# Dodecahedron (EntityTopic, 12)

The dodecahedron is one of the five Platonic solids. It contains 12 pentagons joining three to a vertex.

## Coordinates

The coordinates of a dodecahedron with side length 2/φ (where φ = (1+√5)/2) are:

(±1, ±1, ±1
(0, ±1/φ, ±φ)
(±1/φ, ±φ, 0)
(±φ, 0, ±1/φ)

The first set of coordinates shows that a cube can be inscribed into a dodecahedron.

## Equations

• The hypervolumes of a dodecahedron with side length l are given by:
total edge length = 30l
surface area = 3√(25 + 10√5) · l2
volume = (15 + 7√5)4 · l3

## Incidence matrix

Dual: icosahedron

 # TXID Type Name Va Ea 5a 0 Va = point ; 1 Ea 2 = digon ; 2 5a 5 5 = pentagon ; 3 C1a 20 30 12 = dodecahedron ;

## Usage as facets

 Notable Trishapes Regular: tetrahedron • cube • octahedron • dodecahedron • icosahedron Direct truncates: tetrahedral truncate • cubic truncate • octahedral truncate • dodecahedral truncate • icosahedral truncate Mesotruncates: stauromesohedron • stauroperihedron • stauropantohedron • rhodomesohedron • rhodoperihedron • rhodopantohedron Snubs: snub staurohedron • snub rhodohedron Curved: sphere • torus • cylinder • cone • frustum • crind