Regular dodecahedron Regular dodecahedron A regular dodecahedron or pentagonal dodecahedron is a dodecahedron that is regular, which is composed of twelve regular pentagonal faces, three meeting at each vertex. It is one of the five Platonic solids. It has 12 faces, 20 vertices, 30 edges, and 160 diagonals (60 face diagonals, 100 space diagonals). It is represented by the Schläfli symbol {5,3}. If the edge length of a regular dodecahedron is "a", the radius of a circumscribed sphere (one that touches the regular dodecahedron at all vertices) is and the radius of an inscribed sphere (tangent to each of the regular dodecahedron's
Rhombic dodecahedron Bilinski dodecahedron. It has the same topology but different geometry. The rhombic faces in this form have the golden ratio. Another topologically equivalent variation, sometimes called a deltoidal dodecahedron or trapezoidal dodecahedron, is isohedral with tetrahedral symmetry order 24, distorting rhombic faces into kites (deltoids). It has 8 vertices adjusted in or out in alternate sets of 4, with the limiting case a tetrahedral envelope. Variations can be parametrized by ("a","b"), where "b" is determined from "a" for planar faces. (1,1) is the rhombic solution. As ("a") approaches , ("b") approaches infinity. When projected onto a sphere (see right), it