# Regular Polyhedra

## Summary

**Regular polyhedra** generalize the notion of regular polygons to three dimensions. A regular polyhedron is a polyhedron that consists of polygonal congruent faces, which are assembled in the same way around each vertex. As shown below, there are only five convex regular polyhedra, and they are known collectively as the Platonic solids.

The following table gives the number of vertices, the number of edges, and the number of faces in each of the five Platonic solids. These values satisfy Euler's formula: \(V – E + F = 2\), in which \(V\), \(E\), and \(F\) each denote the number of vertices, the number of edges, and the number of faces, respectively.

\[ \begin{align} \text{polyhedron} &&\text{Vertices} &&\text{Edges} &&\text{Faces} \\ \text{regular tetrahedron} &&4 &&6 &&4 \\ \text{cube} &&8 &&12 &&6 \\ \text{regular octahedron} &&6 &&12 &&8 \\ \text{regular dodecahedron} &&20 &&30 &&12 \\ \text{regular icosahedron} &&12 &&30 &&20 \end{align} \]

## Examples

## How many faces does a regular tetrahedron have?

From the summary above, we know that a regular tetrahedron has 4 vertices, 6 edges, and 4 faces. In fact, it is called a "tetrahedron" because it has 4 faces. \( _ \square \)

## Identify a regular polyhedron that has 6 vertices and 8 faces.

What has 6 vertices, 12 edges, and 8 faces is a regular octahedron. \( _ \square \)

## If a regular polyhedron has 12 vertices, and 30 edges, then how many faces does it have?

Let we \(V\) be the number of vertices, \(E\) the number of edges, and \(F\) the number of faces. Then from Euler's formula, we have

\[ \begin{align} V - E + F &= 2 \\ F & = 2 - V + E \\ & = 2 - 12 + 30 \\ &= 20. \end{align} \]

Thus, a regular polyhedron that has 12 vertices and 30 edges, has 20 faces. \( _ \square \)

## What is the volume of a cube whose side length is 3?

A cube is a regular polyhedron whose edges always meet at the vertices perpendicularly.

Thus, the volume of the cube is

\[ 3 \times 3 \times 3 = 27.\ _\square \]

## What is the sum of the number of faces of all types of regular polyhedra?

There are five types of regular polyhedra: regular tetrahedron, cube, regular octahedron, regular dodecagon, and regular icosahedron.

Since the number of faces of each regular polyhedron are 4, 6, 8, 12, and 20, respectively, the answer is

\[ 4 + 6 + 8 + 12 + 20 = 50.\ _\square \]

**Cite as:**Regular Polyhedra.

*Brilliant.org*. Retrieved from https://brilliant.org/wiki/regular-polyhedra/