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)	${\displaystyle \arccos({\frac {1}{3}})}$	70.529°
	Hexahedron or Cube	{4,3}	(4.4.4)	${\displaystyle \arccos(0)={\frac {\pi }{2}}}$	90°
	Octahedron	{3,4}	(3.3.3.3)	${\displaystyle \arccos(-{\frac {1}{3}})}$	109.471°
	Dodecahedron	{5,3}	(5.5.5)	${\displaystyle \arccos(-{\frac {\sqrt {5}}{5}})}$	116.565°
	Icosahedron	{3,5}	(3.3.3.3.3)	${\displaystyle \arccos(-{\frac {\sqrt {5}}{3}})}$	138.190°
Kepler–Poinsot solids (regular nonconvex)
	Small stellated dodecahedron	{⁠5/2⁠,5}	(⁠5/2⁠.⁠5/2⁠.⁠5/2⁠.⁠5/2⁠.⁠5/2⁠)	${\displaystyle \arccos(-{\frac {\sqrt {5}}{5}})}$	116.565°
	Great dodecahedron	{5,⁠5/2⁠}	⁠(5.5.5.5.5)/2⁠	${\displaystyle \arccos({\frac {\sqrt {5}}{5}})}$	63.435°
	Great stellated dodecahedron	{⁠5/2⁠,3}	(⁠5/2⁠.⁠5/2⁠.⁠5/2⁠)	${\displaystyle \arccos({\frac {\sqrt {5}}{5}})}$	63.435°
	Great icosahedron	{3,⁠5/2⁠}	⁠(3.3.3.3.3)/2⁠	${\displaystyle \arccos({\frac {\sqrt {5}}{3}})}$	41.810°
Quasiregular polyhedra (Rectified regular)
	Tetratetrahedron	r{3,3}	(3.3.3.3)	${\displaystyle \arccos(-{\frac {1}{3}})}$	109.471°
	Cuboctahedron	r{3,4}	(3.4.3.4)	${\displaystyle \arccos(-{\frac {\sqrt {3}}{3}})}$	125.264°
	Icosidodecahedron	r{3,5}	(3.5.3.5)	${\displaystyle \arccos {(-{\frac {1}{15}}{\sqrt {75+30{\sqrt {5}}}})}}$	142.623°
	Dodecadodecahedron	r{⁠5/2⁠,5}	(5.⁠5/2⁠.5.⁠5/2⁠)	${\displaystyle \arccos(-{\frac {\sqrt {5}}{5}})}$	116.565°
	Great icosidodecahedron	r{⁠5/2⁠,3}	(3.⁠5/2⁠.3.⁠5/2⁠)	${\displaystyle \arccos {({\frac {1}{15}}{\sqrt {75+30{\sqrt {5}}}})}}$	37.377°
Ditrigonal polyhedra
	Small ditrigonal icosidodecahedron	a{5,3}	(3.⁠5/2⁠.3.⁠5/2⁠.3.⁠5/2⁠)	${\displaystyle \arccos {(-{\frac {1}{15}}{\sqrt {75+30{\sqrt {5}}}})}}$	142.623°
	Ditrigonal dodecadodecahedron	b{5,⁠5/2⁠}	(5.⁠5/3⁠.5.⁠5/3⁠.5.⁠5/3⁠)	${\displaystyle \arccos({\frac {\sqrt {5}}{5}})}$	63.435°
	Great ditrigonal icosidodecahedron	c{3,⁠5/2⁠}	⁠(3.5.3.5.3.5)/2⁠	${\displaystyle \arccos {({\frac {1}{15}}{\sqrt {75-30{\sqrt {5}}}})}}$	79.188°
Hemipolyhedra
	Tetrahemihexahedron	o{3,3}	(3.4.⁠3/2⁠.4)	${\displaystyle \arccos({\frac {\sqrt {3}}{3}})}$	54.736°
	Cubohemioctahedron	o{3,4}	(4.6.⁠4/3⁠.6)	${\displaystyle \arccos({\frac {\sqrt {3}}{3}})}$	54.736°
	Octahemioctahedron	o{4,3}	(3.6.⁠3/2⁠.6)	${\displaystyle \arccos({\frac {1}{3}})}$	70.529°
	Small dodecahemidodecahedron	o{3,5}	(5.10.⁠5/4⁠.10)	${\displaystyle \arccos {({\frac {1}{15}}{\sqrt {195-6{\sqrt {5}}}})}}$	26.058°
	Small icosihemidodecahedron	o{5,3}	(3.10.⁠3/2⁠.10)	${\displaystyle \arccos(-{\frac {\sqrt {5}}{5}})}$	116.56°
	Great dodecahemicosahedron	o{⁠5/2⁠,5}	(5.6.⁠5/4⁠.6)	${\displaystyle \arccos {({\frac {1}{15}}{\sqrt {75+30{\sqrt {5}}}})}}$	37.377°
	Small dodecahemicosahedron	o{5,⁠5/2⁠}	(⁠5/2⁠.6.⁠5/3⁠.6)	${\displaystyle \arccos {({\frac {1}{15}}{\sqrt {75-30{\sqrt {5}}}})}}$	79.188°
	Great icosihemidodecahedron	o{⁠5/2⁠,3}	(3.⁠10/3⁠.⁠3/2⁠.⁠10/3⁠)	${\displaystyle \arccos {({\frac {1}{15}}{\sqrt {75+30{\sqrt {5}}}})}}$	37.377°
	Great dodecahemidodecahedron	o{3,⁠5/2⁠}	(⁠5/2⁠.⁠10/3⁠.⁠5/3⁠.⁠10/3⁠)	${\displaystyle \arccos({\frac {\sqrt {5}}{5}})}$	63.435°
Quasiregular dual solids
	Rhombic hexahedron (Dual of tetratetrahedron)	—	V(3.3.3.3)	${\displaystyle \arccos(0)={\frac {\pi }{2}}}$	90°
	Rhombic dodecahedron (Dual of cuboctahedron)	—	V(3.4.3.4)	${\displaystyle \arccos(-{\frac {1}{2}})={\frac {2\pi }{3}}}$	120°
	Rhombic triacontahedron (Dual of icosidodecahedron)	—	V(3.5.3.5)	${\displaystyle \arccos(-{\frac {{\sqrt {5}}+1}{4}})={\frac {4\pi }{5}}}$	144°
	Medial rhombic triacontahedron (Dual of dodecadodecahedron)	—	V(5.⁠5/2⁠.5.⁠5/2⁠)	${\displaystyle \arccos(-{\frac {1}{2}})={\frac {2\pi }{3}}}$	120°
	Great rhombic triacontahedron (Dual of great icosidodecahedron)	—	V(3.⁠5/2⁠.3.⁠5/2⁠)	${\displaystyle \arccos({\frac {{\sqrt {5}}-1}{4}})={\frac {2\pi }{5}}}$	72°
Duals of the ditrigonal polyhedra
	Small triambic icosahedron (Dual of small ditrigonal icosidodecahedron)	—	V(3.⁠5/2⁠.3.⁠5/2⁠.3.⁠5/2⁠)	${\displaystyle \arccos(-{\frac {1}{3}})}$	109.471°
	Medial triambic icosahedron (Dual of ditrigonal dodecadodecahedron)	—	V(5.⁠5/3⁠.5.⁠5/3⁠.5.⁠5/3⁠)	${\displaystyle \arccos(-{\frac {1}{3}})}$	109.471°
	Great triambic icosahedron (Dual of great ditrigonal icosidodecahedron)	—	V⁠(3.5.3.5.3.5)/2⁠	${\displaystyle \arccos(-{\frac {1}{3}})}$	109.471°
Duals of the hemipolyhedra
	Tetrahemihexacron (Dual of tetrahemihexahedron)	—	V(3.4.⁠3/2⁠.4)	${\displaystyle \pi -{\frac {\pi }{2}}}$	90°
	Hexahemioctacron (Dual of cubohemioctahedron)	—	V(4.6.⁠4/3⁠.6)	${\displaystyle \pi -{\frac {\pi }{3}}}$	120°
	Octahemioctacron (Dual of octahemioctahedron)	—	V(3.6.⁠3/2⁠.6)	${\displaystyle \pi -{\frac {\pi }{3}}}$	120°
	Small dodecahemidodecacron (Dual of small dodecahemidodecacron)	—	V(5.10.⁠5/4⁠.10)	${\displaystyle \pi -{\frac {\pi }{5}}}$	144°
	Small icosihemidodecacron (Dual of small icosihemidodecacron)	—	V(3.10.⁠3/2⁠.10)	${\displaystyle \pi -{\frac {\pi }{5}}}$	144°
	Great dodecahemicosacron (Dual of great dodecahemicosahedron)	—	V(5.6.⁠5/4⁠.6)	${\displaystyle \pi -{\frac {\pi }{3}}}$	120°
	Small dodecahemicosacron (Dual of small dodecahemicosahedron)	—	V(⁠5/2⁠.6.⁠5/3⁠.6)	${\displaystyle \pi -{\frac {\pi }{3}}}$	120°
	Great icosihemidodecacron (Dual of great icosihemidodecacron)	—	V(3.⁠10/3⁠.⁠3/2⁠.⁠10/3⁠)	${\displaystyle \pi -{\frac {2\pi }{5}}}$	72°
	Great dodecahemidodecacron (Dual of great dodecahemidodecacron)	—	V(⁠5/2⁠.⁠10/3⁠.⁠5/3⁠.⁠10/3⁠)	${\displaystyle \pi -{\frac {2\pi }{5}}}$	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)
• Weisstein, Eric W. "Uniform Polyhedron". MathWorld .