18
8. Assign homework
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Closure:
Assign exit ticket and collect.
Assessments:
 Class participation
 Completion of Activities
 Exit Ticket
 Homework
19
Name:
Date:
Geometry: Three-Dimensional Objects: Spheres: Classwork
tangent
chord
net
point
circle
secant
diameter
height
center
equidistant
radius
great circle
Special Segments in Spheres
1. In space, a sphere is the set of all points ____________ from a given point called its
_____________.
2. A segment whose endpoints are the center of the sphere and a point on the sphere is a
______________.
5. A _________________ of a sphere is a segment whose endpoints are points on the sphere.
6. A chord than contains the sphere’s center is a _________________ of the sphere.
7. A __________________ to a sphere is a line that intersects the sphere in exactly one point.
Intersection of a plane and a sphere
A plane can intersect a sphere in a:
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Geometry: Three-Dimensional Objects: Spheres: Classwork
1. All chords of a sphere are diameters.
2. A radius of a great circle of a sphere is
also a radius of the sphere.
3. Two spheres may intersect in exactly one
point.
4. A sphere is contained in a plane.
5. If a great circle of a sphere is congruent
to a great circle of another sphere, then the
spheres are congruent.
6. The eastern hemisphere of Earth is
congruent to the western hemisphere of the
Earth.
7. In a sphere, all the radii are congruent.
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Geometry: Three-Dimensional Objects: Spheres: Classwork
Surface Area of Sphere
Archimedes, a Greek mathematician, found out that the surface area of a
sphere is the same as the curved surface area of a cylinder having the same
diameter as the sphere and a height same length as the diameter. In other
words, when a sphere is inscribed in a cylinder then the lateral area of the
cylinder equals the surface area of the sphere.
Curved surface area of cylinder = 2rh
= 2r(2r)
= 4r2
Given a sphere with radius r, the surface area = 4 r2
TRY:
Find the surface area of a sphere with radius 12 centimeters long.
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