Polyhedrons
Dan Greenberg
Lori Jordan
Andrew Gloag
Victor Cifarelli
Jim Sconyers
Bill Zahner
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Printed: January 2, 2015
AUTHORS
Dan Greenberg
Lori Jordan
Andrew Gloag
Victor Cifarelli
Jim Sconyers
Bill Zahner
www.ck12.org
C HAPTER
Chapter 1. Polyhedrons
1
Polyhedrons
Here you’ll learn what a polyhedron is and the parts of a polyhedron. You’ll then use these parts in a formula called
Euler’s Theorem.
What if you were given a solid three-dimensional figure, like a carton of ice cream? How could you determine how
the faces, vertices, and edges of that figure are related? After completing this Concept, you’ll be able to use Euler’s
Theorem to answer that question.
Watch This
MEDIA
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Polyhedrons CK-12
Guidance
A polyhedron is a 3-dimensional figure that is formed by polygons that enclose a region in space. Each polygon in
a polyhedron is a face. The line segment where two faces intersect is an edge. The point of intersection of two edges
is a vertex.
Examples of polyhedrons include a cube, prism, or pyramid. Cones, spheres, and cylinders are not polyhedrons
because they have surfaces that are not polygons. The following are more examples of polyhedrons:
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The number of faces (F), vertices (V ) and edges (E) are related in the same way for any polyhedron. Their
relationship was discovered by the Swiss mathematician Leonhard Euler, and is called Euler’s Theorem.
Euler’s Theorem: F +V = E + 2.
Faces +Vertices = Edges + 2
5+6 = 9+2
A regular polyhedron is a polyhedron where all the faces are congruent regular polygons. There are only five
regular polyhedra, called the Platonic solids.
1.
2.
3.
4.
5.
Regular Tetrahedron: A 4-faced polyhedron and all the faces are equilateral triangles.
Cube: A 6-faced polyhedron and all the faces are squares.
Regular Octahedron: An 8-faced polyhedron and all the faces are equilateral triangles.
Regular Dodecahedron: A 12-faced polyhedron and all the faces are regular pentagons.
Regular Icosahedron: A 20-faced polyhedron and all the faces are equilateral triangles.
Example A
Determine if the following solids are polyhedrons. If the solid is a polyhedron, name it and find the number of faces,
edges and vertices its has.
a)
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Chapter 1. Polyhedrons
b)
c)
Answer:
a) The base is a triangle and all the sides are triangles, so this is a triangular pyramid, which is also known as a
tetrahedron. There are 4 faces, 6 edges and 4 vertices.
b) This solid is also a polyhedron. The bases are both pentagons, so it is a pentagonal prism. There are 7 faces, 15
edges, and 10 vertices.
c) The bases are circles. Circles are not polygons, so it is not a polyhedron.
Example B
Find the number of faces, vertices, and edges in an octagonal prism.
There are 10 faces and 16 vertices. Use Euler’s Theorem, to solve for E.
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F +V = E + 2
10 + 16 = E + 2
24 = E
Therefore, there are 24 edges.
Example C
A truncated icosahedron is a polyhedron with 12 regular pentagonal faces, 20 regular hexagonal faces, and 90
edges. This icosahedron closely resembles a soccer ball. How many vertices does it have? Explain your reasoning.
We can use Euler’s Theorem to solve for the number of vertices.
F +V = E + 2
32 +V = 90 + 2
V = 60
Therefore, it has 60 vertices.
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Polyhedrons CK-12
Guided Practice
1. In a six-faced polyhedron, there are 10 edges. How many vertices does the polyhedron have?
2. Markus counts the edges, faces, and vertices of a polyhedron. He comes up with 10 vertices, 5 faces, and 12
edges. Did he make a mistake?
3. Is this a polyhedron? Explain.
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Chapter 1. Polyhedrons
Answers:
1. Solve for V in Euler’s Theorem.
F +V = E + 2
6 +V = 10 + 2
V =6
Therefore, there are 6 vertices.
2. Plug all three numbers into Euler’s Theorem.
F +V = E + 2
5 + 10 = 12 + 2
15 6= 14
Because the two sides are not equal, Markus made a mistake.
3. All of the faces are polygons, so this is a polyhedron. Notice that even though not all of the faces are regular
polygons, the number of faces, vertices, and edges still works with Euler’s Theorem.
Explore More
Complete the table using Euler’s Theorem.
TABLE 1.1:
1.
2.
3.
4.
5.
6.
7.
8.
Name
Rectangular Prism
Octagonal Pyramid
Regular
Icosahedron
Cube
Triangular Pyramid
Octahedron
Heptagonal Prism
Triangular Prism
Faces
6
Edges
12
16
20
12
4
8
5
12
21
9
Vertices
9
12
8
4
14
Determine if the following figures are polyhedra. If so, name the figure and find the number of faces, edges, and
vertices.
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9.
10.
11.
12.
13.
6