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 of a regular dodecahedron centered at the origin and suitably scaled and oriented: where is the golden ratio (also written "τ") ≈ 1.618. The edge length is . The circumradius is . Similar to the symmetry of the vertex coordinates, the equations of the twelve facets of the regular dodecahedron also display symmetry in their coefficients: 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 regular dodecahedron make up three of the four Kepler–Poinsot polyhedra. A rectified regular