Polyhedrons
Bill Zahner
Lori Jordan
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Printed: March 28, 2016
AUTHORS
Bill Zahner
Lori Jordan
www.ck12.org
C HAPTER
Chapter 1. Polyhedrons
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Polyhedrons
Here you’ll learn how to identify polyhedron and regular polyhedron and the connections between the numbers of
faces, edges, and vertices in polyhedron.
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
Click image to the left or use the URL below.
URL: http://www.ck12.org/flx/render/embeddedobject/137626
CK-12 Foundation: Chapter11PolyhedronsA
Learn more about 3-D Solid Properties by watching the video at this link.
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 called a face. The line segment where two faces intersect is called an edge and the point of intersection
of two edges is a vertex. There are no gaps between the edges or vertices in a polyhedron. Examples of polyhedrons
include a cube, prism, or pyramid. Non-polyhedrons are cones, spheres, and cylinders because they have sides that
are not polygons.
A prism is a polyhedron with two congruent bases, in parallel planes, and the lateral sides are rectangles. Prisms
are explored in further detail in another Concept.
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A pyramid is a polyhedron with one base and all the lateral sides meet at a common vertex. The lateral sides are
triangles. Pyramids are explored in further detail in another Concept.
All prisms and pyramids are named by their bases. So, the first prism would be a triangular prism and the second
would be an octagonal prism. The first pyramid would be a hexagonal pyramid and the second would be a square
pyramid. The lateral faces of a pyramid are always triangles.
Euler’s Theorem states that the number of faces (F), vertices (V ), and edges (E) of a polyhedron can be related
such that F +V = E + 2.
A regular polyhedron is a polyhedron where all the faces are congruent regular polygons. There are five regular
polyhedra called the Platonic solids, after the Greek philosopher Plato. These five solids are significant because they
are the only five regular polyhedra. There are only five because the sum of the measures of the angles that meet at
each vertex must be less than 360◦ . Therefore the only combinations are 3, 4 or 5 triangles at each vertex, 3 squares
at each vertex or 3 pentagons. Each of these polyhedra have a name based on the number of sides, except the cube.
•
•
•
•
•
Regular Tetrahedron: A 4-faced polyhedron where all the faces are equilateral triangles.
Cube: A 6-faced polyhedron where all the faces are squares.
Regular Octahedron: An 8-faced polyhedron where all the faces are equilateral triangles.
Regular Dodecahedron: A 12-faced polyhedron where all the faces are regular pentagons.
Regular Icosahedron: A 20-faced polyhedron where all the faces are equilateral triangles.
Example A
Determine if the following solids are polyhedrons. If the solid is a polyhedron, name it and determine the number of
faces, edges and vertices each has.
a)
b)
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Chapter 1. Polyhedrons
c)
Solutions:
a) The base is a triangle and all the sides are triangles, so this is a polyhedron, a triangular pyramid. There are 4
faces, 6 edges and 4 vertices.
b) This solid is also a polyhedron because all the faces are polygons. The bases are both pentagons, so it is a
pentagonal prism. There are 7 faces, 15 edges, and 10 vertices.
c) This is a cylinder and has bases that are circles. Circles are not polygons, so it is not a polyhedron.
Example B
Find the number of faces, vertices, and edges in the octagonal prism.
Because this is a polyhedron, we can use Euler’s Theorem to find either the number of faces, vertices or edges. It is
easiest to count the faces, there are 10 faces. If we count the vertices, there are 16. Using this, we can solve for E in
Euler’s Theorem.
F +V = E + 2
10 + 16 = E + 2
24 = E
There are 24 edges.
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Example C
In a six-faced polyhedron, there are 10 edges. How many vertices does the polyhedron have?
Solve for V in Euler’s Theorem.
F +V = E + 2
6 +V = 10 + 2
V =6
There are 6 vertices.
Watch this video for help with the Examples above.
MEDIA
Click image to the left or use the URL below.
URL: http://www.ck12.org/flx/render/embeddedobject/137627
CK-12 Foundation: Chapter11PolyhedronsB
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.
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.
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Chapter 1. Polyhedrons
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.
9.
10.
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11.
12.
13.
14. A truncated icosahedron is a polyhedron with 12 regular pentagonal faces and 20 regular hexagonal faces and
90 edges. This icosahedron closely resembles a soccer ball. How many vertices does it have? Explain your
reasoning.
For problems 15-17, we are going to connect the Platonic Solids to probability. A six sided die is the shape of a
cube. The probability of any one side landing face up is 61 because each of the six faces is congruent to each other.
15. What shape would we make a die with 12 faces? If we number these faces 1 to 12, and each face has the same
likelihood of landing face up, what is the probability of rolling a multiple of three?
16. I have a die that is a regular octahedron. Each face is labeled with a consecutive prime number starting with
2. What is the largest prime number on my die?
17. Challenge Rebecca wants to design a new die. She wants it to have one red face. The other faces will be
yellow, blue or green. How many faces should her die have and how many of each color does it need so that:
the probability of rolling yellow is eight times the probability of rolling red, the probability of rolling green
is half the probability of rolling yellow and the probability of rolling blue is seven times the probability of
rolling red?
Answers for Explore More Problems
To view the Explore More answers, open this PDF file and look for section 11.1.
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