User:Fropuff/Draft 11

From Wikipedia, the free encyclopedia

This is a table of properties of the five Platonic solids.

Name Tetrahedron Hexahedron
or Cube		 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 12 30 30
Faces 4 6 8 12 20
Euler characteristic 2 2 2 2 2
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
Dual polyhedron tetrahedron octahedron cube icosahedron dodecahedron
Rotational symmetries 12 24 60
Symmetries 24 48 120
Symmetry group Td Oh Ih
Angular deficiency (δ) \pi\, \frac{\pi}{2} \frac{2\pi}{3} \frac{\pi}{5} \frac{\pi}{3}
Dihedral angle (θ) ≈ 70.53° 90° ≈ 109.47° ≈ 116.56° ≈ 138.19°
\tan{\theta\over 2} 1\over{\sqrt 2} 1\, \sqrt 2 \varphi \varphi^2
\cos\theta\, 1\over 3 0\, -{1\over 3} -{1\over{\sqrt 5}} -{{\sqrt 5}\over 3}
Solid angle (Ω) ≈ 0.551286 ≈ 1.57080 ≈ 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 (a) 2\sqrt 2 2\, \sqrt 2 2/\varphi 2\,
Circumradius (R) \sqrt 3 \sqrt 3 1\, \sqrt 3 \xi\varphi
Midradius (ρ) 1\, \sqrt 2 1\over{\sqrt 2} \varphi \varphi
Inradius (r) 1\over{\sqrt 3} 1\, 1\over{\sqrt 3} \varphi/\xi \varphi^2\over 3
Surface area 8\sqrt 3 24\, 4\sqrt 3 60\over{\xi\varphi} 20\sqrt 3
Volume 8\over 3 8\, 4\over 3 4(\varphi+2) {20\over 3}\varphi^2