Table of polyhedron dihedral angles

Last updated February 15, 2021

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)	70.529°
	Hexahedron or Cube	{4,3}	(4.4.4)	arccos (0) = $π$ /2	90°
	Octahedron	{3,4}	(3.3.3.3)	arccos (-1/3)	109.471°
	Dodecahedron	{5,3}	(5.5.5)	arccos (-√5/5)	116.565°
	Icosahedron	{3,5}	(3.3.3.3.3)	arccos (-√5/3)	138.190°
Kepler–Poinsot solids (regular nonconvex)
	Small stellated dodecahedron	{5/2,5}	(5/2.5/2.5/2.5/2.5/2)	arccos (-√5/5)	116.565°
	Great dodecahedron	{5,5/2}	(5.5.5.5.5)/2	arccos (√5/5)	63.435°
	Great stellated dodecahedron	{5/2,3}	(5/2.5/2.5/2)	arccos (√5/5)	63.435°
	Great icosahedron	{3,5/2}	(3.3.3.3.3)/2	arccos (√5/3)	41.810°
Quasiregular polyhedra (Rectified regular)
	Tetratetrahedron	r{3,3}	(3.3.3.3)	arccos (-1/3)	109.471°
	Cuboctahedron	r{3,4}	(3.4.3.4)	arccos (-√3/3)	125.264°
	Icosidodecahedron	r{3,5}	(3.5.3.5)	$\arccos {\left(-{\frac {1}{15}}{\sqrt {75+30{\sqrt {5}}}}\right)}$	142.623°
	Dodecadodecahedron	r{5/2,5}	(5.5/2.5.5/2)	arccos (-√5/5)	116.565°
	Great icosidodecahedron	r{5/2,3}	(3.5/2.3.5/2)	$\arccos {\left({\frac {1}{15}}{\sqrt {75+30{\sqrt {5}}}}\right)}$	37.377°
Ditrigonal polyhedra
	Small ditrigonal icosidodecahedron	a{5,3}	(3.5/2.3.5/2.3.5/2)
	Ditrigonal dodecadodecahedron	b{5,5/2}	(5.5/3.5.5/3.5.5/3)
	Great ditrigonal icosidodecahedron	c{3,5/2}	(3.5.3.5.3.5)/2
Hemipolyhedra
	Tetrahemihexahedron	o{3,3}	(3.4.3/2.4)	arccos (√3/3)	54.736°
	Cubohemioctahedron	o{3,4}	(4.6.4/3.6)	arccos (√3/3)	54.736°
	Octahemioctahedron	o{4,3}	(3.6.3/2.6)	arccos (1/3)	70.529°
	Small dodecahemidodecahedron	o{3,5}	(5.10.5/4.10)	$\arccos {\left({\frac {1}{15}}{\sqrt {195-6{\sqrt {5}}}}\right)}$	26.058°
	Small icosihemidodecahedron	o{5,3}	(3.10.3/2.10)	arccos (-√5/5)	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)	arccos (0) = $π$ /2	90°
	Rhombic dodecahedron (Dual of cuboctahedron)	—	V(3.4.3.4)	arccos (-1/2) = 2 $π$ /3	120°
	Rhombic triacontahedron (Dual of icosidodecahedron)	—	V(3.5.3.5)	arccos (-√5+1/4) = 4 $π$ /5	144°
	Medial rhombic triacontahedron (Dual of dodecadodecahedron)	—	V(5.5/2.5.5/2)	arccos (-1/2) = 2 $π$ /3	120°
	Great rhombic triacontahedron (Dual of great icosidodecahedron)	—	V(3.5/2.3.5/2)	arccos (√5-1/4) = 3 $π$ /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 (Dual of ditrigonal dodecadodecahedron)	—	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)	$π$ − $π$ /2	90°
	Hexahemioctacron (Dual of cubohemioctahedron)	—	V(4.6.4/3.6)	$π$ − $π$ /3	120°
	Octahemioctacron (Dual of octahemioctahedron)	—	V(3.6.3/2.6)	$π$ − $π$ /3	120°
	Small dodecahemidodecacron (Dual of small dodecahemidodecacron)	—	V(5.10.5/4.10)	$π$ − $π$ /5	144°
	Small icosihemidodecacron (Dual of small icosihemidodecacron)	—	V(3.10.3/2.10)	$π$ − $π$ /5	144°
	Great dodecahemicosacron (Dual of great dodecahemicosahedron)	—	V(5.6.5/4.6)	$π$ − $π$ /3	120°
	Small dodecahemicosacron (Dual of small dodecahemicosahedron)	—	V(5/2.6.5/3.6)	$π$ − $π$ /3	120°
	Great icosihemidodecacron (Dual of great icosihemidodecacron)	—	V(3.10/3.3/2.10/3)	$π$ − 2 $π$ /5	72°
	Great dodecahemidodecacron (Dual of great dodecahemidodecacron)	—	V(5/2.10/3.5/3.10/3)	$π$ − 2 $π$ /5	72°

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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)
Weisstein, Eric W. "Uniform Polyhedron". MathWorld .

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