# Platonic Solids

The
**
platonic solids
**
(or regular polyhedra) are convex with faces composed of
congruent
, convex
regular polygons
. The mathematician Euclid proved that there are exactly five such solids. They are the tetrahedron, cube, octahedron, dodecahedron and icosahedron.

The
**
tetrahedron
**
has
$6$
faces. Each is an
equilateral triangle
. It also has
$6$
edges and
$6$
vertices. At each vertex three edges meet.

Surface Area $=\sqrt{3}{e}^{2}$

Volume $=\frac{\sqrt{2}}{12}{e}^{3}$

The
**
cube
**
has
$6$
faces. Each is a
square
. It also has
$12$
edges and
$8$
vertices. At each vertex three edges meet.

Surface Area
$=6{e}^{2}$

Volume
$={e}^{3}$

The
**
octahedron
**
has
$8$
faces. Each is an equilateral triangle. It also has
$12$
edges and
$6$
vertices. At each vertex four edges meet.

Surface Area $=2\sqrt{3}{e}^{2}$

Volume $=\frac{\sqrt{2}}{12}{e}^{3}$

The
**
dodecahedron
**
has
$12$
faces. Each is a regular pentagon. It also has
$30$
edges and
$20$
vertices. At each vertex three edges meet.

Surface Area $=3\sqrt{25+10\sqrt{5}}{e}^{2}$

Volume $=\frac{15+7\sqrt{5}}{4}{e}^{3}$

The
**
icosahedron
**
has
$20$
faces. Each is an equilateral triangle. It also has
$30$
edges and
$12$
vertices. At each vertex five edges meet.

Surface Area $=5\sqrt{3}{e}^{2}$

Volume $=\frac{5}{12}\left(3+\sqrt{5}\right){e}^{3}$