This is a table of properties of the five convex regular polyhedra (also known as Platonic solids).
Name | Tetrahedron | Hexahedron | Octahedron | Dodecahedron | Icosahedron | |
---|---|---|---|---|---|---|
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 | |||
Rotational symmetries | 12 | 24 | 60 | |||
Face solid angle | ||||||
Angular deficiency () | ||||||
Dihedral angle () | ≈ 70.53° | ≈ 109.47° | ≈ 116.56° | ≈ 138.19° | ||
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 () | ||||||
Circumradius () | ||||||
Midradius () | ||||||
Inradius () | ||||||
Surface area | ||||||
Volume |
This is a table of properties of the five convex regular polyhedra (also known as Platonic solids).
Name | Tetrahedron | Hexahedron | Octahedron | Dodecahedron | Icosahedron | |
---|---|---|---|---|---|---|
Picture |
![]() |
![]() |
![]() |
![]() |
![]() |
|
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 | ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() |
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 |
|||
Symmetries | 24 | 48 | 120 | |||
Rotation symmetry group | [3, 3]+ 332 T | [4, 3]+ 432 O | [5, 3]+ 532 I | |||
Rotational symmetries | 12 | 24 | 60 | |||
Face solid angle | ||||||
Angular deficiency () | ||||||
Dihedral angle () | ≈ 70.53° | ≈ 109.47° | ≈ 116.56° | ≈ 138.19° | ||
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 () | ||||||
Circumradius () | ||||||
Midradius () | ||||||
Inradius () | ||||||
Surface area | ||||||
Volume |