# List of uniform polyhedra

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There are 75 non-prismatic uniform polyhedra, not including the infinite families of polygonal prisms and antiprisms. 5 are regular and convex, 4 are regular and nonconvex, 13 are nonregular and convex, and 53 are nonregular and nonconvex.

A large amount of near-miss uniform polyhedra may also be constructed. These polyhedra are isogonal and all of their faces are either regular or almost regular. Some of the near-misses are semi-uniform.

Name Short Name Image Coxeter Diagram Vertex Type Vertices Edges Faces Army Regiment Symmetry Dihedral angles
Tetrahedron tet x3o3o 3.3.3 4 6 4 triangles Tetrahedron Tetrahedron A3 3-3: ${\displaystyle \arccos\left( \frac{1}{3} \right) \approx 70.52878°}$
Truncated tetrahedron tut x3x3o 3.6.6 12 18 4 triangles
4 hexagons
Truncated tetrahedron Truncated tetrahedron 6-3: ${\displaystyle \arccos\left( -\frac{1}{3} \right) \approx 109.47122°}$

6-6: ${\displaystyle \arccos\left( \frac{1}{3} \right) \approx 70.52878°}$

Cube cube x4o3o 4.4.4 8 12 6 squares Cube Cube B3 4-4: ${\displaystyle 90°}$
Truncated cube tic x4x3o 3.8.8 24 36 8 triangles
6 octagons
Truncated cube Truncated cube 8-3: ${\displaystyle \arccos\left( -\frac{\sqrt{3}}{3} \right) \approx 125.26439°}$

8-8: ${\displaystyle 90°}$

Great cubicuboctahedron gocco x4/3x3o4*a 8/3.3.8/3.4 24 48 8 triangles
6 squares
6 octagrams
Great cubicuboctahedron 8/3-3: ${\displaystyle \arccos\left( -\frac{\sqrt{3}}{3} \right) \approx 125.26439°}$

8/3-4: ${\displaystyle 90°}$

Quasirhombicuboctahedron querco x4/3o3x 4.3/2.4.4 24 48 8 triangles
6+12 squares
4-4: ${\displaystyle 45°}$

4-3: ${\displaystyle \arccos\left( \frac{\sqrt{6}}{3} \right) \approx 35.26439°}$

Great rhombihexahedron groh 4.8/3.4/3.8/5 24 48 12 squares
6 octagrams
8/3-4 #1: ${\displaystyle 90°}$

8/3-4 #2: ${\displaystyle 45°}$

Cuboctahedron co o4x3o 3.4.3.4 12 24 8 triangles
6 squares
Cuboctahedron Cuboctahedron 4-3: ${\displaystyle \arccos\left( -\frac{\sqrt{3}}{3} \right) \approx 125.26439°}$
Octahemioctahedron oho x3/2o3x3*a 6.3/2.6.3 12 24 8 triangles
4 hexagons
6-3: ${\displaystyle \arccos\left( \frac{1}{3} \right) \approx 70.52878°}$
Cubohemioctahedron cho (o4/3x3x4*a)/2 6.4/3.6.4 12 24 6 squares
4 hexagons
6-4: ${\displaystyle \arccos\left( \frac{\sqrt{3}}{3} \right) \approx 54.73561°}$
Truncated octahedron toe o4x3x 4.6.6 24 36 6 squares
8 hexagons
Truncated octahedron Truncated octahedron 6-4: ${\displaystyle \arccos\left( -\frac{\sqrt{3}}{3} \right) \approx 125.26439°}$

6-6: ${\displaystyle \arccos\left( -\frac{1}{3} \right) \approx 109.47122°}$

Octahedron oct o4o3x 3.3.3.3 6 12 8 triangles Octahedron Octahedron 3-3: ${\displaystyle \arccos\left( -\frac{1}{3} \right) \approx 109.47122°}$
Tetrahemihexahedron thah (x3/2o3x)/2 4.3/2.4.3 6 12 4 triangles
3 squares
A3 3-3: ${\displaystyle \arccos\left( \frac{\sqrt{3}}{3} \right) \approx 54.73561°}$
Quasitruncated hexahedron quith x4/3x3o 8/3.8/3.3 24 36 8 triangles
6 octagrams
Small rhombicuboctahedron Quasitruncated hexahedron B3 8/3-8/3: ${\displaystyle 90°}$

8/3-3: ${\displaystyle \arccos\left( \frac{\sqrt{3}}{3} \right) \approx 54.73561°}$

Small rhombicuboctahedron sirco x4o3x 3.4.4.4 24 48 8 triangles
6+12 squares
Small rhombicuboctahedron 4-3: ${\displaystyle \arccos\left( -\frac{\sqrt{6}}{3} \right) \approx 144.73561°}$

4-4: ${\displaystyle 135°}$

Small cubicuboctahedron socco x4/3o3x4*a 8.3/2.8.4 24 48 8 triangles
6 squares
6 octagons
8-4: ${\displaystyle 90°}$

8-3: ${\displaystyle \arccos\left( \frac{\sqrt{3}}{3} \right) \approx 54.73561°}$

Small rhombihexahedron sroh 4.8.4/3.8/7 24 48 12 squares
6 octagons
8-4 #1: ${\displaystyle 90°}$

8-4 #2: ${\displaystyle 45°}$

Great rhombicuboctahedron girco x4x3x 4.6.8 48 72 12 squares
8 hexagons
6 octagons
Great rhombicuboctahedron Great rhombicuboctahedron 6-4: ${\displaystyle \arccos\left( -\frac{\sqrt{6}}{3} \right) \approx 144.73561°}$

8-4: ${\displaystyle 135°}$ 8-6: ${\displaystyle \arccos\left( -\frac{\sqrt{3}}{3} \right) \approx 125.26439°}$

Cubitruncated cuboctahedron cotco x4/3x3x4*a 8/3.6.8 48 72 8 hexagons
6 octagons
6 octagrams
Semi-uniform great rhombicuboctahedron Cuboctatruncated cuboctahedron 8/3-6: ${\displaystyle \arccos\left( -\frac{\sqrt{3}}{3} \right) \approx 125.26439°}$

8/3-8: ${\displaystyle 90°}$
8-6: ${\displaystyle \arccos\left( -\frac{\sqrt{3}}{3} \right) \approx 125.26439°}$

Quasitruncated cuboctahedron quitco x4/3x3x 8/3.4.6/5 48 72 12 squares
8 hexagons
6 octagrams
Semi-uniform great rhombicuboctahedron Quasitruncated cuboctahedron 8/3-4: ${\displaystyle 135°}$

8/3-6: ${\displaystyle \arccos\left( \frac{\sqrt{3}}{3} \right) \approx 54.73561°}$
6-4: ${\displaystyle \arccos\left( \frac{\sqrt{6}}{3} \right) \approx 35.26439°}$

Snub cube snic s4s3s 3.3.3.3.4 24 60 8+24 triangles
6 squares
Snub cube Snub cube B3+ 3-3: ${\displaystyle \approx 53.23459°}$

4-3: ${\displaystyle \approx 142.98343°}$

Dodecahedron doe x5o3o 5.5.5 20 30 12 pentagons Dodecahedron Dodecahedron H3 5-5: ${\displaystyle \arccos\left( -\frac{\sqrt{5}}{5} \right) \approx 116.56505°}$
Great stellated dodecahedron gissid x5/2o3o 5/2.5/2.5/2 20 30 12 pentagrams Great stellated dodecahedron 5/2-5/2: ${\displaystyle \arccos\left( \frac{\sqrt{5}}{5} \right) \approx 63.43495°}$
Small ditrigonary icosidodecahedron sidtid x5/2o3o3*a 5/2.3.5/2.3.5/2.3 20 60 20 triangles
12 pentagrams
Small ditrigonary icosidodecahedron 5/2-3: ${\displaystyle \arccos\left( -\sqrt{\frac{5+2\sqrt{5}}{15}} \right) \approx 142.62263°}$
Ditrigonary dodecadodecahedron ditdid x5/3o3o5*a 5/3.5.5/3.5.5/3.5 20 60 12 pentagons
12 pentagrams
5-5/2: ${\displaystyle \arccos\left( \frac{\sqrt{5}}{5} \right) \approx 63.43495°}$
Great ditrigonary icosidodecahedron gidtid x3/2o3o5*a (5.3.5.3.5.3)/2 20 60 20 triangles
12 pentagons
5-3: ${\displaystyle \arccos\left( -\sqrt{\frac{5+2\sqrt{5}}{15}} \right) \approx 142.62263°}$
Truncated dodecahedron tid x5x3o 3.10.10 60 90 20 triangles
12 decagons
Truncated dodecahedron Truncated dodecahedron 10-3: ${\displaystyle \arccos\left( -\sqrt{\frac{5+2\sqrt{5}}{15}} \right) \approx 142.62263°}$

10-10: ${\displaystyle \arccos\left( -\frac{\sqrt{5}}{5} \right) \approx 116.56505°}$

Great ditrigonal dodecicosidodecahedron gidditdid x5/3x3o5*a 10/3.3.10/3.5 60 120 20 triangles
12 pentagons
12 decagrams
Great ditrigonal dodecicosidodecahedron 3-10/3: ${\displaystyle \arccos\left( -\sqrt{\frac{5+2\sqrt{5}}{15}} \right) \approx 142.62263°}$

5-10/3: ${\displaystyle \arccos\left( -\frac{\sqrt{5}}{5} \right) \approx 116.56505°}$

Great icosicosidodecahedron giid o3/2x3x5*a 6.3/2.6.5 60 120 20 triangles
12 pentagons
20 hexagons
5-6: ${\displaystyle \arccos\left( \sqrt{\frac{5+2\sqrt{5}}{15}} \right) \approx 79.18768°}$

3-6: ${\displaystyle \arccos\left( -\frac{\sqrt{5}}{3} \right) \approx 41.81032°}$

Great dodecicosahedron giddy 6.10/3.6/5.10/7 60 120 20 hexagons
12 decagrams
6-10/3 #1: ${\displaystyle \arccos\left( -\sqrt{\frac{5+2\sqrt{5}}{15}} \right) \approx 142.62263°}$

6-10/3 #2: ${\displaystyle \arccos\left( \sqrt{\frac{5+2\sqrt{5}}{15}} \right) \approx 37.37737°}$

Small inverted retrosnub icosicosidodecahedron sirsid s5/2s3/2s3/2*a (3.3.3.3.3.5/3)/2 60 180 60 triangles
20 hexagrams
12 pentagrams
Semi-uniform truncated dodecahedron Small inverted retrosnub icosicosidodecahedron 5/2–3: ${\displaystyle \arccos\left(\sqrt{\frac{15-2\sqrt5-2\sqrt{30\sqrt5-65}}{15}}\right) ≈ 44.45753°}$

3–3: ${\displaystyle \arccos\left(\frac{\sqrt{3+2\sqrt5}}{3}\right) ≈ 24.33196°}$

Icosidodecahedron id o5x3o 3.5.3.5 30 60 20 triangles
12 pentagons
Icosidodecahedron Icosidodecahedron 3-5: ${\displaystyle \arccos\left( -\sqrt{\frac{5+2\sqrt{5}}{15}} \right) \approx 142.62263°}$
Small dodecahemidodecahedron

sidhid

(x5/4o5x5*a)/2 10.5/4.10.5 30 60 12 pentagons
6 decagons
5-10: ${\displaystyle \arccos\left( \frac{\sqrt{5}}{5} \right) \approx 63.43495°}$
Small icosihemidodecahedron seihid (x3/2o3x5*a)/2 10.3/2.10.3 30 60 20 triangles
6 decagons
3-10: ${\displaystyle \arccos\left( \sqrt{\frac{5+2\sqrt{5}}{15}} \right) \approx 37.37737°}$
Dodecadodecahedron did o5/2x5o 5/2.5.5/2.5 30 60 12 pentagons
12 pentagrams
Dodecadodecahedron 5-5/2: ${\displaystyle \arccos\left( -\frac{\sqrt{5}}{5} \right) \approx 116.56505°}$
Great dodecahemicosahedron gidhei (o5/4x3x5*a)/2 6.5/4.6.5 30 60 12 pentagons
10 hexagons
5-6: ${\displaystyle \arccos\left( \sqrt{\frac{5+2\sqrt{5}}{15}} \right) \approx 37.37737°}$
Small dodecahemicosahedron sidhei (x5/3o5/2x3*a)/2 6.5/3.6.5/2 30 60 12 pentagrams
10 hexagons
5-5/2: ${\displaystyle \arccos\left( \sqrt{\frac{5-2\sqrt{5}}{15}} \right) \approx 79.18768°}$
Great icosidodecahedron gid o5/2x3o 5/2.3.5/2.3 30 60 20 triangles
12 pentagrams
Great icosidodecahedron 3-5/2: ${\displaystyle \arccos\left( \sqrt{\frac{5+2\sqrt{5}}{15}} \right) \approx 37.37737°}$
Great icosihemidodecahedron geihid (o3/2x5/3x3*a)/2 10/3.3/2.10/3.3 30 60 20 triangles
6 decagrams
3-10: ${\displaystyle \arccos\left( \sqrt{\frac{5+2\sqrt{5}}{15}} \right) \approx 37.37737°}$
Great dodecahemidodecahedron gidhid (x5/3x5/3o5/2*a)/2 10/3.5/3.10/3.5/2 30 60 12 pentagrams
6 decagrams
5/2-10: ${\displaystyle \arccos\left( \frac{\sqrt{5}}{5} \right) \approx 63.43495°}$
Truncated icosahedron ti o5x3x 5.6.6 60 90 12 pentagons
20 hexagons
Truncated icosahedron Truncated icosahedron 5-6: ${\displaystyle \arccos\left( -\sqrt{\frac{5+2\sqrt{5}}{15}} \right) \approx 142.62263°}$

6-6: ${\displaystyle \arccos\left( -\frac{\sqrt{5}}{3} \right) \approx 138.18968°}$

Truncated great dodecahedron tigid o5/2x5x 10.10.5/2 60 90 12 pentagrams
12 decagons
Semi-uniform truncated icosahedron Truncated great dodecahedron 10-5/2: ${\displaystyle \arccos\left( -\frac{\sqrt{5}}{5} \right) \approx 116.56505°}$

10-10: ${\displaystyle \arccos\left( \frac{\sqrt{5}}{5} \right) \approx 63.43495°}$

Great dodecicosidodecahedron gaddid x5/3x5/2o3*a 10/3.5/2.10/3.3 60 120 20 triangles
12 pentagrams
12 decagrams
Great dodecicosidodecahedron 5/2-10/3: ${\displaystyle \arccos\left( -\frac{\sqrt{5}}{5} \right) \approx 116.56505°}$

3-10/3: ${\displaystyle \arccos\left( -\sqrt{\frac{2-\sqrt{5}}{15}} \right) \approx 100.81232°}$

Quasirhombicosidodecahedron qrid x5/3o3x 4.5/3.4.3 60 120 20 triangles
30 squares
12 pentagrams
4-3: ${\displaystyle \arccos\left( \frac{\sqrt{15}-\sqrt{3}}{6} \right) \approx 69.09484°}$

5/2-4: ${\displaystyle \arccos\left( \frac{5-\sqrt{5}}{10} \right) \approx 58.28253°}$

Great rhombidodecahedron gird 4.10/3.4/3.10/7 60 120 30 squares
12 decagrams
4-10/3 #1: ${\displaystyle \arccos\left( \frac{5-\sqrt{5}}{10} \right) \approx 58.28253°}$

4-10/3 #2: ${\displaystyle \arccos\left( \frac{5+\sqrt{5}}{10} \right) \approx 31.71747°}$

12 pentagons
12 pentagrams
Semi-uniform truncated icosahedron Rhombidodecadodecahedron 4-5/2: ${\displaystyle \arccos\left( -\sqrt{\frac{5+\sqrt{5}}{10}} \right) \approx 148.28253°}$

4-5: ${\displaystyle \arccos\left( -\sqrt{\frac{5-\sqrt{5}}{10}} \right) \approx 121.71747°}$

Icosidodecadodecahedron ided o5/3x3x5*a 6.5/3.6.5 60 120 12 pentagons
12 pentagrams
20 hexagons
5-6: ${\displaystyle \arccos\left( -\sqrt{\frac{5-2\sqrt{5}}{15}} \right) \approx 79.18768°}$

5/2-6: ${\displaystyle \arccos\left( -\sqrt{\frac{5+2\sqrt{5}}{15}} \right) \approx 37.37737°}$

Rhombicosahedron ri 6.4.6/5.4/3 60 120 30 squares
20 hexagons
4-6 #1: ${\displaystyle \arccos\left( \frac{\sqrt{3}-\sqrt{15}}{6} \right) \approx 110.90516°}$

4-6 #1: ${\displaystyle \arccos\left( \frac{\sqrt{3}+\sqrt{15}}{6} \right) \approx 20.90516°}$

Small snub icosicosidodecahedron seside s5/2s3s3*a 3.3.3.3.3.5/2 60 180 60 triangles
20 hexagrams
12 pentagrams
Semi-uniform truncated icosahedron Small snub icosicosidodecahedron 5/2-3: ${\displaystyle \arccos\left(-\sqrt{\frac{15-2\sqrt5+2\sqrt{30\sqrt5-65}}{15}}\right) \approx 161.02258°}$

3-3: ${\displaystyle \arccos\left(-\frac{\sqrt{3+2\sqrt5}}{3}\right) \approx 155.66804°}$

Icosahedron ike o5o3x 3.3.3.3.3 12 30 20 triangles Icosahedron Icosahedron 3-3: ${\displaystyle \arccos\left( -\frac{\sqrt{5}}{3} \right) \approx 138.18969°}$
Great dodecahedron gad o5/2o5x (5.5.5.5.5)/2 12 30 12 pentagons 5-5: ${\displaystyle \arccos\left( \frac{\sqrt{5}}{5} \right) \approx 63.43495°}$
Small stellated dodecahedron sissid x5/2o5o 5/2.5/2.5/2.5/2.5/2 12 30 12 pentagrams Small stellated dodecahedron 5/2-5/2: ${\displaystyle \arccos\left( -\frac{\sqrt{5}}{5} \right) \approx 116.56505°}$
Great icosahedron gike o5/2o3x (3.3.3.3.3)/2 12 30 20 triangles 3-3: ${\displaystyle \arccos\left( \frac{\sqrt{5}}{3} \right) \approx 41.81031°}$
Quasitruncated small stellated dodecahedron quit sissid x5/3x5o 10/3.10/3.5 60 90 12 pentagons
12 decagrams
Small rhombicosidodecahedron Quasitruncated small stellated dodecahedron 10/3-10/3: ${\displaystyle \arccos\left( -\frac{\sqrt{5}}{5} \right) \approx 116.56505°}$

5-10/3: ${\displaystyle \arccos\left( \frac{\sqrt{5}}{5} \right) \approx 63.43495°}$

Small rhombicosidodecahedron srid x5o3x 3.4.5.4 60 120 20 triangles
30 squares
12 pentagons
Small rhombicosidodecahedron 4-3: ${\displaystyle \arccos\left( -\frac{\sqrt{3}+\sqrt{15}}{6} \right) \approx 159.09484°}$

4-5: ${\displaystyle \arccos\left( -\sqrt{\frac{5+\sqrt{5}}{10}} \right) \approx 148.28253°}$

Small dodecicosidodecahedron saddid x3/2o5x5*a 10.3/2.10.5 60 120 20 triangles
12 pentagons
12 decagons
5-10: ${\displaystyle \arccos\left( \frac{\sqrt{5}}{5} \right) \approx 63.43495°}$

3-10: ${\displaystyle \arccos\left( \sqrt{\frac{5+2\sqrt{5}}{15}} \right) \approx 37.37737°}$

Small rhombidodecahedron sird 10.4.10/9.4/3 60 120 30 squares
12 decagons
4-10 #1: ${\displaystyle \arccos\left( -\sqrt{\frac{5-\sqrt{5}}{10}} \right) \approx 121.71747°}$

4-10 #2: ${\displaystyle \arccos\left( -\sqrt{\frac{5+\sqrt{5}}{10}} \right) \approx 148.28253°}$

Truncated great icosahedron tiggy o5/2x3x 6.6.5/2 60 90 12 pentagrams
20 hexagons
Semi-uniform small rhombicosidodecahedron Truncated great icosahedron 6-5/2: ${\displaystyle \arccos\left( -\sqrt{\frac{5-2\sqrt{5}}{15}} \right) \approx 100.81232°}$

6-6: ${\displaystyle \arccos\left( \frac{\sqrt{5}}{3} \right) \approx 41.81031°}$

Small icosicosidodecahedron siid x5/2o3x3*a 6.5/2.6.3 60 120 20 triangles
12 pentagrams
20 hexagons
Semi-uniform small rhombicosidodecahedron Small icosicosidodecahedron 5/2-6: ${\displaystyle \arccos\left( -\sqrt{\frac{5+2\sqrt{5}}{15}} \right) \approx 142.62263°}$

3-6: ${\displaystyle \arccos\left( -\frac{\sqrt{5}}{3} \right) \approx 38.18969°}$

Small ditrigonal dodecicosidodecahedron sidditdid x5/3o3x5*a 10.5/3.10.3 60 120 20 triangles
12 pentagrams
12 decagons
3-10: ${\displaystyle \arccos\left( \sqrt{\frac{5-2\sqrt{5}}{15}} \right) \approx 100.81232°}$

5/2-10: ${\displaystyle \arccos\left( \frac{\sqrt{5}}{4} \right) \approx 63.43495°}$

Small dodecicosahedron siddy 10.6.10/9.6/5 60 120 20 hexagons
12 decagons
6-10 #1: ${\displaystyle \arccos\left(\sqrt{\frac{5-2\sqrt5}{15}}\right) \approx 79.18768°}$

6-10 #2: ${\displaystyle \arccos\left(\sqrt{\frac{5+2\sqrt5}{15}}\right) ≈ 37.37737°}$

Quasitruncated great stellated dodecahedron quit gissid x5/3x3o 10/3.10/3.3 60 90 20 triangles
12 decagrams
Quasitruncated great stellated dodecahedron 10/3–3: ${\displaystyle \arccos\left(\sqrt{\frac{5-2\sqrt5}{15}}\right) \approx 79.18768°}$

10/3–10/3: ${\displaystyle \arccos\left(\frac{\sqrt5}{5}\right) \approx 63.43494°}$

Great dirhombicosidodecahedron gidrid s3s5/2s3/2s5/3*aØ*c *bØ*d (4.5/3.4.3.4.5/2.4.3/2)/2 60 240 40 triangles
60 squares
24 pentagrams
Semi-uniform small rhombicosidodecahedron Great dirhombicosidodecahedron 5/2–4: ${\displaystyle \arccos\left(\sqrt{\frac{5-2\sqrt5}{5}}\right) ≈ 71.03929°}$

3–4: ${\displaystyle \arccos\left(\frac{\sqrt3}{3}\right) ≈ 54.73561°}$

Great snub dodecicosidodecahedron gisdid s5/3s5/2s3*a 3.3.3.5/3.3.5/2 60 180 20+60 triangles
24 pentagrams
Great snub dodecicosidodecahedron H3+ 5/2–3 #1: ${\displaystyle \arccos\left(-\sqrt{\frac{5+2\sqrt5-4\sqrt{5\sqrt5-10}}{15}}\right) ≈ 125.77490°}$

3–3: ${\displaystyle \arccos\left(-\frac13\right) ≈ 109.47122°}$
5/2–3 #2: ${\displaystyle \arccos\left(\sqrt{\frac{5+2\sqrt5+4\sqrt{5\sqrt5-10}}{15}}\right) ≈ 16.30368°}$

Great rhombicosidodecahedron grid x5x3x 4.6.10 120 180 30 squares
20 hexagons
12 decagons
Great rhombicosidodecahedron Great rhombicosidodecahedron H3 6–4: ${\displaystyle \arccos\left(-\frac{\sqrt3+\sqrt{15}}{6}\right) ≈ 159.09484°}$

10–4: ${\displaystyle \arccos\left(-\sqrt{\frac{5+\sqrt5}{10}}\right) ≈ 148.28253°}$
10–6: ${\displaystyle \arccos\left(-\sqrt{\frac{5+2\sqrt5}{15}}\right) ≈ 142.62263°}$

Icositruncated dodecadodecahedron idtid x5/3x3x5*a 10/3.6.10 120 180 20 hexagons
12 decagons
12 decagrams
Semi-uniform great rhombicosidodecahedron Icosidodecatruncated icosidodecahedron 10/3–6: ${\displaystyle \arccos\left(-\sqrt{\frac{5+2\sqrt5}{15}}\right) ≈ 142.62263°}$

10–10/3: ${\displaystyle \arccos\left(-\frac{\sqrt5}{5}\right) ≈ 116.56505°}$
10–6: ${\displaystyle \arccos\left(-\sqrt{\frac{5-2\sqrt5}{15}}\right) ≈ 100.81232°}$

Quasitruncated dodecadodecahedron quitdid x5/3x5x 10/3.4.10/9 120 180 30 squares
12 decagons
12 decagrams
Semi-uniform great rhombicosidodecahedron Quasitruncated dodecadodecahedron 10/3–4: ${\displaystyle \arccos\left(-\sqrt{\frac{5+\sqrt5}{10}}\right) ≈ 148.28253°}$

10–10/3: ${\displaystyle \arccos\left(\frac{\sqrt5}{5}\right) ≈ 63.43495°}$
10–4: ${\displaystyle \arccos\left(\sqrt{\frac{5-\sqrt5}{10}}\right) ≈ 58.28253°}$

Great quasitruncated icosidodecahedron gaquatid x5/3x3x 10/3.4.6 120 180 30 squares
20 hexagons
12 decagrams
Semi-uniform great rhombicosidodecahedron Great quasitruncated icosidodecahedron 10/3–4: ${\displaystyle \arccos\left(-\sqrt{\frac{5-\sqrt5}{10}}\right) ≈ 121.71747°}$

10/3–6: ${\displaystyle \arccos\left(\sqrt{\frac{5-2\sqrt5}{15}}\right) ≈ 79.18768°}$
6–4: ${\displaystyle \arccos\left(\frac{\sqrt{15}-\sqrt3}{6}\right) ≈ 69.09484°}$

Snub dodecahedron snid s5s3s 3.3.3.3.5 60 150 20+60 triangles
12 pentagons
Snub dodecahedron Snub dodecahedron H3+ 3-3: ${\displaystyle \approx 164.17537°}$

5-3: ${\displaystyle \approx 152.92992°}$

Snub dodecadodecahedron siddid s5/2s5s 3.3.5/2.3.5 60 150 60 triangles
12 pentagons
12 pentagrams
Non-uniform snub dodecahedron Snub dodecadodecahedron 5–3: ${\displaystyle \approx 129.79515°}$

3–3: ${\displaystyle \approx 151.48799°}$ 5/2–3: ${\displaystyle \approx 157.77792°}$

Inverted snub dodecadodecahedron isdid s5/3s5s 3.5/3.3.3.5 60 150 60 triangles
12 pentagons
12 pentagrams
Non-uniform snub dodecahedron Inverted snub dodecadodecahedron 3–3: ${\displaystyle \approx 130.49074°}$

5-3: ${\displaystyle \approx 68.64088°}$ 5/2-3: ${\displaystyle \approx 11.12448°}$

Great snub icosidodecahedron gosid s5/2s3s 3.3.3.3.5/2 60 150 20+60 triangles
12 pentagrams
Non-uniform snub dodecahedron Great snub icosidodecahedron 5/2–3: ${\displaystyle \approx 138.82237°}$

3–3: ${\displaystyle \approx 126.82315°}$

Great inverted snub icosidodecahedron gisid s5/3s3s 3.3.3.3.5/3 60 150 20+60 triangles
12 pentagrams
Non-uniform snub dodecahedron Great inverted snub icosidodecahedron 3–3: ${\displaystyle \approx 89.78760°}$

5/2–3: ${\displaystyle \approx 21.61047°}$

Snub icosidodecadodecahedron sided s5/3s3s5*a 3.3.3.5.5/3 60 180 20+60 triangles
12 pentagons
12 pentagrams
Non-uniform snub dodecahedron Snub icosidodecadodecahedron 3–3: ${\displaystyle \approx 146.78125°}$

5–3: ${\displaystyle \approx 120.43401°}$
5/2–3: ${\displaystyle \approx 7.35214°}$

Great retrosnub icosidodecahedron girsid s5/3s3/2s (3.3.3.3.5/2)/2 60 150 20+60 triangles
12 pentagrams
Non-uniform snub dodecahedron Great inverted retrosnub icosidodecahedron 5/2–3: ${\displaystyle \approx 67.31029°}$

3–3: ${\displaystyle \approx 21.72466°}$

## Exotic polyhedroids

Name Short Name Image Coxeter Diagram Vertex Type Vertices Edges Faces Army Regiment Symmetry Dihedral angles
Small complex icosidodecahedron(ike + gad) cid o5/3o3x5*a (3.5.3.5.3.5.3.5.3.5)/2 12 30*2 20 triangles
12 pentagons
Icosahedron Icosahedron H3
Great complex icosidodecahedron(sissid + gike) gacid o5/3x3o5*a (3.5/2.3.5/2.3.5/2.3.5/2.3.5/2)/2 12 30*2 20 triangles
12 pentagrams
Icosahedron Great icosahedron
Small complex rhombicosidodecahedron(sidtid + 5 cubes) sicdatrid x3o5/2x 3(3.4.5/2.4) 20 60*2 20 triangles Dodecahedron sidtid