# Polyhedron Data

Steven Dutch, Professor Emeritus, Natural and Applied Sciences, Universityof Wisconsin - Green Bay

## Faces, Edges, Vertices

Numbers in parentheses denote the number of sides of each face. For example, 12(5) forthe dodecahedron means the solid has 12 pentagon faces. Faces, edges and vertices obey Euler'sRule: F + V = E + 2.

polyhedron Faces Edges Vertices
Platonic Solids
Tetrahedron 4(3) 6 4
Cube 6(4) 12 8
Octahedron 8(3) 12 6
Dodecahedron 12(5) 30 20
Icosahedron 20(3) 30 12
Archimedean Solids
Truncated Tetrahedron 4(3) 4(6) 18 12
Truncated Cube 8(3) 6(8) 36 24
Cuboctahedron 8(3) 6(4) 24 12
Truncated Octahedron 8(6) 6(4) 36 24
Rhombicuboctahedron 8(3) 18(4) 48 24
"Truncated Cuboctahedron" 12(4) 8(6) 6(8) 72 48
Snub Cube 32(3) 6(4) 60 24
Truncated dodecahedron 20(3) 12(10) 90 60
Icosidodecahedron 20(3) 12(5) 60 30
Truncated Icosahedron 12(5) 20(6) 90 60
Rhombicosidodecahedron 20(3) 30(4) 12(5) 120 60
"Truncated Icosidodecahedron" 30(4) 20(6) 12(10) 180 120
Snub Dodecahedron 80(3) 12(5) 150 60
Prisms and Antiprisms
n-Prism n(4) 2(n) 3n 2n
n-Antiprism 2n(3) 2(n) 4n 2n

## Dimension Data

Edge lengths, inradii and circumradii are given with respect to the interradius,the radius of a sphere that touches the midpoints of each edge. The inradius is the best overall estimator of size. The edge length can vary widely for polyhedra of about the samesize, depending on how complex the solid is. Volume is the volume for edge length 1; forother edges multiply by the edge length cubed.

(Edge =1)
Platonic Solids
Tetrahedron 2.8284 0.5774 1.7321 109.47 0.11785
Cube 1.4142 0.7071 1.2247 70.53 1.00000
Octahedron 2 0.8165 1.4142 90 0.47140
Dodecahedron 0.7639 0.8507 1.0705 41.82 7.66312
Icosahedron 1.2361 0.9342 1.1756 63.43 2.18170
Archimedean Solids
Truncated Tetrahedron 0.9418 0.9045 1.1055 50.47 0.394
Truncated Cube 0.5858 0.9597 1.0420 32.65 17.76
Cuboctahedron 1.1547 0.8660 1.1547 60.00 2.37
Truncated Octahedron 0.6667 0.9487 1.0541 36.87 12.71
Rhombicuboctahedron 0.7654 0.9340 1.0707 41.88 8.74
"Truncated Cuboctahedron" 0.4419 0.9765 1.0241 24.92 45.63
Snub Cube 0.8018 0.9282 1.0773 43.68 7.68
Truncated dodecahedron 0.3416 0.9857 1.0145 19.40 97.45
Icosidodecahedron 0.6498 0.9511 1.0515 36.00 14.31
Truncated Icosahedron 0.4120 0.9794 1.0120 23.28 57.56
Rhombicosidodecahedron 0.4595 0.9747 1.0260 25.87 42.01
"Truncated Icosidodecahedron" 0.2653 0.9914 1.0087 15.10 21.79
Snub Dodecahedron 0.4769 0.9727 1.0280 26.82 37.72
Prisms and Antiprisms
n-Prism 2 sin180/n 1/sqrt(1 + sin^2(180/n)) sqrt(1 + sin^2(180/n)) arctan(sin 180/n)
n-Antiprism 4 sin 180/2n 1/sqrt(3 - 2cos 180/n) sqrt(3 -
2cos 180/n)
arctan(2 sin(180/2n))

Created 1 March 1999, Last Update 1 March 1999