Picture
Name
Schläfli
Vertex/Face
exact dihedral angle
dihedral angle
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 —
V(3.3.3.3)
arccos
(
0
)
=
π
2
{\displaystyle \arccos(0)={\frac {\pi }{2}}}
90°
Rhombic dodecahedron —
V(3.4.3.4)
arccos
(
−
1
2
)
=
2
π
3
{\displaystyle \arccos(-{\frac {1}{2}})={\frac {2\pi }{3}}}
120°
Rhombic triacontahedron —
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 —
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 —
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 —
V(3.5 / 2 .3.5 / 2 .3.5 / 2 )
arccos
(
−
1
3
)
{\displaystyle \arccos(-{\frac {1}{3}})}
109.471°
Medial triambic icosahedron —
V(5.5 / 3 .5.5 / 3 .5.5 / 3 )
arccos
(
−
1
3
)
{\displaystyle \arccos(-{\frac {1}{3}})}
109.471°
Great triambic icosahedron —
V(3.5.3.5.3.5) / 2
arccos
(
−
1
3
)
{\displaystyle \arccos(-{\frac {1}{3}})}
109.471°
Duals of the hemipolyhedra
Tetrahemihexacron —
V(3.4.3 / 2 .4)
π
−
π
2
{\displaystyle \pi -{\frac {\pi }{2}}}
90°
Hexahemioctacron —
V(4.6.4 / 3 .6)
π
−
π
3
{\displaystyle \pi -{\frac {\pi }{3}}}
120°
Octahemioctacron —
V(3.6.3 / 2 .6)
π
−
π
3
{\displaystyle \pi -{\frac {\pi }{3}}}
120°
Small dodecahemidodecacron —
V(5.10.5 / 4 .10)
π
−
π
5
{\displaystyle \pi -{\frac {\pi }{5}}}
144°
Small icosihemidodecacron —
V(3.10.3 / 2 .10)
π
−
π
5
{\displaystyle \pi -{\frac {\pi }{5}}}
144°
Great dodecahemicosacron —
V(5.6.5 / 4 .6)
π
−
π
3
{\displaystyle \pi -{\frac {\pi }{3}}}
120°
Small dodecahemicosacron —
V(5 / 2 .6.5 / 3 .6)
π
−
π
3
{\displaystyle \pi -{\frac {\pi }{3}}}
120°
Great icosihemidodecacron —
V(3.10 / 3 .3 / 2 .10 / 3 )
π
−
2
π
5
{\displaystyle \pi -{\frac {2\pi }{5}}}
72°
Great dodecahemidodecacron —
V(5 / 2 .10 / 3 .5 / 3 .10 / 3 )
π
−
2
π
5
{\displaystyle \pi -{\frac {2\pi }{5}}}
72°
