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9.1: Polyhedrons

Last updated

Jun 15, 2022
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- 9: Solid Figures
- 9.2: Faces, Edges, and Vertices of Solids

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3-D figures formed by polygons enclosing regions in space.

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.

f-d_91a81d101450f612b406dcbf3024bc51c5e0c3e5fd1bc61db8da1e34+IMAGE_TINY+IMAGE_TINY.png — Figure $\PageIndex{1}$

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:

f-d_fa878ef60b960bd6f1562c3ac60b15f53a594f113655d50b854b82eb+IMAGE_TINY+IMAGE_TINY.png — Figure $\PageIndex{2}$

f-d_337d522665f20bbff90b3f9a3127f63b0cd095e15083ad7ea1861635+IMAGE_TINY+IMAGE_TINY.png — Figure $\PageIndex{3}$

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: $F+V=E+2$ .

f-d_550cc1ce1962dbbdf1cf6cec098e0eedf1f135df4f918297fd4175ce+IMAGE_TINY+IMAGE_TINY.png — Figure $\PageIndex{4}$

$Faces+Vertices=Edges+2$

$5+6=9+2$

A regular polyhedron is a polyhedron where all the faces are congruent regular polygons. There are only five regular polyhedra, called the Platonic solids.

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.

f-d_49bc4b842ef6632bb34ca98ae4bf64a358fde57545f076976c79cea8+IMAGE_TINY+IMAGE_TINY.png — Figure $\PageIndex{5}$

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?

Example $\PageIndex{1}$

Figure $\PageIndex{6}$
Figure $\PageIndex{7}$
Figure $\PageIndex{8}$

Solution

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.

Example $\PageIndex{2}$

In a six-faced polyhedron, there are 10 edges. How many vertices does the polyhedron have?

Solution

Solve for $V$ in Euler’s Theorem.

$\begin{aligned} F+V&=E+2 \\ 6+V&=10+2 \\ V&=6\end{aligned}$

Therefore, there are 6 vertices.

Example $\PageIndex{3}$

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?

Solution

Plug all three numbers into Euler’s Theorem.

$\begin{aligned} F+V&=E+2 \\ 5+10&=12+2 \\ 15 &\neq 14 \end{aligned}$

Because the two sides are not equal, Markus made a mistake.

Example $\PageIndex{4}$

Find the number of faces, vertices, and edges in an octagonal prism.

f-d_df896d80b04745665ba854217d4381a3b32087709616f72c1a2a4d96+IMAGE_TINY+IMAGE_TINY.png — Figure $\PageIndex{9}$

Solution

There are 10 faces and 16 vertices. Use Euler’s Theorem, to solve for $E$ .

$\begin{aligned} F+V&=E+2 \\ 10+16&=E+2 \\ 24&=E \end{aligned}$

Therefore, there are 24 edges.

Example $\PageIndex{5}$

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.

f-d_6def0c116e07990393eb5902bfd8e689d813dfdef551ebe9ad9a2a75+IMAGE_TINY+IMAGE_TINY.png — Figure $\PageIndex{10}$

Solution

We can use Euler's Theorem to solve for the number of vertices.

$\begin{aligned} F+V&=E+2 \\ 32+V&=90+2 \\ V&=60\end{aligned}$

Therefore, it has 60 vertices.

Review

Complete the table using Euler’s Theorem.

	*Name*	*Faces*	*Edges*	*Vertices*
1.	Rectangular Prism	6	12
2.	Octagonal Pyramid		16	9
3.	Regular Icosahedron	20		12
4.	Cube		12	8
5.	Triangular Pyramid	4		4
6.	Octahedron	8	12
7.	Heptagonal Prism		21	14
8.	Triangular Prism	5	9

Determine if the following figures are polyhedra. If so, name the figure and find the number of faces, edges, and vertices.

Figure $\PageIndex{11}$
Figure $\PageIndex{12}$
Figure $\PageIndex{13}$
Figure $\PageIndex{14}$
Figure $\PageIndex{15}$

Review (Answers)

To see the Review answers, open this PDF file and look for section 11.1.

Additional Resources

Video: Polyhedrons Principles - Basic

Activities: Polyhedrons Discussion Questions

Study Aids: Polyhedra Study Guide

Practice: Polyhedrons

Real World: Roly Poly Polyhedron!

Review

Review (Answers)

Additional Resources

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