Table of polyhedron dihedral angles
The dihedral angles for the edge-transitive polyhedra are:
| Picture | Name | Schläfli symbol |
Vertex/Face configuration |
exact dihedral angle (radians) |
dihedral angle – exact in bold, else approximate (degrees) |
|---|---|---|---|---|---|
| Platonic solids (regular convex) | |||||
| Tetrahedron | {3,3} | (3.3.3) | 70.529° | ||
| Hexahedron or Cube | {4,3} | (4.4.4) | 90° | ||
| Octahedron | {3,4} | (3.3.3.3) | 109.471° | ||
| Dodecahedron | {5,3} | (5.5.5) | 116.565° | ||
| Icosahedron | {3,5} | (3.3.3.3.3) | 138.190° | ||
| Kepler–Poinsot polyhedra (regular nonconvex) | |||||
| Small stellated dodecahedron | {5/2,5} | (5/2.5/2.5/2.5/2.5/2) | 116.565° | ||
| Great dodecahedron | {5,5/2} | (5.5.5.5.5)/2 | 63.435° | ||
| Great stellated dodecahedron | {5/2,3} | (5/2.5/2.5/2) | 63.435° | ||
| Great icosahedron | {3,5/2} | (3.3.3.3.3)/2 | 41.810° | ||
| Quasiregular polyhedra (Rectified regular) | |||||
| Tetratetrahedron | r{3,3} | (3.3.3.3) | 109.471° | ||
| Cuboctahedron | r{3,4} | (3.4.3.4) | 125.264° | ||
| Icosidodecahedron | r{3,5} | (3.5.3.5) | 142.623° | ||
| Dodecadodecahedron | r{5/2,5} | (5.5/2.5.5/2) | 116.565° | ||
| Great icosidodecahedron | r{5/2,3} | (3.5/2.3.5/2) | 37.377° | ||
| Ditrigonal polyhedra | |||||
| Small ditrigonal icosidodecahedron | a{5,3} | (3.5/2.3.5/2.3.5/2) | 142.623° | ||
| Ditrigonal dodecadodecahedron | b{5,5/2} | (5.5/3.5.5/3.5.5/3) | 63.435° | ||
| Great ditrigonal icosidodecahedron | c{3,5/2} | (3.5.3.5.3.5)/2 | 79.188° | ||
| Hemipolyhedra | |||||
| Tetrahemihexahedron | o{3,3} | (3.4.3/2.4) | 54.736° | ||
| Cubohemioctahedron | o{3,4} | (4.6.4/3.6) | 54.736° | ||
| Octahemioctahedron | o{4,3} | (3.6.3/2.6) | 70.529° | ||
| Small dodecahemidodecahedron | o{3,5} | (5.10.5/4.10) | 26.058° | ||
| Small icosihemidodecahedron | o{5,3} | (3.10.3/2.10) | 116.565° | ||
| Great dodecahemicosahedron | o{5/2,5} | (5.6.5/4.6) | 37.377° | ||
| Small dodecahemicosahedron | o{5,5/2} | (5/2.6.5/3.6) | 79.188° | ||
| Great icosihemidodecahedron | o{5/2,3} | (3.10/3.3/2.10/3) | 37.377° | ||
| Great dodecahemidodecahedron | o{3,5/2} | (5/2.10/3.5/3.10/3) | 63.435° | ||
| Quasiregular dual solids | |||||
| Rhombic hexahedron (Dual of tetratetrahedron) |
— | V(3.3.3.3) | 90° | ||
| Rhombic dodecahedron (Dual of cuboctahedron) |
— | V(3.4.3.4) | 120° | ||
| Rhombic triacontahedron (Dual of icosidodecahedron) |
— | V(3.5.3.5) | 144° | ||
| Medial rhombic triacontahedron (Dual of dodecadodecahedron) |
— | V(5.5/2.5.5/2) | 120° | ||
| Great rhombic triacontahedron (Dual of great icosidodecahedron) |
— | V(3.5/2.3.5/2) | 72° | ||
| Duals of the ditrigonal polyhedra | |||||
| Small triambic icosahedron (Dual of small ditrigonal icosidodecahedron) |
— | V(3.5/2.3.5/2.3.5/2) | 109.471° | ||
| Medial triambic icosahedron (Dual of ditrigonal dodecadodecahedron) |
— | V(5.5/3.5.5/3.5.5/3) | 109.471° | ||
| Great triambic icosahedron (Dual of great ditrigonal icosidodecahedron) |
— | V(3.5.3.5.3.5)/2 | 109.471° | ||
| Duals of the hemipolyhedra | |||||
| Tetrahemihexacron (Dual of tetrahemihexahedron) |
— | V(3.4.3/2.4) | 90° | ||
| Hexahemioctacron (Dual of cubohemioctahedron) |
— | V(4.6.4/3.6) | 120° | ||
| Octahemioctacron (Dual of octahemioctahedron) |
— | V(3.6.3/2.6) | 120° | ||
| Small dodecahemidodecacron (Dual of small dodecahemidodecacron) |
— | V(5.10.5/4.10) | 144° | ||
| Small icosihemidodecacron (Dual of small icosihemidodecacron) |
— | V(3.10.3/2.10) | 144° | ||
| Great dodecahemicosacron (Dual of great dodecahemicosahedron) |
— | V(5.6.5/4.6) | 120° | ||
| Small dodecahemicosacron (Dual of small dodecahemicosahedron) |
— | V(5/2.6.5/3.6) | 120° | ||
| Great icosihemidodecacron (Dual of great icosihemidodecacron) |
— | V(3.10/3.3/2.10/3) | 72° | ||
| Great dodecahemidodecacron (Dual of great dodecahemidodecacron) |
— | V(5/2.10/3.5/3.10/3) | 72° | ||
References
- Coxeter, Regular Polytopes (1963), Macmillan 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)