# Dodecahedron

Dodecahedron
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Type Platonic
Face polygon pentagon
Faces 12
Edges 30
Vertices 20
Faces per vertex 3
Vertices per face 5
Symmetry group icosahedral (Ih)
of order 120
Dual polyhedron icosahedron
Properties regular, convex

A dodecahedron is literally a polyhedron with 12 faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve pentagonal faces, with three meeting at each vertex. It has twenty vertices and thirty edges. Its dual polyhedron is the icosahedron.

## Area and volume

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

${\displaystyle A=3{\sqrt {25+10{\sqrt {5}}}}a^{2}}$
${\displaystyle V={\begin{matrix}{1 \over 4}\end{matrix}}(15+7{\sqrt {5}})a^{3}}$

## Canonical coordinates

Canonical coordinates for the vertices of a dodecahedron centered at the origin are {(0,±1/φ,±φ), (±1/φ,±φ,0), (±φ,0,±1/φ), (±1,±1,±1)}, where φ = (1+√5)/2 is the golden mean.

The dihedral angle of a dodecahedron is approximately 116.565 degrees.

## Geometric relations

The term dodecahedron is also used for other polyhedra with twelve faces, most notably the rhombic dodecahedron which is dual to the cuboctahedron and occurs in nature as a crystal form. The normal dodecahedron is sometimes called the pentagonal dodecahedron to distinguish it.

Five cubes can be made from these, with their edges as diagonals of the dodecahedron's faces, and together these comprise 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.

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

### Icosahedron vs dodecahedron

Despite appearances, 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%).

## Uses

• If each edge of a dodecahedron are replaced by one ohm resistors, the resistance between opposite vertices is 7/6 ohms, and 19/30 ohm between adjacent vertices.
• Especially in roleplaying, this solid is known as a d12, one of the more common Polyhedral dice.