Name |
Tetrahedron |
Hexahedron |
Octahedron |
Dodecahedron |
Icosahedron |
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Picture |
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Schläfli symbol |
{3, 3} | {4, 3} | {3, 4} | {5, 3} | {3, 5} | |
Face polygon |
triangle | square | triangle | pentagon | triangle | |
Vertex figure |
triangle | triangle | square | triangle | pentagon | |
Vertices () |
4 | 8 | 6 | 20 | 12 | |
Edges () |
6 |
12 |
30 |
|
Faces () |
4 | 6 | 8 | 12 | 20 | |
Euler characteristic () |
2 |
2 |
Coxeter number () |
4 |
6 |
10 |
|
Vertex configuration |
3.3.3 | 4.4.4 | 3.3.3.3 | 5.5.5 | 3.3.3.3.3 | |
Wythoff symbol |
3 | 3 2 | 3 | 4 2 | 4 | 3 2 | 3 | 5 2 | 5 | 3 2 | q | p 2 |
Coxeter diagram |
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Dual polyhedron |
tetrahedron | octahedron | hexahedron | icosahedron | dodecahedron | |
Symmetry group |
tetrahedral group [3, 3] *332 Td A3 |
octahedral group [4, 3] *432 Oh BC3 |
icosahedral group [5, 3] *532 Ih H3 |
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Symmetries |
24 | 48 | 120 | |
Rotation symmetry group |
[3, 3]+ 332 T |
[4, 3]+ 432 O |
[5, 3]+ 532 I |
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Rotational symmetries |
12 | 24 | 60 | |
Face solid angle |
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Angular deficiency () |
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Dihedral angle () |
≈ 70.53° | | ≈ 109.47° | ≈ 116.56° | ≈ 138.19° | |
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Solid angle () |
≈ 0.551286 | | ≈ 1.35935 | ≈ 2.96174 | ≈ 2.63455 | |
Standard vertex coordinates |
(1,1,−1) (1,−1,1) (−1,1,1) (−1,−1,−1) |
(±1,±1,±1) |
(±1,0,0) (0,±1,0) (0,0,±1) |
(±1,±1,±1) (0,±1/φ,±φ) (±1/φ,±φ,0) (±φ,0,±1/φ) |
(0,±1,±φ) (±1,±φ,0) (±φ,0,±1) |
Edge length () |
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Circumradius () |
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Midradius () |
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Inradius () |
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Surface area |
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Volume |
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