Table of polyhedron dihedral angles
From Wikipedia, the free encyclopedia
The dihedral angles for the edge-uniform polyhedra are:
| Picture | Name | Schläfli symbol |
Vertex/Face configuration |
exact dihedral angle (radians) |
approximate dihedral angle (degrees) |
|---|---|---|---|---|---|
| Platonic solids | |||||
 |
Tetrahedron | {3,3} | (3)3 | arccos(1/3) | 70.53° |
 |
Hexahedron or Cube | {4,3} | (4)3 | π/2 | 90° |
 |
Octahedron | {3,4} | (3)4 | π − arccos(1/3) | 109.47° |
 |
Dodecahedron | {5,3} | (5)3 | π − arctan(2) | 116.56° |
 |
Icosahedron | {3,5} | (3)5 | π − arccos(√5/3) | 138.19° |
| Kepler-Poinsot solids | |||||
 |
Small stellated dodecahedron | {5/2,5} | (5/2)5 | π − arctan(2) | 116.56° |
 |
Great dodecahedron | {5,5/2} | (5)5/2 | arctan(2) | 63.435° |
 |
Great stellated dodecahedron | {5/2,3} | (5/2)3 | arctan(2) | 63.435° |
 |
Great icosahedron | {3,5/2} | (3)5/2 | arcsin(2/3) | 41.810° |
| Quasiregular solids (Rectified regular) | |||||
 |
Tetratetrahedron |  |
(3.3.3.3) | π − arccos(1/3) | 109.47° |
 |
Cuboctahedron |  |
(3.4.3.4) | π − arccos(1/sqr(3)) | 125.264° |
 |
Icosidodecahedron |  |
(3.5.3.5) |  |
142.623° |
 |
Dodecadodecahedron |  |
(5.5/2.5.5/2) | ||
 |
Great icosidodecahedron |  |
(3.5/2.3.5/2) | ||
| Quasiregular dual solids | |||||
|
Dual of tetratetrahedron | - | V(3.3.3.3) | π − π/2 | 90° |
 |
Rhombic dodecahedron (Dual of cuboctahedron) |
- | V(3.4.3.4) | π − π/3 | 120° |
 |
Rhombic triacontahedron (Dual of icosidodecahedron) |
- | V(3.5.3.5) | π − π/5 | 144° |
[edit] References
- Coxeter, Regular Polytopes (1963), Macmillian Company
- Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8 (Table I: Regular Polytopes, (i) The nine regular polyhedra {p,q} in ordinary space)
- Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X. (Section 3-7 to 3-9)
