# Icosahedron

An icosahedron is a space figure with 20 faces that are polygons. The prefix "icosa" means twenty. In the figure below are 3 different types of icosahedrons.

Although not common, a dice in the shape of an icosahedron is sometimes used in games.

## Properties of a regular icosahedron

A regular icosahedron is an icosahedron whose faces are all congruent regular polygons. Otherwise, it is irregular. Regular icosahedrons are studied more often.

A regular icosahedron, such as the one shown above is one of 5 Platonic solids, which are types of regular polyhedra. Regular icosahedrons, have 20 congruent faces that are congruent equilateral triangles, 30 congruent edges, and 20 vertices; an edge is a line segment formed by the intersection of two adjacent faces; a vertex for a regular icosahedron is a point where 5 edges meet.

## Surface area of a regular icosahedron

We can find the area of one of the faces and multiply it by twenty to find the total surface area of a regular icosahedron. An equilateral triangle with side length e (also the length of the edges of a regular icosahedron) has an area, A, of

The total surface area, S, of a regular icosahedron in terms of its edges, e, is:

## Volume of a regular icosahedron

The volume, V, of a regular icosahedron is

where e is the length of the edge.

Example:

If the total surface area of a regular icosahedron is , what is its volume?

We can find e by substituting the given value in for the total surface area to get

100 = e2

e = 10

Substituting the length of the edge into the volume formula: