Regular dodecahedron dodecahedron forms an icosidodecahedron. The regular dodecahedron has icosahedral symmetry I, Coxeter group [5,3], order 120, with an abstract group structure of "A" × "Z". When a regular 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...), which ratio is approximately , or in exact terms: or . A regular dodecahedron has 12 faces and 20 vertices,
Regular dodecahedron faces) is while the midradius, which touches the middle of each edge, is These quantities may also be expressed as where "ϕ" is the golden ratio. Note that, given a regular dodecahedron of edge length one, "r" is the radius of a circumscribing sphere about a cube of edge length "ϕ", and "r" is the apothem of a regular pentagon of edge length "ϕ". The surface area "A" and the volume "V" of a regular dodecahedron of edge length "a" are: Additionally, the surface area and volume of a regular dodecahedron are related to the golden ratio. A dodecahedron with