# 9.1: Polyhedrons

• • Contributed by CK12
• CK12

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. 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: Figure $$\PageIndex{2}$$ 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: $$F+V=E+2$$. 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.

1. Regular Tetrahedron: A 4-faced polyhedron and all the faces are equilateral triangles.
2. Cube: A 6-faced polyhedron and all the faces are squares.
3. Regular Octahedron: An 8-faced polyhedron and all the faces are equilateral triangles.
4. Regular Dodecahedron: A 12-faced polyhedron and all the faces are regular pentagons.
5. Regular Icosahedron: A 20-faced polyhedron and all the faces are equilateral triangles. 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 $$\PageIndex{1}$$

1. Figure $$\PageIndex{6}$$
2. Figure $$\PageIndex{7}$$
3. 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. 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. 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.

1. Figure $$\PageIndex{11}$$
2. Figure $$\PageIndex{12}$$
3. Figure $$\PageIndex{13}$$
4. Figure $$\PageIndex{14}$$
5. Figure $$\PageIndex{15}$$