Definitions

# Dodecahedron

[doh-dek-uh-hee-druhn, doh-dek-]
A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex. It has twenty (20) vertices and thirty (30) edges. Its dual polyhedron is the icosahedron. If one were to make every one of the Platonic solids with edges of the same length, the dodecahedron would be the largest.

## Area and volume

The area A and the volume V of a regular dodecahedron of edge length a are:

$A = 3sqrt\left\{25+10sqrt\left\{5\right\}\right\} a^2 approx 20.64572a^2$

$V = frac\left\{1\right\}\left\{4\right\} \left(15+7sqrt\left\{5\right\}\right) a^3 approx 7.66311896a^3$

## Cartesian coordinates

The following Cartesian coordinates define the vertices of a dodecahedron centered at the origin:
(±1, ±1, ±1)
(0, ±1/φ, ±φ)
(±1/φ, ±φ, 0)
(±φ, 0, ±1/φ)
where φ = (1+√5)/2 is the golden ratio (also written τ). The side length is 2/φ = √5−1. The containing sphere has a radius of √3.

The dihedral angle of a dodecahedron is 2arctan(φ) or approximately 116.565 degrees.

## Geometric relations

The regular dodecahedron is the third in an infinite set of truncated trapezohedra which can be constructed by truncating the two axial vertices of a pentagonal trapezohedron.

The stellations of the dodecahedron make up three of the four Kepler-Poinsot polyhedra.

A rectified dodecahedron forms an icosidodecahedron.

The regular dodecahedron has 120 symmetries, forming the group $A_5times Z_2$.

### Vertex arrangement

The dodecahedron shares its vertex arrangement with four nonconvex uniform polyhedrons and three uniform compounds.

Five cubes fit within, with their edges as diagonals of the dodecahedron's faces, and together these make up the regular polyhedral compound of five cubes. Since two tetrahedra can fit on alternate cube vertices, five and ten tetrahedra can also fit in a dodecahedron.

 Great stellated dodecahedron Small ditrigonal icosidodecahedron Ditrigonal dodecadodecahedron Great ditrigonal icosidodecahedron Compound of five cubes Compound of five tetrahedra Compound of ten tetrahedra

### Icosahedron vs dodecahedron

When a dodecahedron is inscribed in a sphere, it occupies more of the sphere's volume (66.49%) than an icosahedron inscribed in the same sphere (60.54%).

A regular dodecahedron with edge length 1 has more than three and a half times the volume of an icosahedron with the same length edges (7.663... compared with 2.181...).

## Stellations

The 3 stellations of the dodecahedron are all regular (nonconvex) polyhedra: (Kepler-Poinsot polyhedra polyhedra)

0 1 2 3
Stellation
Dodecahedron

Small stellated dodecahedron

Great dodecahedron

Great stellated dodecahedron
Facet diagram

## Other dodecahedra

The term dodecahedron is also used for other polyhedra with twelve faces, most notably the rhombic dodecahedron which is dual to the cuboctahedron (an Archimedean solid) and occurs in nature as a crystal form. The Platonic solid dodecahedron can be called a pentagonal dodecahedron or a regular dodecahedron to distinguish it. The pyritohedron is an irregular pentagonal dodecahedron.

Other dodecahedra include: