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.
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: