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User:Fropuff/Tables/Platonic solids

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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