Geometry (B): READ: Polyhedron and Euler's Formula Reading

The Polyhedron

Learning Objectives

Identify polyhedra.
Understand the properties of polyhedra.
Use Euler’s formula solve problems.
Identify regular (Platonic) polyhedra.

Introduction

In earlier chapters you learned that a polygon is a two-dimensional (planar) figure that is made of three or more points joined together by line segments. Examples of polygons include triangles, quadrilaterals, pentagons, or octagons. In general, an $n-\;\mathrm{gon}$ is a polygon with n sides. So a triangle is a $3-\;\mathrm{gon}$ , or $3-$ sided polygon, a pentagon is a $5-gon$ , or $5-$ sided polygon.

You can use polygons to construct a 3-dimensional figure called a polyhedron (plural: polyhedra). A polyhedron is a 3-dimensional figure that is made up of polygon faces. A cube is an example of a polyhedron and its faces are squares (quadrilaterals).

Polyhedron or Not

A polyhedron has the following properties:

It is a $3-$ dimensional figure.
It is made of polygons and only polygons. Each polygon is called a face of the polyhedron.
Polygon faces join together along segments called edges.
Each edge joins exactly two faces.
Edges meet in points called vertices.
There are no gaps between edges or vertices.

Example 1

Is the figure a polyhedron?

Yes. A figure is a polyhedron if it has all of the properties of a polyhedron. This figure:

Is $3-$ dimensional.
Is constructed entirely of flat polygons (triangles and rectangles).
Has faces that meet in edges and edges that meet in vertices.
Has no gaps between edges.
Has no non-polygon faces (e.g., curves).
Has no concave faces.

Since the figure has all of the properties of a polyhedron, it is a polyhedron.

Example 2

Is the figure a polyhedron?

No. This figure has faces, edges, and vertices, but all of its surfaces are not flat polygons. Look at the end surface marked A. It is flat, but it has a curved edge so it is not a polygon. Surface B is not flat (planar).

Example 3

Is the figure a polyhedron?

No. The figure is made up of polygons and it has faces, edges, and vertices. But the faces do not fit together—the figure has gaps. The figure also has an overlap that creates a concave surface. For these reasons, the figure is not a polyhedron.

Face, Vertex, Edge, Base

As indicated above, a polyhedron joins faces together along edges, and edges together at vertices. The following statements are true of any polyhedron:

Each edge joins exactly two faces.
Each edge joins exactly two vertices.

To see why this is true, take a look at this prism. Each of its edges joins two faces along a single line segment. Each of its edges includes exactly two vertices.

Let’s count the number of faces, edges, and vertices in a few typical polyhedra. The square pyramid gets its name from its base, which is a square. It has $5\;\mathrm{faces}$ , $8\;\mathrm{edges}$ , and $5\;\mathrm{vertices}$ .

Other figures have a different number of faces, edges, and vertices.

If we make a table that summarizes the data from each of the figures we get:

Figure	Vertices	Faces	Edges
Square pyramid	$5$	$5$	$8$
Rectangular prism	$8$	$6$	$12$
Octahedron	$6$	$8$	$12$
Pentagonal prism	$10$	$7$	$15$

Do you see a pattern? Calculate the sum of the number of vertices and edges. Then compare that sum to the number of edges:

Figure	$V$	$F$	$E$	$V + F$
square pyramid	$5$	$5$	$8$	$10$
rectangular prism	$8$	$6$	$12$	$14$
octahedron	$6$	$8$	$12$	$14$
pentagonal prism	$10$	$7$	$15$	$17$

Do you see the pattern? The formula that summarizes this relationship is named after mathematician Leonhard Euler. Euler’s formula says, for any polyhedron:

Euler's Formula for Polyhedra

$\text{vertices} + \text{faces} = \text{edges} + 2$

$v + f = e + 2$

You can use Euler’s formula to find the number of edges, faces, or vertices in a polyhedron.

Example 4

Count the number of faces, edges, and vertices in the figure. Does it conform to Euler’s formula?

There are $6\;\mathrm{faces}$ , $12\;\mathrm{edges}$ , and $8\;\mathrm{vertices}$ . Using the formula:

$v + f & = e + 2\\ 8 + 6 & = 12 + 2$

So the figure conforms to Euler’s formula.

Example 5

In a $6-$ faced polyhedron, there are $10\;\mathrm{edges}$ . How many vertices does the polyhedron have?

Use Euler's formula.

$v + f & = e + 2 && \text{Euler’s formula}\\ v + 6 & = 10 + 2 && \text{Substitute values for}\ f\ \text{and} \ e \\ v & = 6 && \text{Solve}$

There are $6\;\mathrm{vertices}$ in the figure.

Example 6

A 3-dimensional figure has $10\;\mathrm{vertices}$ , $5\;\mathrm{faces}$ , and $12\;\mathrm{edges}$ . It is a polyhedron? How do you know?

Use Euler's formula.

$v + f & = e + 2&& \text{Euler’s formula}\\ 10 + 5 & \neq 12 + 2 && \text{Substitute values for} \ v, f,\ \text{and}\ e \\ 15 & \neq 14&& \text{Evaluate}$

The equation does not hold so Euler’s formula does not apply to this figure. Since all polyhedra conform to Euler’s formula, this figure must not be a polyhedron.

Regular Polyhedra

Polyhedra can be named and classified in a number of ways—by side, by angle, by base, by number of faces, and so on. Perhaps the most important classification is whether or not a polyhedron is regular or not. You will recall that a regular polygon is a polygon whose sides and angles are all congruent.

A polyhedron is regular if it has the following characteristics:

All faces are the same.
All faces are congruent regular polygons.
The same number of faces meet at every vertex.
The figure has no gaps or holes.
The figure is convex—it has no indentations.

Example 7

Is a cube a regular polyhedron?

All faces of a cube are regular polygons—squares. The cube is convex because it has no indented surfaces. The cube is simple because it has no gaps. Therefore, a cube is a regular polyhedron.

A polyhedron is semi-regular if all of its faces are regular polygons and the same number of faces meet at every vertex.

Semi-regular polyhedra often have two different kinds of faces, both of which are regular polygons.
Prisms with a regular polygon base are one kind of semi-regular polyhedron.
Not all semi-regular polyhedra are prisms. An example of a non-prism is shown below.

Completely irregular polyhedra also exist. They are made of different kinds of regular and irregular polygons.

So now a question arises. Given that a polyhedron is regular if all of its faces are congruent regular polygons, it is convex and contains no gaps or holes. How many regular polyhedra actually exist?

In fact, you may be surprised to learn that only five regular polyhedra can be made. They are known as the Platonic (or noble) solids.

Note that no matter how you try, you can’t construct any other regular polyhedra besides the ones above.

Example 8

How many faces, edges, and vertices does a tetrahedron (see above) have?

$\;\mathrm{Faces}: 4, \;\mathrm{edges}: 6, \;\mathrm{vertices}: 4$

Example 9

Which regular polygon does an icosahedron feature?

An equilateral triangle

The following questions are for your own review. The answers below are for you to check your work and understanding.

Review Questions

Identify each of the following three-dimensional figures:

Below is a list of the properties of a polyhedron. Two of the properties are not correct. Find the incorrect ones and correct them.
- It is a $3$ dimensional figure.
- Some of its faces are polygons.
- Polygon faces join together along segments called edges.
- Each edge joins three faces.
- There are no gaps between edges and vertices.
Complete the table and verify Euler’s formula for each of the figures in the problem.

Figure	# vertices	# edges	# faces
Pentagonal prism
Rectangular pyramid
Triangular prism
Trapezoidal prism

Review Answers

Identify each of the following three dimensional figures:

pentagonal prism
rectangular pyramid
triangular prism
triangular pyramid
trapezoidal prism
Below is a list of the properties of a polyhedron. Two of the properties are not correct. Find the incorrect ones and correct them.
- It is a $3$ dimensional figure.
- Some of its faces are polygons. All of its faces are polygons.
- Polygon faces join together along segments called edges.
- Each edge joins three faces. Each edge joins two faces.
- There are no gaps between edges and vertices.
Complete the table and verify Euler’s formula for each of the figures in the problem.

Figure	# vertices	# edges	# faces
Pentagonal prism	$10$	$15$	$7$
Rectangular pyramid	$5$	$8$	$5$
Triangular prism	$6$	$9$	$5$
Trapezoidal prism	$8$	$12$	$6$

In all cases
$\;\mathrm{vertices} + \;\mathrm{faces} = \;\mathrm{edges} + 2$

Last modified: Tuesday, June 29, 2010, 12:02 PM