Steven Dutch, Natural and Applied Sciences, University
of Wisconsin - Green Bay

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

Polyhedron | Faces | Edges | Vertices |
---|---|---|---|

Platonic Solids | |||

Tetrahedron&251 | 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 |

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 same
size, depending on how complex the solid is. Volume is the volume for edge length 1; for
other edges multiply by the edge length cubed.

Polyhedron | Edge Length | Inradius | Circumradius | Central Angle | Volume (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)) |

Return to Symmetry Index

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*Created 1 March 1999, Last Update 1 March 1999*

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