Dodecahedron

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In geometry, a dodecahedron (Greek δωδεκάεδρον, from δώδεκα 'twelve' + εδρον 'base', 'seat' or 'face') is any polyhedron with twelve flat faces, but usually a regular dodecahedron is meant: a Platonic solid. It is composed of 12 regular pentagonal faces, with three meeting at each vertex, and is represented by the Schläfli symbol {5,3}. It has 20 vertices and 30 edges. Its dual polyhedron is the icosahedron, with Schläfli symbol {3,5}.

A large number of other (nonregular) polyhedra also have 12 sides, but are given other names. The most frequently named other dodecahedron is the rhombic dodecahedron.

Contents

Dimensions

If the edge length of a regular dodecahedron is a, the radius of a circumscribed sphere (one that touches the dodecahedron at all vertices) is

and the radius of an inscribed sphere (tangent to each of the dodecahedron's faces) is

while the midradius, which touches the middle of each edge, is

Area and volume

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

Cartesian coordinates

The following Cartesian coordinates define the vertices of a dodecahedron centered at the origin:

where φ = (1+√5)/2 is the golden ratio (also written τ) = ~1.618. The edge length is 2/φ = √5–1. The containing sphere has a radius of √3.