Table of polyhedron dihedral angles
In-game article clicks load inline without leaving the challenge.
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) | arccos ( 1 3 ) {\displaystyle \arccos({\frac {1}{3}})} | 70.529° | |
| Hexahedron or Cube | {4,3} | (4.4.4) | arccos ( 0 ) = π 2 {\displaystyle \arccos(0)={\frac {\pi }{2}}} | 90° | |
| Octahedron | {3,4} | (3.3.3.3) | arccos ( − 1 3 ) {\displaystyle \arccos(-{\frac {1}{3}})} | 109.471° | |
| Dodecahedron | {5,3} | (5.5.5) | arccos ( − 5 5 ) {\displaystyle \arccos(-{\frac {\sqrt {5}}{5}})} | 116.565° | |
| Icosahedron | {3,5} | (3.3.3.3.3) | arccos ( − 5 3 ) {\displaystyle \arccos(-{\frac {\sqrt {5}}{3}})} | 138.190° | |
| Kepler–Poinsot polyhedra (regular nonconvex) | |||||
| Small stellated dodecahedron | {5/2,5} | (5/2.5/2.5/2.5/2.5/2) | arccos ( − 5 5 ) {\displaystyle \arccos(-{\frac {\sqrt {5}}{5}})} | 116.565° | |
| Great dodecahedron | {5,5/2} | (5.5.5.5.5)/2 | arccos ( 5 5 ) {\displaystyle \arccos({\frac {\sqrt {5}}{5}})} | 63.435° | |
| Great stellated dodecahedron | {5/2,3} | (5/2.5/2.5/2) | arccos ( 5 5 ) {\displaystyle \arccos({\frac {\sqrt {5}}{5}})} | 63.435° | |
| Great icosahedron | {3,5/2} | (3.3.3.3.3)/2 | arccos ( 5 3 ) {\displaystyle \arccos({\frac {\sqrt {5}}{3}})} | 41.810° | |
| Quasiregular polyhedra (Rectified regular) | |||||
| Tetratetrahedron | r{3,3} | (3.3.3.3) | arccos ( − 1 3 ) {\displaystyle \arccos(-{\frac {1}{3}})} | 109.471° | |
| Cuboctahedron | r{3,4} | (3.4.3.4) | arccos ( − 3 3 ) {\displaystyle \arccos(-{\frac {\sqrt {3}}{3}})} | 125.264° | |
| Icosidodecahedron | r{3,5} | (3.5.3.5) | arccos ( − 1 15 75 + 30 5 ) {\displaystyle \arccos {(-{\frac {1}{15}}{\sqrt {75+30{\sqrt {5}}}})}} | 142.623° | |
| Dodecadodecahedron | r{5/2,5} | (5.5/2.5.5/2) | arccos ( − 5 5 ) {\displaystyle \arccos(-{\frac {\sqrt {5}}{5}})} | 116.565° | |
| Great icosidodecahedron | r{5/2,3} | (3.5/2.3.5/2) | arccos ( 1 15 75 + 30 5 ) {\displaystyle \arccos {({\frac {1}{15}}{\sqrt {75+30{\sqrt {5}}}})}} | 37.377° | |
| Ditrigonal polyhedra | |||||
| Small ditrigonal icosidodecahedron | a{5,3} | (3.5/2.3.5/2.3.5/2) | arccos ( − 1 15 75 + 30 5 ) {\displaystyle \arccos {(-{\frac {1}{15}}{\sqrt {75+30{\sqrt {5}}}})}} | 142.623° | |
| Ditrigonal dodecadodecahedron | b{5,5/2} | (5.5/3.5.5/3.5.5/3) | arccos ( 5 5 ) {\displaystyle \arccos({\frac {\sqrt {5}}{5}})} | 63.435° | |
| Great ditrigonal icosidodecahedron | c{3,5/2} | (3.5.3.5.3.5)/2 | arccos ( 1 15 75 − 30 5 ) {\displaystyle \arccos {({\frac {1}{15}}{\sqrt {75-30{\sqrt {5}}}})}} | 79.188° | |
| Hemipolyhedra | |||||
| Tetrahemihexahedron | o{3,3} | (3.4.3/2.4) | arccos ( 3 3 ) {\displaystyle \arccos({\frac {\sqrt {3}}{3}})} | 54.736° | |
| Cubohemioctahedron | o{3,4} | (4.6.4/3.6) | arccos ( 3 3 ) {\displaystyle \arccos({\frac {\sqrt {3}}{3}})} | 54.736° | |
| Octahemioctahedron | o{4,3} | (3.6.3/2.6) | arccos ( 1 3 ) {\displaystyle \arccos({\frac {1}{3}})} | 70.529° | |
| Small dodecahemidodecahedron | o{3,5} | (5.10.5/4.10) | arccos ( 1 15 195 − 6 5 ) {\displaystyle \arccos {({\frac {1}{15}}{\sqrt {195-6{\sqrt {5}}}})}} | 26.058° | |
| Small icosihemidodecahedron | o{5,3} | (3.10.3/2.10) | arccos ( − 5 5 ) {\displaystyle \arccos(-{\frac {\sqrt {5}}{5}})} | 116.565° | |
| Great dodecahemicosahedron | o{5/2,5} | (5.6.5/4.6) | arccos ( 1 15 75 + 30 5 ) {\displaystyle \arccos {({\frac {1}{15}}{\sqrt {75+30{\sqrt {5}}}})}} | 37.377° | |
| Small dodecahemicosahedron | o{5,5/2} | (5/2.6.5/3.6) | arccos ( 1 15 75 − 30 5 ) {\displaystyle \arccos {({\frac {1}{15}}{\sqrt {75-30{\sqrt {5}}}})}} | 79.188° | |
| Great icosihemidodecahedron | o{5/2,3} | (3.10/3.3/2.10/3) | arccos ( 1 15 75 + 30 5 ) {\displaystyle \arccos {({\frac {1}{15}}{\sqrt {75+30{\sqrt {5}}}})}} | 37.377° | |
| Great dodecahemidodecahedron | o{3,5/2} | (5/2.10/3.5/3.10/3) | arccos ( 5 5 ) {\displaystyle \arccos({\frac {\sqrt {5}}{5}})} | 63.435° | |
| Quasiregular dual solids | |||||
| Rhombic hexahedron (Dual of tetratetrahedron) | — | V(3.3.3.3) | arccos ( 0 ) = π 2 {\displaystyle \arccos(0)={\frac {\pi }{2}}} | 90° | |
| Rhombic dodecahedron (Dual of cuboctahedron) | — | V(3.4.3.4) | arccos ( − 1 2 ) = 2 π 3 {\displaystyle \arccos(-{\frac {1}{2}})={\frac {2\pi }{3}}} | 120° | |
| Rhombic triacontahedron (Dual of icosidodecahedron) | — | V(3.5.3.5) | arccos ( − 5 + 1 4 ) = 4 π 5 {\displaystyle \arccos(-{\frac {{\sqrt {5}}+1}{4}})={\frac {4\pi }{5}}} | 144° | |
| Medial rhombic triacontahedron (Dual of dodecadodecahedron) | — | V(5.5/2.5.5/2) | arccos ( − 1 2 ) = 2 π 3 {\displaystyle \arccos(-{\frac {1}{2}})={\frac {2\pi }{3}}} | 120° | |
| Great rhombic triacontahedron (Dual of great icosidodecahedron) | — | V(3.5/2.3.5/2) | arccos ( 5 − 1 4 ) = 2 π 5 {\displaystyle \arccos({\frac {{\sqrt {5}}-1}{4}})={\frac {2\pi }{5}}} | 72° | |
| Duals of the ditrigonal polyhedra | |||||
| Small triambic icosahedron (Dual of small ditrigonal icosidodecahedron) | — | V(3.5/2.3.5/2.3.5/2) | arccos ( − 1 3 ) {\displaystyle \arccos(-{\frac {1}{3}})} | 109.471° | |
| Medial triambic icosahedron (Dual of ditrigonal dodecadodecahedron) | — | V(5.5/3.5.5/3.5.5/3) | arccos ( − 1 3 ) {\displaystyle \arccos(-{\frac {1}{3}})} | 109.471° | |
| Great triambic icosahedron (Dual of great ditrigonal icosidodecahedron) | — | V(3.5.3.5.3.5)/2 | arccos ( − 1 3 ) {\displaystyle \arccos(-{\frac {1}{3}})} | 109.471° | |
| Duals of the hemipolyhedra | |||||
| Tetrahemihexacron (Dual of tetrahemihexahedron) | — | V(3.4.3/2.4) | π − π 2 {\displaystyle \pi -{\frac {\pi }{2}}} | 90° | |
| Hexahemioctacron (Dual of cubohemioctahedron) | — | V(4.6.4/3.6) | π − π 3 {\displaystyle \pi -{\frac {\pi }{3}}} | 120° | |
| Octahemioctacron (Dual of octahemioctahedron) | — | V(3.6.3/2.6) | π − π 3 {\displaystyle \pi -{\frac {\pi }{3}}} | 120° | |
| Small dodecahemidodecacron (Dual of small dodecahemidodecacron) | — | V(5.10.5/4.10) | π − π 5 {\displaystyle \pi -{\frac {\pi }{5}}} | 144° | |
| Small icosihemidodecacron (Dual of small icosihemidodecacron) | — | V(3.10.3/2.10) | π − π 5 {\displaystyle \pi -{\frac {\pi }{5}}} | 144° | |
| Great dodecahemicosacron (Dual of great dodecahemicosahedron) | — | V(5.6.5/4.6) | π − π 3 {\displaystyle \pi -{\frac {\pi }{3}}} | 120° | |
| Small dodecahemicosacron (Dual of small dodecahemicosahedron) | — | V(5/2.6.5/3.6) | π − π 3 {\displaystyle \pi -{\frac {\pi }{3}}} | 120° | |
| Great icosihemidodecacron (Dual of great icosihemidodecacron) | — | V(3.10/3.3/2.10/3) | π − 2 π 5 {\displaystyle \pi -{\frac {2\pi }{5}}} | 72° | |
| Great dodecahemidodecacron (Dual of great dodecahemidodecacron) | — | V(5/2.10/3.5/3.10/3) | π − 2 π 5 {\displaystyle \pi -{\frac {2\pi }{5}}} | 72° |
- 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). . Dover Publications, Inc. ISBN 0-486-23729-X.