# Table of polyhedron dihedral angles

The dihedral angles for the edge-transitive polyhedra are:

Picture Name Schläfli
symbol
Vertex/Face
configuration
exact dihedral angle
dihedral angle
– exact in bold,
else approximate
(degrees)
Platonic solids (regular convex)
Tetrahedron {3,3} (3.3.3) arccos(1/3) 70.53°
Hexahedron or Cube {4,3} (4.4.4) π/2 90°
Octahedron {3,4} (3.3.3.3) π − arccos(1/3) 109.47°
Dodecahedron {5,3} (5.5.5) π − arctan(2) 116.56°
Icosahedron {3,5} (3.3.3.3.3) π − arccos(5/3) 138.19°
Kepler–Poinsot solids (regular nonconvex)
Small stellated dodecahedron {5/2,5} (5/2.5/2.5/2.5/2.5/2) π − arctan(2) 116.56°
Great dodecahedron {5,5/2} (5.5.5.5.5)/2 arctan(2) 63.435°
Great stellated dodecahedron {5/2,3} (5/2.5/2.5/2) arctan(2) 63.435°
Great icosahedron {3,5/2} (3.3.3.3.3)/2 arcsin(2/3) 41.810°
Quasiregular polyhedra (Rectified regular)
Tetratetrahedron r{3,3} (3.3.3.3) π − arccos(1/3) 109.47°
Cuboctahedron r{3,4} (3.4.3.4) π − arccos(1/3) 125.264°
Icosidodecahedron r{3,5} (3.5.3.5) ${\displaystyle \pi -\arccos {\left({\sqrt {\frac {5+2{\sqrt {5}}}{15}}}\right)}}$ 142.623°
Dodecadodecahedron r{5/2,5} (5.5/2.5.5/2) π − arctan(2) 116.56°
Great icosidodecahedron r{5/2,3} (3.5/2.3.5/2) ${\displaystyle \arccos {\left({\sqrt {\frac {5+2{\sqrt {5}}}{15}}}\right)}}$ 37.377°
Ditrigonal polyhedra
Small ditrigonal icosidodecahedron a{5,3} (3.5/2.3.5/2.3.5/2)
Great ditrigonal icosidodecahedron c{3,5/2} (3.5.3.5.3.5)/2
Hemipolyhedra
Tetrahemihexahedron o{3,3} (3.4.3/2.4) 54.73°
Cubohemioctahedron o{3,4} (4.6.4/3.6) 54.73°
Octahemioctahedron o{4,3} (3.6.3/2.6) 70.53°
Small dodecahemidodecahedron o{3,5} (5.10.5/4.10) 26.063°
Small icosihemidodecahedron o{5,3} (3.10.3/2.10) 116.56°
Great dodecahemicosahedron o{5/2,5} (5.6.5/4.6)
Small dodecahemicosahedron o{5,5/2} (5/2.6.5/3.6)
Great icosihemidodecahedron o{5/2,3} (3.10/3.3/2.10/3)
Great dodecahemidodecahedron o{3,5/2} (5/2.10/3.5/3.10/3)
Quasiregular dual solids
Rhombic hexahedron
(Dual of tetratetrahedron)
V(3.3.3.3) ππ/2 90°
Rhombic dodecahedron
(Dual of cuboctahedron)
V(3.4.3.4) ππ/3 120°
Rhombic triacontahedron
(Dual of icosidodecahedron)
V(3.5.3.5) ππ/5 144°
Medial rhombic triacontahedron
V(5.5/2.5.5/2) ππ/3 120°
Great rhombic triacontahedron
(Dual of great icosidodecahedron)
V(3.5/2.3.5/2) π2π/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)
Medial triambic icosahedron
V(5.5/3.5.5/3.5.5/3)
Great triambic icosahedron
(Dual of great ditrigonal icosidodecahedron)
V(3.5.3.5.3.5)/2
Duals of the hemipolyhedra
Tetrahemihexacron
(Dual of tetrahemihexahedron)
V(3.4.3/2.4) ππ/2 90°
Hexahemioctacron
(Dual of cubohemioctahedron)
V(4.6.4/3.6) ππ/3 120°
Octahemioctacron
(Dual of octahemioctahedron)
V(3.6.3/2.6) ππ/3 120°
Small dodecahemidodecacron
(Dual of small dodecahemidodecacron)
V(5.10.5/4.10) ππ/5 144°
Small icosihemidodecacron
(Dual of small icosihemidodecacron)
V(3.10.3/2.10) ππ/5 144°
Great dodecahemicosacron
(Dual of great dodecahemicosahedron)
V(5.6.5/4.6) ππ/3 120°
Small dodecahemicosacron
(Dual of small dodecahemicosahedron)
V(5/2.6.5/3.6) ππ/3 120°
Great icosihemidodecacron
(Dual of great icosihemidodecacron)
V(3.10/3.3/2.10/3) π2π/5 72°
Great dodecahemidodecacron
(Dual of great dodecahemidodecacron)
V(5/2.10/3.5/3.10/3) π2π/5 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)