List of Johnson solids

In geometry, polyhedra are three-dimensional objects where points are connected by lines to form polygons. The points, lines, and polygons of a polyhedron are referred to as its vertices, edges, and faces respectively.^[1] A polyhedron is considered to be convex if:^[2]

• The shortest path between any two of its vertices lies either within its interior or on its boundary
• None of its faces are coplanar—they do not share the same plane, and do not "lie flat"
• None of its edges are colinear—they are not segments of the same line.

A polyhedron is said to be regular if each of its faces are both equilateral and equiangular.^[3] Regular polyhedra with the additional property of vertex-transitivity are called uniform polyhedra.^[4] A Johnson solid (or Johnson–Zalgaller solid) is any convex polyhedron with only regular polygons as its faces. Some authors exclude uniform polyhedra—which include the Platonic solids and Archimedean solids, as well as prisms and antiprisms—from their definition.^[5]

The Johnson solids are named for the mathematician Norman Johnson (1930–2017), who published a list of 92 convex polyhedra conforming with the above definition in 1966. Moreover, Johnson conjectured that the list was complete, and there could not be any other examples. Johnson's conjecture was later proven by the Russian-Israeli mathematician Victor Zalgaller (1920–2020) in 1969.^[6] The first six Johnson solids are the square pyramid, pentagonal pyramid, triangular cupola, square cupola, pentagonal cupola, and pentagonal rotunda. These solids may be applied to construct another polyhedron that has the same properties, a process known as augmentation; attaching prism or antiprism to those is known as elongation or gyroelongation, respectively. Some others may be constructed by diminishment, the removal of those from the component of polyhedra, or by snubification, a construction by cutting loose the edges, lifting the faces and rotate in certain angle, after which adding the equilateral triangles between them.^[7]

Every polyhedra has own characteristics, including symmetry and measurement. An object is said to be symmetrical if there is such transformation preserving the immunity to change. All of those transformations may be composed in a concept of group, alongside the number of elements, known as order. In two-dimensional space, these transformations include rotating around the center of a polygon and reflecting an object around the perpendicular bisector of a polygon. A polygon that is rotated symmetrically in ${\textstyle {\frac {360^{\circ }}{n}}}$ is denoted by ${\displaystyle C_{n}}$, a cyclic group of order ${\displaystyle n}$; combining with the reflection symmetry results in the symmetry of dihedral group ${\displaystyle D_{n}}$ of order ${\displaystyle 2n}$.^[8] In three-dimensional symmetry point groups, the transformation of polyhedra's symmetry includes the rotation around the line passing through the base center, known as axis of symmetry, and reflection relative to perpendicular planes passing through the bisector of a base; this is known as the pyramidal symmetry ${\displaystyle C_{nv}}$ of order ${\displaystyle 2n}$. Relatedly, polyhedra that preserve their symmetry by reflecting it across a horizontal plane are known as prismatic symmetry ${\displaystyle D_{nh}}$ of order ${\displaystyle 4n}$. The antiprismatic symmetry ${\displaystyle D_{nd}}$ of order ${\displaystyle 4n}$ preserves the symmetry by rotating its half bottom and reflection across the horizontal plane.^[9] The symmetry group ${\displaystyle C_{nh}}$ of order ${\displaystyle 2n}$ preserves the symmetry by rotation around the axis of symmetry and reflection on horizontal plane; one case that preserves the symmetry by one full rotation and one reflection horizontal plane is ${\displaystyle C_{1h}}$ of order 2, or simply denoted as ${\displaystyle C_{s}}$.^[10] The mensuration of polyhedra includes the surface area and volume. An area is a two-dimensional measurement calculated by the product of length and width, and the surface area is the overall area of all faces of polyhedra that is measured by summing all of them.^[11] A volume is a measurement of the region in three-dimensional space.^[12]

The following table contains the 92 Johnson solids of the edge length ${\displaystyle a}$. Each of the columns includes the enumeration of Johnson solid, denoted as ${\displaystyle J_{n}}$,^[13] the number of vertices, edges, and faces, symmetry, surface area ${\displaystyle A}$ and volume ${\displaystyle V}$.

Table of all 92 Johnson solids

${\displaystyle J_{n}}$	Solid name	Image	Vertices	Edges	Faces	Symmetry group and its order[14]	Surface area and volume[15]
1	Equilateral square pyramid		5	8	5	${\displaystyle C_{4v}}$ of order 8	${\displaystyle {\begin{aligned}A&=\left(1+{\sqrt {3}}\right)a^{2}\\&\approx 2.7321a^{2}\\V&={\frac {\sqrt {2}}{6}}a^{3}\\&\approx 0.2357a^{3}\end{aligned}}}$
2	Pentagonal pyramid		6	10	6	${\displaystyle C_{5v}}$ of order 10	${\displaystyle {\begin{aligned}A&={\frac {a^{2}}{2}}{\sqrt {{\frac {5}{2}}\left(10+{\sqrt {5}}+{\sqrt {75+30{\sqrt {5}}}}\right)}}\\&\approx 3.8855a^{2}\\V&=\left({\frac {5+{\sqrt {5}}}{24}}\right)a^{3}\\&\approx 0.3015a^{3}\end{aligned}}}$
3	Triangular cupola		9	15	8	${\displaystyle C_{3v}}$ of order 6	${\displaystyle {\begin{aligned}A&=\left(3+{\frac {5{\sqrt {3}}}{2}}\right)a^{2}\\&\approx 7.3301a^{2}\\V&=\left({\frac {5}{3{\sqrt {2}}}}\right)a^{3}\\&\approx 1.1785a^{3}\end{aligned}}}$
4	Square cupola		12	20	10	${\displaystyle C_{4v}}$ of order 8	${\displaystyle {\begin{aligned}A&=\left(7+2{\sqrt {2}}+{\sqrt {3}}\right)a^{2}\\&\approx 11.5605a^{2}\\V&=\left(1+{\frac {2{\sqrt {2}}}{3}}\right)a^{3}\\&\approx 1.9428a^{3}\end{aligned}}}$
5	Pentagonal cupola		15	25	12	${\displaystyle C_{5v}}$ of order 10	${\displaystyle {\begin{aligned}A&=\left({\frac {1}{4}}\left(20+5{\sqrt {3}}+{\sqrt {5\left(145+62{\sqrt {5}}\right)}}\right)\right)a^{2}\\&\approx 16.5798a^{2}\\V&=\left({\frac {1}{6}}\left(5+4{\sqrt {5}}\right)\right)a^{3}\\&\approx 2.3241a^{3}\end{aligned}}}$
6	Pentagonal rotunda		20	35	17	${\displaystyle C_{5v}}$ of order 10	${\displaystyle {\begin{aligned}A&=\left({\frac {1}{2}}\left(5{\sqrt {3}}+{\sqrt {10\left(65+29{\sqrt {5}}\right)}}\right)\right)a^{2}\\&\approx 22.3472a^{2}\\V&=\left({\frac {1}{12}}\left(45+17{\sqrt {5}}\right)\right)a^{3}\\&\approx 6.9178a^{3}\end{aligned}}}$
7	Elongated triangular pyramid		7	12	7	${\displaystyle C_{3v}}$ of order 6	${\displaystyle {\begin{aligned}A&=\left(3+{\sqrt {3}}\right)a^{2}\\&\approx 4.7321a^{2}\\V&=\left({\frac {1}{12}}\left({\sqrt {2}}+3{\sqrt {3}}\right)\right)a^{3}\\&\approx 0.5509a^{3}\end{aligned}}}$
8	Elongated square pyramid		9	16	9	${\displaystyle C_{4v}}$ of order 8	${\displaystyle {\begin{aligned}A&=\left(5+{\sqrt {3}}\right)a^{2}\\&\approx 6.7321a^{2}\\V&=\left(1+{\frac {\sqrt {2}}{6}}\right)a^{3}\\&\approx 1.2357a^{3}\end{aligned}}}$
9	Elongated pentagonal pyramid		11	20	11	${\displaystyle C_{5v}}$ of order 10	${\displaystyle {\begin{aligned}A&={\frac {20+5{\sqrt {3}}+{\sqrt {25+10{\sqrt {5}}}}}{4}}a^{2}\\&\approx 8.8855a^{2}\\V&=\left({\frac {5+{\sqrt {5}}+6{\sqrt {25+10{\sqrt {5}}}}}{24}}\right)a^{3}\\&\approx 2.022a^{3}\end{aligned}}}$
10	Gyroelongated square pyramid		9	20	13	${\displaystyle C_{4v}}$ of order 8	${\displaystyle {\begin{aligned}A&=(1+3{\sqrt {3}})a^{2}\\&\approx 6.1962a^{2}\\V&={\frac {1}{6}}\left({\sqrt {2}}+2{\sqrt {4+3{\sqrt {2}}}}\right)a^{3}\\&\approx 1.1927a^{3}\end{aligned}}}$
11	Gyroelongated pentagonal pyramid		11	25	16	${\displaystyle C_{5v}}$ of order 10	${\displaystyle {\begin{aligned}A&={\frac {1}{4}}\left(15{\sqrt {3}}+{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 8.2157a^{2}\\V&={\frac {1}{24}}\left(25+9{\sqrt {5}}\right)a^{3}\\&\approx 1.8802a^{3}\end{aligned}}}$
12	Triangular bipyramid		5	9	6	${\displaystyle D_{3h}}$ of order 12	${\displaystyle {\begin{aligned}A&={\frac {3{\sqrt {3}}}{2}}a^{2}\\&\approx 2.5981a^{2}\\V&={\frac {\sqrt {2}}{6}}a^{3}\\&\approx 0.2358a^{3}\end{aligned}}}$
13	Pentagonal bipyramid		7	15	10	${\displaystyle D_{5h}}$ of order 20	${\displaystyle {\begin{aligned}A&={\frac {5{\sqrt {3}}}{2}}a^{2}\\&\approx 4.3301a^{2}\\V&={\frac {1}{12}}\left(5+{\sqrt {5}}\right)a^{3}\\&\approx 0.603a^{3}\end{aligned}}}$
14	Elongated triangular bipyramid		8	15	9	${\displaystyle D_{3h}}$ of order 12	${\displaystyle {\begin{aligned}A&={\frac {3}{2}}\left(2+{\sqrt {3}}\right)a^{2}\\&\approx 5.5981a^{2}\\V&={\frac {1}{12}}\left(2{\sqrt {2}}+3{\sqrt {3}}\right)a^{3}\\&\approx 0.6687a^{3}\end{aligned}}}$
15	Elongated square bipyramid		10	20	12	${\displaystyle D_{4h}}$ of order 16	${\displaystyle {\begin{aligned}A&=2\left(2+{\sqrt {3}}\right)a^{2}\\&\approx 7.4641a^{2}\\V&={\frac {1}{3}}\left(3+{\sqrt {2}}\right)a^{3}\\&\approx 1.4714a^{3}\end{aligned}}}$
16	Elongated pentagonal bipyramid		12	25	15	${\displaystyle D_{5h}}$ of order 20	${\displaystyle {\begin{aligned}A&={\frac {5}{2}}\left(2+{\sqrt {3}}\right)a^{2}\\&\approx 9.3301a^{2}\\V&={\frac {1}{12}}\left(5+{\sqrt {5}}+3{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{3}\\&\approx 2.3235a^{3}\end{aligned}}}$
17	Gyroelongated square bipyramid		10	24	16	${\displaystyle D_{4d}}$ of order 16	${\displaystyle {\begin{aligned}A&=4{\sqrt {3}}a^{2}\\&\approx 6.9282a^{2}\\V&={\frac {1}{12}}\left(5+{\sqrt {5}}+3{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{3}\\&\approx 2.3235a^{3}\end{aligned}}}$
18	Elongated triangular cupola		15	27	14	${\displaystyle C_{3v}}$ of order 6	${\displaystyle {\begin{aligned}A&={\frac {1}{2}}\left(18+5{\sqrt {3}}\right)a^{2}\\&\approx 13.3301a^{2}\\V&={\frac {1}{3}}\left({\sqrt {2}}+{\sqrt {4+3{\sqrt {2}}}}\right)a^{3}\\&\approx 1.4284a^{3}\end{aligned}}}$
19	Elongated square cupola		20	36	18	${\displaystyle C_{4v}}$ of order 8	${\displaystyle {\begin{aligned}A&=(15+2{\sqrt {2}}+{\sqrt {3}})a^{2}\\&\approx 19.5605a^{2}\\V&=\left(3+{\frac {8{\sqrt {2}}}{3}}\right)a^{3}\\&\approx 6.7712a^{3}\end{aligned}}}$
20	Elongated pentagonal cupola		25	45	22	${\displaystyle C_{5v}}$ of order 10	${\displaystyle {\begin{aligned}A&={\frac {1}{4}}\left(60+5{\sqrt {3}}+10{\sqrt {5+2{\sqrt {5}}}}+{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 26.5798a^{2}\\V&={\frac {1}{6}}\left(5+4{\sqrt {5}}+15{\sqrt {5+2{\sqrt {5}}}}\right)a^{3}\\&\approx 10.0183a^{3}\end{aligned}}}$
21	Elongated pentagonal rotunda		30	55	27	${\displaystyle C_{5v}}$ of order 10	${\displaystyle {\begin{aligned}A&={\frac {1}{2}}a^{2}\left(20+5{\sqrt {3}}+5{\sqrt {5+2{\sqrt {5}}}}+3{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)\\&\approx 32.3472a^{2}\\V&={\frac {1}{12}}a^{3}\left(45+17{\sqrt {5}}+30{\sqrt {5+2{\sqrt {5}}}}\right)\\&\approx 14.612a^{3}\end{aligned}}}$
22	Gyroelongated triangular cupola		15	33	20	${\displaystyle C_{3v}}$ of order 6	${\displaystyle {\begin{aligned}A&={\frac {1}{2}}\left(6+11{\sqrt {3}}\right)a^{2}\\&\approx 12.5263a^{2}\\V&={\frac {1}{3}}{\sqrt {{\frac {61}{2}}+18{\sqrt {3}}+30{\sqrt {1+{\sqrt {3}}}}}}a^{3}\\&\approx 3.5161a^{3}\end{aligned}}}$
23	Gyroelongated square cupola		20	44	26	${\displaystyle C_{4v}}$ of order 8	${\displaystyle {\begin{aligned}A&=(7+2{\sqrt {2}}+5{\sqrt {3}})a^{2}\\&\approx 18.4887a^{2}\\V&=\left(1+{\frac {2}{3}}{\sqrt {2}}+{\frac {2}{3}}{\sqrt {4+2{\sqrt {2}}+2{\sqrt {146+103{\sqrt {2}}}}}}\right)a^{3}\\&\approx 6.2108a^{3}\end{aligned}}}$
24	Gyroelongated pentagonal cupola		25	55	32	${\displaystyle C_{5v}}$ of order 10	${\displaystyle {\begin{aligned}A&={\frac {1}{4}}\left(20+25{\sqrt {3}}+10{\sqrt {5+2{\sqrt {5}}}}+{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 25.2400a^{2}\\V&=\left({\frac {5}{6}}+{\frac {2}{3}}{\sqrt {5}}+{\frac {5}{6}}{\sqrt {2{\sqrt {650+290{\sqrt {5}}}}-2{\sqrt {5}}-2}}\right)a^{3}\\&\approx 9.0733a^{3}\end{aligned}}}$
25	Gyroelongated pentagonal rotunda		30	65	37	${\displaystyle C_{5v}}$ of order 10	${\displaystyle {\begin{aligned}A&={\frac {1}{2}}\left(15{\sqrt {3}}+\left(5+3{\sqrt {5}}\right){\sqrt {5+2{\sqrt {5}}}}\right)a^{2}\\&\approx 31.0075a^{2}\\V&=\left({\frac {45}{12}}+{\frac {17}{12}}{\sqrt {5}}+{\frac {5}{6}}{\sqrt {2{\sqrt {650+290{\sqrt {5}}}}-2{\sqrt {5}}-2}}\right)a^{3}\\&\approx 13.6671a^{3}\end{aligned}}}$
26	Gyrobifastigium		8	14	8	${\displaystyle D_{2d}}$ of order 8	${\displaystyle {\begin{aligned}A&=\left(4+{\sqrt {3}}\right)a^{2}\\&\approx 5.7321a^{2}\\V&=\left({\frac {\sqrt {3}}{2}}\right)a^{3}\\&\approx 0.866a^{3}\end{aligned}}}$
27	Triangular orthobicupola		12	24	14	${\displaystyle D_{3h}}$ of order 12	${\displaystyle {\begin{aligned}A&=2\left(3+{\sqrt {3}}\right)a^{2}\\&\approx 9.4641a^{2}\\V&={\frac {5{\sqrt {2}}}{3}}a^{3}\\&\approx 2.357a^{3}\end{aligned}}}$
28	Square orthobicupola		16	32	18	${\displaystyle D_{4h}}$ of order 16	${\displaystyle {\begin{aligned}A&=2(5+{\sqrt {3}})a^{2}\\&\approx 13.4641a^{2}\\V&=\left(2+{\frac {4{\sqrt {2}}}{3}}\right)a^{3}\\&\approx 3.8856a^{3}\end{aligned}}}$
29	Square gyrobicupola		16	32	18	${\displaystyle D_{4d}}$ of order 16	${\displaystyle {\begin{aligned}A&=2(5+{\sqrt {3}})a^{2}\\&\approx 13.4641a^{2}\\V&=\left(2+{\frac {4{\sqrt {2}}}{3}}\right)a^{3}\\&\approx 3.8856a^{3}\end{aligned}}}$
30	Pentagonal orthobicupola		20	40	22	${\displaystyle D_{5h}}$ of order 20	${\displaystyle {\begin{aligned}A&=\left(10+{\sqrt {{\frac {5}{2}}\left(10+{\sqrt {5}}+{\sqrt {75+30{\sqrt {5}}}}\right)}}\right)a^{2}\\&\approx 17.7711a^{2}\\V&={\frac {1}{3}}\left(5+4{\sqrt {5}}\right)a^{3}\\&\approx 4.6481a^{3}\end{aligned}}}$
31	Pentagonal gyrobicupola		20	40	22	${\displaystyle D_{5d}}$ of order 20	${\displaystyle {\begin{aligned}A&=\left(10+{\sqrt {{\frac {5}{2}}\left(10+{\sqrt {5}}+{\sqrt {75+30{\sqrt {5}}}}\right)}}\right)a^{2}\\&\approx 17.7711a^{2}\\V&={\frac {1}{3}}\left(5+4{\sqrt {5}}\right)a^{3}\\&\approx 4.6481a^{3}\end{aligned}}}$
32	Pentagonal orthocupolarotunda		25	50	27	${\displaystyle C_{5v}}$ of order 10	${\displaystyle {\begin{aligned}A&=\left(5+{\frac {1}{4}}{\sqrt {1900+490{\sqrt {5}}+210{\sqrt {75+30{\sqrt {5}}}}}}\right)a^{2}\\&\approx 23.5385a^{2}\\V&={\frac {5}{12}}\left(11+5{\sqrt {5}}\right)a^{3}\\&\approx 9.2418a^{3}\end{aligned}}}$
33	Pentagonal gyrocupolarotunda		25	50	27	${\displaystyle C_{5v}}$ of order 10	${\displaystyle {\begin{aligned}A&=\left(5+{\frac {15}{4}}{\sqrt {3}}+{\frac {7}{4}}{\sqrt {25+10{\sqrt {5}}}}\right)a^{2}\\&\approx 23.5385a^{2}\\V&={\frac {5}{12}}\left(11+5{\sqrt {5}}\right)a^{3}\\&\approx 9.2418a^{3}\end{aligned}}}$
34	Pentagonal orthobirotunda		30	60	32	${\displaystyle D_{5h}}$ of order 20	${\displaystyle {\begin{aligned}A&=\left((5{\sqrt {3}}+3{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 29.306a^{2}\\V&={\frac {1}{6}}(45+17{\sqrt {5}})a^{3}\\&\approx 13.8355a^{3}\end{aligned}}}$
35	Elongated triangular orthobicupola		18	36	20	${\displaystyle D_{3h}}$ of order 12	${\displaystyle {\begin{aligned}A&=2(6+{\sqrt {3}})a^{2}\\&\approx 15.4641a^{2}\\V&=\left({\frac {5{\sqrt {2}}}{3}}+{\frac {3{\sqrt {3}}}{2}}\right)a^{3}\\&\approx 4.9551a^{3}\end{aligned}}}$
36	Elongated triangular gyrobicupola		18	36	20	${\displaystyle D_{3d}}$ of order 12	${\displaystyle {\begin{aligned}A&=2(6+{\sqrt {3}})a^{2}\\&\approx 15.4641a^{2}\\V&=\left({\frac {5{\sqrt {2}}}{3}}+{\frac {3{\sqrt {3}}}{2}}\right)a^{3}\\&\approx 4.9551a^{3}\end{aligned}}}$
37	Elongated square gyrobicupola		24	48	26	${\displaystyle D_{4d}}$ of order 16	${\displaystyle {\begin{aligned}A&=2(9+{\sqrt {3}})a^{2}\\&\approx 21.4641a^{2}\\V&=\left(4+{\frac {10{\sqrt {2}}}{3}}\right)a^{3}\\&\approx 8.714a^{3}\end{aligned}}}$
38	Elongated pentagonal orthobicupola		30	60	32	${\displaystyle D_{5h}}$ of order 20	${\displaystyle {\begin{aligned}A&=\left(20+{\sqrt {{\frac {5}{2}}\left(10+{\sqrt {5}}+{\sqrt {75+30{\sqrt {5}}}}\right)}}\right)a^{2}\\&\approx 27.7711a^{2}\\V&={\frac {1}{6}}\left(10+8{\sqrt {5}}+15{\sqrt {5+2{\sqrt {5}}}}\right)a^{3}\\&\approx 12.3423a^{3}\end{aligned}}}$
39	Elongated pentagonal gyrobicupola		30	60	32	${\displaystyle D_{5d}}$ of order 20	${\displaystyle {\begin{aligned}A&=\left(20+{\sqrt {{\frac {5}{2}}\left(10+{\sqrt {5}}+{\sqrt {75+30{\sqrt {5}}}}\right)}}\right)a^{2}\\&\approx 27.7711a^{2}\\V&={\frac {1}{6}}\left(10+8{\sqrt {5}}+15{\sqrt {5+2{\sqrt {5}}}}\right)a^{3}\\&\approx 12.3423a^{3}\end{aligned}}}$
40	Elongated pentagonal orthocupolarotunda		35	70	37	${\displaystyle C_{5v}}$ of order 10	${\displaystyle {\begin{aligned}A&={\frac {1}{4}}\left(60+{\sqrt {10\left(190+49{\sqrt {5}}+21{\sqrt {75+30{\sqrt {5}}}}\right)}}\right)a^{2}\\&\approx 33.5385a^{2}\\V&={\frac {5}{12}}\left(11+5{\sqrt {5}}+6{\sqrt {5+2{\sqrt {5}}}}\right)a^{3}\\&\approx 16.936a^{3}\end{aligned}}}$
41	Elongated pentagonal gyrocupolarotunda		35	70	37	${\displaystyle C_{5v}}$ of order 10	${\displaystyle {\begin{aligned}A&={\frac {1}{4}}\left(60+{\sqrt {10\left(190+49{\sqrt {5}}+21{\sqrt {75+30{\sqrt {5}}}}\right)}}\right)a^{2}\\&\approx 33.5385a^{2}\\V&={\frac {5}{12}}\left(11+5{\sqrt {5}}+6{\sqrt {5+2{\sqrt {5}}}}\right)a^{3}\\&\approx 16.936a^{3}\end{aligned}}}$
42	Elongated pentagonal orthobirotunda		40	80	42	${\displaystyle D_{5h}}$ of order 20	${\displaystyle {\begin{aligned}A&=\left(10+{\sqrt {30\left(10+3{\sqrt {5}}+{\sqrt {75+30{\sqrt {5}}}}\right)}}\right)a^{2}\\&\approx 39.306a^{2}\\V&={\frac {1}{6}}\left(45+17{\sqrt {5}}+15{\sqrt {5+2{\sqrt {5}}}}\right)a^{3}\\&\approx 21.5297a^{3}\end{aligned}}}$
43	Elongated pentagonal gyrobirotunda		40	80	42	${\displaystyle D_{5d}}$ of order 20	${\displaystyle {\begin{aligned}A&=\left(10+{\sqrt {30\left(10+3{\sqrt {5}}+{\sqrt {75+30{\sqrt {5}}}}\right)}}\right)a^{2}\\&\approx 39.306a^{2}\\V&={\frac {1}{6}}\left(45+17{\sqrt {5}}+15{\sqrt {5+2{\sqrt {5}}}}\right)a^{3}\\&\approx 21.5297a^{3}\end{aligned}}}$
44	Gyroelongated triangular bicupola		18	42	26	${\displaystyle D_{3}}$ of order 6	${\displaystyle {\begin{aligned}A&=\left(6+5{\sqrt {3}}\right)a^{2}\\&\approx 14.6603a^{2}\\V&={\sqrt {2}}\left({\frac {5}{3}}+{\sqrt {1+{\sqrt {3}}}}\right)a^{3}\\&\approx 4.6946a^{3}\end{aligned}}}$
45	Gyroelongated square bicupola		24	56	34	${\displaystyle D_{4}}$ of order 8	${\displaystyle {\begin{aligned}A&=\left(10+6{\sqrt {3}}\right)a^{2}\\&\approx 20.3923a^{2}\\V&=\left(2+{\frac {4}{3}}{\sqrt {2}}+{\frac {2}{3}}{\sqrt {4+2{\sqrt {2}}+2{\sqrt {146+103{\sqrt {2}}}}}}\right)a^{3}\\&\approx 8.1536a^{3}\end{aligned}}}$
46	Gyroelongated pentagonal bicupola		30	70	42	${\displaystyle D_{5}}$ of order 10	${\displaystyle {\begin{aligned}A&={\frac {1}{2}}\left(20+15{\sqrt {3}}+{\sqrt {25+10{\sqrt {5}}}}\right)a^{2}\\&\approx 26.4313a^{2}\\V&=\left({\frac {5}{3}}+{\frac {4}{3}}{\sqrt {5}}+{\frac {5}{6}}{\sqrt {2{\sqrt {650+290{\sqrt {5}}}}-2{\sqrt {5}}-2}}\right)a^{3}\\&\approx 11.3974a^{3}\end{aligned}}}$
47	Gyroelongated pentagonal cupolarotunda		35	80	47	${\displaystyle C_{5}}$ of order 5	${\displaystyle {\begin{aligned}A&={\frac {1}{4}}\left(20+35{\sqrt {3}}+7{\sqrt {25+10{\sqrt {5}}}}\right)a^{2}\\&\approx 32.1988a^{2}\\V&=\left({\frac {55}{12}}+{\frac {25}{12}}{\sqrt {5}}+{\frac {5}{6}}{\sqrt {2{\sqrt {650+290{\sqrt {5}}}}-2{\sqrt {5}}-2}}\right)a^{3}\\&\approx 15.9911a^{3}\end{aligned}}}$
48	Gyroelongated pentagonal birotunda		40	90	52	${\displaystyle D_{5}}$ of order 10	${\displaystyle {\begin{aligned}A&=\left(10{\sqrt {3}}+3{\sqrt {25+10{\sqrt {5}}}}\right)a^{2}\\&\approx 37.9662a^{2}\\V&=\left({\frac {45}{6}}+{\frac {17}{6}}{\sqrt {5}}+{\frac {5}{6}}{\sqrt {2{\sqrt {650+290{\sqrt {5}}}}-2{\sqrt {5}}-2}}\right)a^{3}\\&\approx 20.5848a^{3}\end{aligned}}}$
49	Augmented triangular prism		7	13	8	${\displaystyle C_{2v}}$ of order 4	${\displaystyle {\begin{aligned}A&={\frac {1}{2}}(4+3{\sqrt {3}})a^{2}\\&\approx 4.5981a^{2}\\V&={\frac {1}{12}}(2{\sqrt {2}}+3{\sqrt {3}})a^{3}\\&\approx 0.6687a^{3}\end{aligned}}}$
50	Biaugmented triangular prism		8	17	11	${\displaystyle C_{2v}}$ of order 4	${\displaystyle {\begin{aligned}A&={\frac {1}{2}}(2+5{\sqrt {3}})a^{2}\\&\approx 5.3301a^{2}\\V&={\sqrt {{\frac {59}{144}}+{\frac {1}{\sqrt {6}}}}}a^{3}\\&\approx 0.9044a^{3}\end{aligned}}}$
51	Triaugmented triangular prism		9	21	14	${\displaystyle D_{3h}}$ of order 12	${\displaystyle {\begin{aligned}A&={\frac {7{\sqrt {3}}}{2}}a^{2}\\&\approx 6.0622a^{2}\\V&={\frac {2{\sqrt {2}}+{\sqrt {3}}}{4}}a^{3}\\&\approx 1.1401a^{3}\end{aligned}}}$
52	Augmented pentagonal prism		11	19	10	${\displaystyle C_{2v}}$ of order 4	${\displaystyle {\begin{aligned}A&={\frac {1}{2}}\left(8+2{\sqrt {3}}+{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 9.173a^{2}\\V&={\frac {1}{12}}{\sqrt {233+90{\sqrt {5}}+12{\sqrt {50+20{\sqrt {5}}}}}}a^{3}\\&\approx 1.9562a^{3}\end{aligned}}}$
53	Biaugmented pentagonal prism		12	23	13	${\displaystyle C_{2v}}$ of order 4	${\displaystyle {\begin{aligned}A&={\frac {1}{2}}a^{2}\left(6+4{\sqrt {3}}+{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)\\&\approx 9.9051a^{2}\\V&={\frac {1}{12}}a^{3}{\sqrt {257+90{\sqrt {5}}+24{\sqrt {50+20{\sqrt {5}}}}}}\\&\approx 2.1919a^{3}\end{aligned}}}$
54	Augmented hexagonal prism		13	22	11	${\displaystyle C_{2v}}$ of order 4	${\displaystyle {\begin{aligned}A&=(5+4{\sqrt {3}})a^{2}\\&\approx 11.9282a^{2}\\V&={\frac {1}{6}}\left({\sqrt {2}}+9{\sqrt {3}}\right)a^{3}\\&\approx 2.8338a^{3}\end{aligned}}}$
55	Parabiaugmented hexagonal prism		14	26	14	${\displaystyle D_{2h}}$ of order 8	${\displaystyle {\begin{aligned}A&=(4+5{\sqrt {3}})a^{2}\\&\approx 12.6603a^{2}\\V&={\frac {1}{6}}\left(2{\sqrt {2}}+9{\sqrt {3}}\right)a^{3}\\&\approx 3.0695a^{3}\end{aligned}}}$
56	Metabiaugmented hexagonal prism		14	26	14	${\displaystyle C_{2v}}$ of order 4	${\displaystyle {\begin{aligned}A&=(4+5{\sqrt {3}})a^{2}\\&\approx 12.6603a^{2}\\V&={\frac {1}{6}}\left(2{\sqrt {2}}+9{\sqrt {3}}\right)a^{3}\\&\approx 3.0695a^{3}\end{aligned}}}$
57	Triaugmented hexagonal prism		15	30	17	${\displaystyle D_{3h}}$ of order 12	${\displaystyle {\begin{aligned}A&=3\left(1+2{\sqrt {3}}\right)a^{2}\\&\approx 13.3923a^{2}\\V&=\left({\frac {1}{\sqrt {2}}}+{\frac {3{\sqrt {3}}}{2}}\right)a^{3}\\&\approx 3.3052a^{3}\end{aligned}}}$
58	Augmented dodecahedron		21	35	16	${\displaystyle C_{5v}}$ of order 10	${\displaystyle {\begin{aligned}A&={\frac {1}{4}}\left(5{\sqrt {3}}+11{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 21.0903a^{2}\\V&={\frac {1}{24}}\left(95+43{\sqrt {5}}\right)a^{3}\\&\approx 7.9646a^{3}\end{aligned}}}$
59	Parabiaugmented dodecahedron		22	40	20	${\displaystyle D_{5d}}$ of order 20	${\displaystyle {\begin{aligned}A&={\frac {5}{2}}\left({\sqrt {3}}+{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 21.5349a^{2}\\V&={\frac {1}{6}}\left(25+11{\sqrt {5}}\right)a^{3}\\&\approx 8.2661a^{3}\end{aligned}}}$
60	Metabiaugmented dodecahedron		22	40	20	${\displaystyle C_{2v}}$ of order 4	${\displaystyle {\begin{aligned}A&={\frac {5}{2}}\left({\sqrt {3}}+{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 21.5349a^{2}\\V&={\frac {1}{6}}\left(25+11{\sqrt {5}}\right)a^{3}\\&\approx 8.2661a^{3}\end{aligned}}}$
61	Triaugmented dodecahedron		23	45	24	${\displaystyle C_{3v}}$ of order 6	${\displaystyle {\begin{aligned}A&={\frac {3}{4}}\left(5{\sqrt {3}}+3{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 21.9795a^{2}\\V&={\frac {5}{8}}\left(7+3{\sqrt {5}}\right)a^{3}\\&\approx 8.5676a^{3}\end{aligned}}}$
62	Metabidiminished icosahedron		10	20	12	${\displaystyle C_{2v}}$ of order 4	${\displaystyle {\begin{aligned}A&={\frac {1}{2}}\left(5{\sqrt {3}}+{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 7.7711a^{2}\\V&={\frac {1}{6}}\left(5+2{\sqrt {5}}\right)a^{3}\\&\approx 1.5787a^{3}\end{aligned}}}$
63	Tridiminished icosahedron		9	15	8	${\displaystyle C_{3v}}$ of order 6	${\displaystyle {\begin{aligned}A&={\frac {1}{4}}\left(5{\sqrt {3}}+3{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{3}\\&\approx 7.3265a^{3}\\V&=\left({\frac {5}{8}}+{\frac {7{\sqrt {5}}}{24}}\right)a^{3}\\&\approx 1.2772a^{3}\end{aligned}}}$
64	Augmented tridiminished icosahedron		10	18	10	${\displaystyle C_{3v}}$ of order 6	${\displaystyle {\begin{aligned}A&={\frac {1}{4}}\left(7{\sqrt {3}}+3{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 8.1925a^{2}\\V&={\frac {1}{24}}\left(15+2{\sqrt {2}}+7{\sqrt {5}}\right)a^{3}\\&\approx 1.395a^{3}\end{aligned}}}$
65	Augmented truncated tetrahedron		15	27	14	${\displaystyle C_{3v}}$ of order 6	${\displaystyle {\begin{aligned}A&={\frac {1}{2}}\left(6+13{\sqrt {3}}\right)a^{2}\\&\approx 14.2583a^{2}\\V&={\frac {11}{2{\sqrt {2}}}}a^{3}\\&\approx 3.8891a^{3}\end{aligned}}}$
66	Augmented truncated cube		28	48	22	${\displaystyle C_{4v}}$ of order 8	${\displaystyle {\begin{aligned}A&=(15+10{\sqrt {2}}+3{\sqrt {3}})a^{2}\\&\approx 34.3383a^{2}\\V&=\left(8+{\frac {16{\sqrt {2}}}{3}}\right)a^{3}\\&\approx 15.5425a^{3}\end{aligned}}}$
67	Biaugmented truncated cube		32	60	30	${\displaystyle D_{4h}}$ of order 16	${\displaystyle {\begin{aligned}A&=2\left(9+4{\sqrt {2}}+2{\sqrt {3}}\right)a^{2}\\&\approx 36.2419a^{2}\\V&=(9+6{\sqrt {2}})a^{3}\\&\approx 17.4853a^{3}\end{aligned}}}$
68	Augmented truncated dodecahedron		65	105	42	${\displaystyle C_{5v}}$ of order 10	${\displaystyle {\begin{aligned}A&={\frac {1}{4}}\left(20+25{\sqrt {3}}+110{\sqrt {5+2{\sqrt {5}}}}+{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 102.1821a^{2}\\V&=\left({\frac {505}{12}}+{\frac {81{\sqrt {5}}}{4}}\right)a^{3}\\&\approx 87.3637a^{3}\end{aligned}}}$
69	Parabiaugmented truncated dodecahedron		70	120	52	${\displaystyle D_{5d}}$ of order 20	${\displaystyle {\begin{aligned}A&={\frac {1}{2}}\left(20+15{\sqrt {3}}+50{\sqrt {5+2{\sqrt {5}}}}+{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 103.3734a^{2}\\V&={\frac {1}{12}}\left(515+251{\sqrt {5}}\right)a^{3}\\&\approx 89.6878a^{3}\end{aligned}}}$
70	Metabiaugmented truncated dodecahedron		70	120	52	${\displaystyle C_{2v}}$ of order 4	${\displaystyle {\begin{aligned}A&={\frac {1}{2}}\left(20+15{\sqrt {3}}+50{\sqrt {5+2{\sqrt {5}}}}+{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 103.3734a^{2}\\V&={\frac {1}{12}}\left(515+251{\sqrt {5}}\right)a^{3}\\&\approx 89.6878a^{3}\end{aligned}}}$
71	Triaugmented truncated dodecahedron		75	135	62	${\displaystyle C_{3v}}$ of order 6	${\displaystyle {\begin{aligned}A&={\frac {1}{4}}\left(60+35{\sqrt {3}}+90{\sqrt {5+2{\sqrt {5}}}}+3{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 104.5648a^{2}\\V&={\frac {7}{12}}\left(75+37{\sqrt {5}}\right)a^{3}\\&\approx 92.0118a^{3}\end{aligned}}}$
72	Gyrate rhombicosidodecahedron		60	120	62	${\displaystyle C_{5v}}$ of order 10	${\displaystyle {\begin{aligned}A&=\left(30+5{\sqrt {3}}+3{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 59.306a^{2}\\V&=\left(20+{\frac {29{\sqrt {5}}}{3}}\right)a^{3}\\&\approx 41.6153a^{3}\end{aligned}}}$
73	Parabigyrate rhombicosidodecahedron		60	120	62	${\displaystyle D_{5d}}$ of order 20	${\displaystyle {\begin{aligned}A&=\left(30+5{\sqrt {3}}+3{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 59.306a^{2}\\V&=\left(20+{\frac {29{\sqrt {5}}}{3}}\right)a^{3}\\&\approx 41.6153a^{3}\end{aligned}}}$
74	Metabigyrate rhombicosidodecahedron		60	120	62	${\displaystyle C_{2v}}$ of order 4	${\displaystyle {\begin{aligned}A&=\left(30+5{\sqrt {3}}+3{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 59.306a^{2}\\V&=\left(20+{\frac {29{\sqrt {5}}}{3}}\right)a^{3}\\&\approx 41.6153a^{3}\end{aligned}}}$
75	Trigyrate rhombicosidodecahedron		60	120	62	${\displaystyle C_{3v}}$ of order 6	${\displaystyle {\begin{aligned}A&=\left(30+5{\sqrt {3}}+3{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 59.306a^{2}\\V&=\left(20+{\frac {29{\sqrt {5}}}{3}}\right)a^{3}\\&\approx 41.6153a^{3}\end{aligned}}}$
76	Diminished rhombicosidodecahedron		55	105	52	${\displaystyle C_{5v}}$ of order 10	${\displaystyle {\begin{aligned}A&={\frac {1}{4}}\left(100+15{\sqrt {3}}+10{\sqrt {5+2{\sqrt {5}}}}+11{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 58.1147a^{2}\\V&=\left({\frac {115}{6}}+9{\sqrt {5}}\right)a^{3}\\&\approx 39.2913a^{3}\end{aligned}}}$
77	Paragyrate diminished rhombicosidodecahedron		55	105	52	${\displaystyle C_{5v}}$ of order 10	${\displaystyle {\begin{aligned}A&={\frac {1}{4}}\left(100+15{\sqrt {3}}+10{\sqrt {5+2{\sqrt {5}}}}+11{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 58.1147a^{2}\\V&=\left({\frac {115}{6}}+9{\sqrt {5}}\right)a^{3}\\&\approx 39.2913a^{3}\end{aligned}}}$
78	Metagyrate diminished rhombicosidodecahedron		55	105	52	${\displaystyle C_{s}}$ of order 2	${\displaystyle {\begin{aligned}A&={\frac {1}{4}}\left(100+15{\sqrt {3}}+10{\sqrt {5+2{\sqrt {5}}}}+11{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 58.1147a^{2}\\V&=\left({\frac {115}{6}}+9{\sqrt {5}}\right)a^{3}\\&\approx 39.2913a^{3}\end{aligned}}}$
79	Bigyrate diminished rhombicosidodecahedron		55	105	52	${\displaystyle C_{s}}$ of order 2	${\displaystyle {\begin{aligned}A&={\frac {1}{4}}\left(100+15{\sqrt {3}}+10{\sqrt {5+2{\sqrt {5}}}}+11{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 58.1147a^{2}\\V&=\left({\frac {115}{6}}+9{\sqrt {5}}\right)a^{3}\\&\approx 39.2913a^{3}\end{aligned}}}$
80	Parabidiminished rhombicosidodecahedron		50	90	42	${\displaystyle D_{5d}}$ of order 20	${\displaystyle {\begin{aligned}A&={\frac {5}{2}}\left(8+{\sqrt {3}}+2{\sqrt {5+2{\sqrt {5}}}}+{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 56.9233a^{2}\\V&={\frac {5}{3}}\left(11+5{\sqrt {5}}\right)a^{3}\\&\approx 36.9672a^{3}\end{aligned}}}$
81	Metabidiminished rhombicosidodecahedron		50	90	42	${\displaystyle C_{2v}}$ of order 4	${\displaystyle {\begin{aligned}A&={\frac {5}{2}}\left(8+{\sqrt {3}}+2{\sqrt {5+2{\sqrt {5}}}}+{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 56.9233a^{2}\\V&={\frac {5}{3}}\left(11+5{\sqrt {5}}\right)a^{3}\\&\approx 36.9672a^{3}\end{aligned}}}$
82	Gyrate bidiminished rhombicosidodecahedron		50	90	42	${\displaystyle C_{s}}$ of order 2	${\displaystyle {\begin{aligned}A&={\frac {5}{2}}\left(8+{\sqrt {3}}+2{\sqrt {5+2{\sqrt {5}}}}+{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 56.9233a^{2}\\V&={\frac {5}{3}}\left(11+5{\sqrt {5}}\right)a^{3}\\&\approx 36.9672a^{3}\end{aligned}}}$
83	Tridiminished rhombicosidodecahedron		45	75	32	${\displaystyle C_{3v}}$ of order 6	${\displaystyle {\begin{aligned}A&={\frac {1}{4}}\left(60+5{\sqrt {3}}+30{\sqrt {5+2{\sqrt {5}}}}+9{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 55.732a^{2}\\V&=\left({\frac {35}{2}}+{\frac {23{\sqrt {5}}}{3}}\right)a^{3}\\&\approx 34.6432a^{3}\end{aligned}}}$
84	Snub disphenoid		8	18	12	${\displaystyle D_{2d}}$ of order 8	${\displaystyle {\begin{aligned}A&=3{\sqrt {3}}a^{2}\\&\approx 5.1962a^{2}\\V&\approx 0.8595a^{3}\end{aligned}}}$
85	Snub square antiprism		16	40	26	${\displaystyle D_{4d}}$ of order 16	${\displaystyle {\begin{aligned}A&=2\left(1+3{\sqrt {3}}\right)a^{2}\\&\approx 12.3923a^{2}\\V&\approx 3.6012a^{3}\end{aligned}}}$
86	Sphenocorona		10	22	14	${\displaystyle C_{2v}}$ of order 4	${\displaystyle {\begin{aligned}A&=(2+3{\sqrt {3}})a^{2}\\&\approx 7.1962a^{2}\\V&={\frac {1}{2}}a^{3}{\sqrt {1+3{\sqrt {\frac {3}{2}}}+{\sqrt {13+3{\sqrt {6}}}}}}\\&\approx 1.5154a^{3}\end{aligned}}}$
87	Augmented sphenocorona		11	26	17	${\displaystyle C_{s}}$ of order 2	${\displaystyle {\begin{aligned}A&=(1+4{\sqrt {3}})a^{2}\\&\approx 7.9282a^{2}\\V&={\frac {1}{2}}a^{3}{\sqrt {1+3{\sqrt {\frac {3}{2}}}+{\sqrt {13+3{\sqrt {6}}}}}}+{\frac {1}{3{\sqrt {2}}}}\\&\approx 1.7511a^{3}\end{aligned}}}$
88	Sphenomegacorona		12	28	18	${\displaystyle C_{2v}}$ of order 4	${\displaystyle {\begin{aligned}A&=2\left(1+2{\sqrt {3}}\right)a^{2}\\&\approx 8.9282a^{2}\\V&\approx 1.9481a^{3}\end{aligned}}}$
89	Hebesphenomegacorona		14	33	21	${\displaystyle C_{2v}}$ of order 4	${\displaystyle {\begin{aligned}A&={\frac {3}{2}}\left(2+3{\sqrt {3}}\right)a^{2}\\&\approx 10.7942a^{2}\\V&\approx 2.9129a^{3}\end{aligned}}}$
90	Disphenocingulum		16	38	24	${\displaystyle D_{2d}}$ of order 8	${\displaystyle {\begin{aligned}A&=(4+5{\sqrt {3}})a^{2}\\&\approx 12.6603a^{2}\\V&\approx 3.7776a^{3}\end{aligned}}}$
91	Bilunabirotunda		14	26	14	${\displaystyle D_{2h}}$ of order 8	${\displaystyle {\begin{aligned}A&=\left(2+2{\sqrt {3}}+{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 12.346a^{2}\\V&={\frac {1}{12}}\left(17+9{\sqrt {5}}\right)a^{3}\\&\approx 3.0937a^{3}\end{aligned}}}$
92	Triangular hebespenorotunda		18	36	20	${\displaystyle C_{3v}}$ of order 6	${\displaystyle {\begin{aligned}A&={\frac {1}{4}}\left(12+19{\sqrt {3}}+3{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 16.3887a^{2}\\V&=\left({\frac {5}{2}}+{\frac {7{\sqrt {5}}}{6}}\right)a^{3}\\&\approx 5.1087a^{3}\end{aligned}}}$