List of Johnson solids

In geometry, a Johnson solid is a strictly convex polyhedron, each face of which is a regular polygon, but which is not uniform, i.e., not a Platonic solid, Archimedean solid, prism or antiprism. In 1966, Norman Johnson published a list which included all 92 solids, and gave them their names and numbers. He did not prove that there were only 92, but he did conjecture that there were no others. Victor Zalgaller proved in 1969 that Johnson's list was complete.

Other polyhedra can be constructed that have only approximately regular planar polygon faces, and are informally called near-miss Johnson solids; there can be no definitive count of them.

The various sections that follow have tables listing all 92 Johnson solids, and values for some of their most important properties. Each table allows sorting by column so that numerical values, or the names of the solids, can be sorted in order.

Vertices, edges, faces, and symmetry

J_n	Solid name	V	E	F	F₃	F₄	F₅	F₆	F₈	F₁₀	Symmetry group	Order
1	Square pyramid	5	8	5	4	1					C_4v, [4], (*44)	8
2	Pentagonal pyramid	6	10	6	5		1				C_5v, [5], (*55)	10
3	Triangular cupola	9	15	8	4	3		1			C_3v, [3], (*33)	6
4	Square cupola	12	20	10	4	5			1		C_4v, [4], (*44)	8
5	Pentagonal cupola	15	25	12	5	5	1			1	C_5v, [5], (*55)	10
6	Pentagonal rotunda	20	35	17	10		6			1	C_5v, [5], (*55)	10
7	Elongated triangular pyramid	7	12	7	4	3					C_3v, [3], (*33)	6
8	Elongated square pyramid	9	16	9	4	5					C_4v, [4], (*44)	8
9	Elongated pentagonal pyramid	11	20	11	5	5	1				C_5v, [5], (*55)	10
10	Gyroelongated square pyramid	9	20	13	12	1					C_4v, [4], (*44)	8
11	Gyroelongated pentagonal pyramid	11	25	16	15		1				C_5v, [5], (*55)	10
12	Triangular bipyramid	5	9	6	6						D_3h, [3,2], (*223)	12
13	Pentagonal bipyramid	7	15	10	10						D_5h, [5,2], (*225)	20
14	Elongated triangular bipyramid	8	15	9	6	3					D_3h, [3,2], (*223)	12
15	Elongated square bipyramid	10	20	12	8	4					D_4h, [4,2], (*224)	16
16	Elongated pentagonal bipyramid	12	25	15	10	5					D_5h, [5,2], (*225)	20
17	Gyroelongated square bipyramid	10	24	16	16						D_4d, [2⁺,8], (2*4)	16
18	Elongated triangular cupola	15	27	14	4	9		1			C_3v, [3], (*33)	6
19	Elongated square cupola	20	36	18	4	13			1		C_4v, [4], (*44)	8
20	Elongated pentagonal cupola	25	45	22	5	15	1			1	C_5v, [5], (*55)	10
21	Elongated pentagonal rotunda	30	55	27	10	10	6			1	C_5v, [5], (*55)	10
22	Gyroelongated triangular cupola	15	33	20	16	3		1			C_3v, [3], (*33)	6
23	Gyroelongated square cupola	20	44	26	20	5			1		C_4v, [4], (*44)	8
24	Gyroelongated pentagonal cupola	25	55	32	25	5	1			1	C_5v, [5], (*55)	10
25	Gyroelongated pentagonal rotunda	30	65	37	30		6			1	C_5v, [5], (*55)	10
26	Gyrobifastigium	8	14	8	4	4					D_2d, [2⁺,4], (2*2)	8
27	Triangular orthobicupola	12	24	14	8	6					D_3h, [3,2], (*223)	12
28	Square orthobicupola	16	32	18	8	10					D_4h, [4,2], (*224)	16
29	Square gyrobicupola	16	32	18	8	10					D_4d, [2⁺,8], (2*4)	16
30	Pentagonal orthobicupola	20	40	22	10	10	2				D_5h, [5,2], (*225)	20
31	Pentagonal gyrobicupola	20	40	22	10	10	2				D_5d, [2⁺,10], (2*5)	20
32	Pentagonal orthocupolarotunda	25	50	27	15	5	7				C_5v, [5], (*55)	10
33	Pentagonal gyrocupolarotunda	25	50	27	15	5	7				C_5v, [5], (*55)	10
34	Pentagonal orthobirotunda	30	60	32	20		12				D_5h, [5,2], (*225)	20
35	Elongated triangular orthobicupola	18	36	20	8	12					D_3h, [3,2], (*223)	12
36	Elongated triangular gyrobicupola	18	36	20	8	12					D_3d, [2⁺,6], (2*3)	12
37	Elongated square gyrobicupola	24	48	26	8	18					D_4d, [2⁺,8], (2*4)	16
38	Elongated pentagonal orthobicupola	30	60	32	10	20	2				D_5h, [5,2], (*225)	20
39	Elongated pentagonal gyrobicupola	30	60	32	10	20	2				D_5d, [2⁺,10], (2*5)	20
40	Elongated pentagonal orthocupolarotunda	35	70	37	15	15	7				C_5v, [5], (*55)	10
41	Elongated pentagonal gyrocupolarotunda	35	70	37	15	15	7				C_5v, [5], (*55)	10
42	Elongated pentagonal orthobirotunda	40	80	42	20	10	12				D_5h, [5,2], (*225)	20
43	Elongated pentagonal gyrobirotunda	40	80	42	20	10	12				D_5d, [2⁺,10], (2*5)	20
44	Gyroelongated triangular bicupola	18	42	26	20	6					D₃, [3,2]⁺,(223)	6
45	Gyroelongated square bicupola	24	56	34	24	10					D₄, [4,2]⁺, (224)	8
46	Gyroelongated pentagonal bicupola	30	70	42	30	10	2				D₅, [5,2]⁺, (225)	10
47	Gyroelongated pentagonal cupolarotunda	35	80	47	35	5	7				C₅, [5]⁺, (55)	5
48	Gyroelongated pentagonal birotunda	40	90	52	40		12				D₅, [5,2]⁺, (225)	10
49	Augmented triangular prism	7	13	8	6	2					C_2v, [2], (*22)	4
50	Biaugmented triangular prism	8	17	11	10	1					C_2v, [2], (*22)	4
51	Triaugmented triangular prism	9	21	14	14						D_3h, [3,2], (*223)	12
52	Augmented pentagonal prism	11	19	10	4	4	2				C_2v, [2], (*22)	4
53	Biaugmented pentagonal prism	12	23	13	8	3	2				C_2v, [2], (*22)	4
54	Augmented hexagonal prism	13	22	11	4	5		2			C_2v, [2], (*22)	4
55	Parabiaugmented hexagonal prism	14	26	14	8	4		2			D_2h, [2,2], (*222)	8
56	Metabiaugmented hexagonal prism	14	26	14	8	4		2			C_2v, [2], (*22)	4
57	Triaugmented hexagonal prism	15	30	17	12	3		2			D_3h, [3,2], (*223)	12
58	Augmented dodecahedron	21	35	16	5		11				C_5v, [5], (*55)	10
59	Parabiaugmented dodecahedron	22	40	20	10		10				D_5d, [2⁺,10], (2*5)	20
60	Metabiaugmented dodecahedron	22	40	20	10		10				C_2v, [2], (*22)	4
61	Triaugmented dodecahedron	23	45	24	15		9				C_3v, [3], (*33)	6
62	Metabidiminished icosahedron	10	20	12	10		2				C_2v, [2], (*22)	4
63	Tridiminished icosahedron	9	15	8	5		3				C_3v, [3], (*33)	6
64	Augmented tridiminished icosahedron	10	18	10	7		3				C_3v, [3], (*33)	6
65	Augmented truncated tetrahedron	15	27	14	8	3		3			C_3v, [3], (*33)	6
66	Augmented truncated cube	28	48	22	12	5			5		C_4v, [4], (*44)	8
67	Biaugmented truncated cube	32	60	30	16	10			4		D_4h, [4,2], (*224)	16
68	Augmented truncated dodecahedron	65	105	42	25	5	1			11	C_5v, [5], (*55)	10
69	Parabiaugmented truncated dodecahedron	70	120	52	30	10	2			10	D_5d, [2⁺,10], (2*5)	20
70	Metabiaugmented truncated dodecahedron	70	120	52	30	10	2			10	C_2v, [2], (*22)	4
71	Triaugmented truncated dodecahedron	75	135	62	35	15	3			9	C_3v, [3], (*33)	6
72	Gyrate rhombicosidodecahedron	60	120	62	20	30	12				C_5v, [5], (*55)	10
73	Parabigyrate rhombicosidodecahedron	60	120	62	20	30	12				D_5d, [2⁺,10], (2*5)	20
74	Metabigyrate rhombicosidodecahedron	60	120	62	20	30	12				C_2v, [2], (*22)	4
75	Trigyrate rhombicosidodecahedron	60	120	62	20	30	12				C_3v, [3], (*33)	6
76	Diminished rhombicosidodecahedron	55	105	52	15	25	11			1	C_5v, [5], (*55)	10
77	Paragyrate diminished rhombicosidodecahedron	55	105	52	15	25	11			1	C_5v, [5], (*55)	10
78	Metagyrate diminished rhombicosidodecahedron	55	105	52	15	25	11			1	C_s, [ ], (*11)	2
79	Bigyrate diminished rhombicosidodecahedron	55	105	52	15	25	11			1	C_s, [ ], (*11)	2
80	Parabidiminished rhombicosidodecahedron	50	90	42	10	20	10			2	D_5d, [2⁺,10], (2*5)	20
81	Metabidiminished rhombicosidodecahedron	50	90	42	10	20	10			2	C_2v, [2], (*22)	4
82	Gyrate bidiminished rhombicosidodecahedron	50	90	42	10	20	10			2	C_s, [ ], (*11)	2
83	Tridiminished rhombicosidodecahedron	45	75	32	5	15	9			3	C_3v, [3], (*33)	6
84	Snub disphenoid	8	18	12	12						D_2d, [2⁺,4], (2*2)	8
85	Snub square antiprism	16	40	26	24	2					D_4d, [2⁺,8], (2*4)	16
86	Sphenocorona	10	22	14	12	2					C_2v, [2], (*22)	4
87	Augmented sphenocorona	11	26	17	16	1					C_s, [ ], (*11)	2
88	Sphenomegacorona	12	28	18	16	2					C_2v, [2], (*22)	4
89	Hebesphenomegacorona	14	33	21	18	3					C_2v, [2], (*22)	4
90	Disphenocingulum	16	38	24	20	4					D_2d, [2⁺,4], (2*2)	8
91	Bilunabirotunda	14	26	14	8	2	4				D_2h, [2,2], (*222)	8
92	Triangular hebesphenorotunda	18	36	20	13	3	3	1			C_3v, [3], (*33)	6

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Legend:

J_n – Johnson solid number
Net – Flattened (unfolded) image
V – Number of vertices
E – Number of edges
F – Number of faces (total)
F₃–F₁₀ – Number of faces by side counts

The square pyramid $J 1$ has the fewest vertices (5), the fewest edges (8), and the fewest faces (5).

The triaugmented truncated dodecahedron $J 71$ has the most vertices (75) and the most edges (135). It also has the highest number of faces (62), along with the gyrate rhombicosidodecahedron $J 72$ , the parabigyrate rhombicosidodecahedron $J 73$ , the metabigyrate rhombicosidodecahedron $J 74$ , and the trigyrate rhombicosidodecahedron $J 75$ .

Surface area

Since all faces of Johnson solids are regular polygons with 3, 4, 5, 6, 8, or 10 sides, and since all these polygons have the same edge length $a$ , the surface area of a Johnson solid can be calculated as

A=\sum _{n=3,4,5,6,8,10}F_{n}A_{n}

where the $F n$ are the polygonal face counts in the previous table and

A_{n}=\left({\frac {n}{4}}\cot {\frac {\pi }{n}}\right)a^{2}

is the area of a regular polygon with $n$ sides of length $a$ . In terms of radicals, one has

A_{3}={\frac {1}{4}}{\sqrt {3}}\,a^{2}

A_{4}=a^{2}

A_{5}={\frac {1}{4}}{\sqrt {5(5+2{\sqrt {5}})}}\,a^{2}

A_{6}={\frac {3}{2}}{\sqrt {3}}\,a^{2}

A_{8}=2(1+{\sqrt {2}})\,a^{2}

A_{10}={\frac {5}{2}}{\sqrt {5+2{\sqrt {5}}}}\,a^{2},

resulting in the following table of surface areas.

J_n	Solid name	A/a² (approx.)	A/a² (exact)
1	Square pyramid	2.732050808	$1+{\sqrt {3}}$
2	Pentagonal pyramid	3.885540910	${\frac {1}{4}}\left(5{\sqrt {3}}+{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)$
3	Triangular cupola	7.330127019	${\frac {1}{2}}\left(6+5{\sqrt {3}}\right)$
4	Square cupola	11.560477932	$7+2{\sqrt {2}}+{\sqrt {3}}$
5	Pentagonal cupola	16.579749753	${\frac {1}{4}}\left(20+5{\sqrt {3}}+10{\sqrt {5+2{\sqrt {5}}}}+{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)$
6	Pentagonal rotunda	22.347200265	${\frac {1}{2}}\left(5{\sqrt {3}}+5{\sqrt {5+2{\sqrt {5}}}}+3{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)$
7	Elongated triangular pyramid	4.732050808	$3+{\sqrt {3}}$
8	Elongated square pyramid	6.732050808	$5+{\sqrt {3}}$
9	Elongated pentagonal pyramid	8.885540910	${\frac {1}{4}}\left(20+5{\sqrt {3}}+{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)$
10	Gyroelongated square pyramid	6.196152423	$1+3{\sqrt {3}}$
11	Gyroelongated pentagonal pyramid	8.215667929	${\frac {1}{4}}\left(15{\sqrt {3}}+{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)$
12	Triangular dipyramid	2.598076211	${\frac {3{\sqrt {3}}}{2}}$
13	Pentagonal dipyramid	4.330127019	${\frac {5{\sqrt {3}}}{2}}$
14	Elongated triangular dipyramid	5.598076211	${\frac {3}{2}}\left(2+{\sqrt {3}}\right)$
15	Elongated square dipyramid	7.464101615	$2\left(2+{\sqrt {3}}\right)$
16	Elongated pentagonal dipyramid	9.330127019	${\frac {5}{2}}\left(2+{\sqrt {3}}\right)$
17	Gyroelongated square dipyramid	6.928203230	$4{\sqrt {3}}$
18	Elongated triangular cupola	13.330127019	${\frac {1}{2}}\left(18+5{\sqrt {3}}\right)$
19	Elongated square cupola	19.560477932	$15+2{\sqrt {2}}+{\sqrt {3}}$
20	Elongated pentagonal cupola	26.579749753	${\frac {1}{4}}\left(60+5{\sqrt {3}}+10{\sqrt {5+2{\sqrt {5}}}}+{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)$
21	Elongated pentagonal rotunda	32.347200265	${\frac {1}{2}}\left(20+5{\sqrt {3}}+5{\sqrt {5+2{\sqrt {5}}}}+3{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)$
22	Gyroelongated triangular cupola	12.526279442	${\frac {1}{2}}\left(6+11{\sqrt {3}}\right)$
23	Gyroelongated square cupola	18.488681163	$7+2{\sqrt {2}}+5{\sqrt {3}}$
24	Gyroelongated pentagonal cupola	25.240003791	${\frac {1}{4}}\left(20+25{\sqrt {3}}+10{\sqrt {5+2{\sqrt {5}}}}+{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)$
25	Gyroelongated pentagonal rotunda	31.007454303	${\frac {1}{2}}\left(15{\sqrt {3}}+5{\sqrt {5+2{\sqrt {5}}}}+3{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)$
26	Gyrobifastigium	5.732050808	$4+{\sqrt {3}}$
27	Triangular orthobicupola	9.464101615	$2\left(3+{\sqrt {3}}\right)$
28	Square orthobicupola	13.464101615	$2\left(5+{\sqrt {3}}\right)$
29	Square gyrobicupola	13.464101615	$2\left(5+{\sqrt {3}}\right)$
30	Pentagonal orthobicupola	17.771081820	${\frac {1}{2}}\left(20+5{\sqrt {3}}+{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)$
31	Pentagonal gyrobicupola	17.771081820	${\frac {1}{2}}\left(20+5{\sqrt {3}}+{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)$
32	Pentagonal orthocupolarotunda	23.538532333	${\frac {1}{4}}\left(20+15{\sqrt {3}}+7{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)$
33	Pentagonal gyrocupolarotunda	23.538532333	${\frac {1}{4}}\left(20+15{\sqrt {3}}+7{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)$
34	Pentagonal orthobirotunda	29.305982845	$5{\sqrt {3}}+3{\sqrt {5\left(5+2{\sqrt {5}}\right)}}$
35	Elongated triangular orthobicupola	15.464101615	$2\left(6+{\sqrt {3}}\right)$
36	Elongated triangular gyrobicupola	15.464101615	$2\left(6+{\sqrt {3}}\right)$
37	Elongated square gyrobicupola	21.464101615	$2\left(9+{\sqrt {3}}\right)$
38	Elongated pentagonal orthobicupola	27.771081820	${\frac {1}{2}}\left(40+5{\sqrt {3}}+{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)$
39	Elongated pentagonal gyrobicupola	27.771081820	${\frac {1}{2}}\left(40+5{\sqrt {3}}+{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)$
40	Elongated pentagonal orthocupolarotunda	33.538532333	${\frac {1}{4}}\left(60+15{\sqrt {3}}+7{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)$
41	Elongated pentagonal gyrocupolarotunda	33.538532333	${\frac {1}{4}}\left(60+15{\sqrt {3}}+7{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)$
42	Elongated pentagonal orthobirotunda	39.305982845	$10+5{\sqrt {3}}+3{\sqrt {5\left(5+2{\sqrt {5}}\right)}}$
43	Elongated pentagonal gyrobirotunda	39.305982845	$10+5{\sqrt {3}}+3{\sqrt {5\left(5+2{\sqrt {5}}\right)}}$
44	Gyroelongated triangular bicupola	14.660254038	$6+5{\sqrt {3}}$
45	Gyroelongated square bicupola	20.392304845	$2\left(5+3{\sqrt {3}}\right)$
46	Gyroelongated pentagonal bicupola	26.431335858	${\frac {1}{2}}\left(20+15{\sqrt {3}}+{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)$
47	Gyroelongated pentagonal cupolarotunda	32.198786370	${\frac {1}{4}}\left(20+35{\sqrt {3}}+7{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)$
48	Gyroelongated pentagonal birotunda	37.966236883	$10{\sqrt {3}}+3{\sqrt {5\left(5+2{\sqrt {5}}\right)}}$
49	Augmented triangular prism	4.598076211	${\frac {1}{2}}\left(4+3{\sqrt {3}}\right)$
50	Biaugmented triangular prism	5.330127019	${\frac {1}{2}}\left(2+5{\sqrt {3}}\right)$
51	Triaugmented triangular prism	6.062177826	${\frac {7{\sqrt {3}}}{2}}$
52	Augmented pentagonal prism	9.173005609	${\frac {1}{2}}\left(8+2{\sqrt {3}}+{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)$
53	Biaugmented pentagonal prism	9.905056416	${\frac {1}{2}}\left(6+4{\sqrt {3}}+{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)$
54	Augmented hexagonal prism	11.928203230	$5+4{\sqrt {3}}$
55	Parabiaugmented hexagonal prism	12.660254038	$4+5{\sqrt {3}}$
56	Metabiaugmented hexagonal prism	12.660254038	$4+5{\sqrt {3}}$
57	Triaugmented hexagonal prism	13.392304845	$3\left(1+2{\sqrt {3}}\right)$
58	Augmented dodecahedron	21.090314916	${\frac {1}{4}}\left(5{\sqrt {3}}+11{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)$
59	Parabiaugmented dodecahedron	21.534901025	${\frac {5}{2}}\left({\sqrt {3}}+{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)$
60	Metabiaugmented dodecahedron	21.534901025	${\frac {5}{2}}\left({\sqrt {3}}+{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)$
61	Triaugmented dodecahedron	21.979487134	${\frac {3}{4}}\left(5{\sqrt {3}}+3{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)$
62	Metabidiminished icosahedron	7.771081820	${\frac {1}{2}}\left(5{\sqrt {3}}+{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)$
63	Tridiminished icosahedron	7.326495711	${\frac {1}{4}}\left(5{\sqrt {3}}+3{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)$
64	Augmented tridiminished icosahedron	8.192521115	${\frac {1}{4}}\left(7{\sqrt {3}}+3{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)$
65	Augmented truncated tetrahedron	14.258330249	${\frac {1}{2}}\left(6+13{\sqrt {3}}\right)$
66	Augmented truncated cube	34.338288046	$15+10{\sqrt {2}}+3{\sqrt {3}}$
67	Biaugmented truncated cube	36.241911729	$2\left(9+4{\sqrt {2}}+2{\sqrt {3}}\right)$
68	Augmented truncated dodecahedron	102.182092220	${\frac {1}{4}}\left(20+25{\sqrt {3}}+110{\sqrt {5+2{\sqrt {5}}}}+{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)$
69	Parabiaugmented truncated dodecahedron	103.373424287	${\frac {1}{2}}\left(20+15{\sqrt {3}}+50{\sqrt {5+2{\sqrt {5}}}}+{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)$
70	Metabiaugmented truncated dodecahedron	103.373424287	${\frac {1}{2}}\left(20+15{\sqrt {3}}+50{\sqrt {5+2{\sqrt {5}}}}+{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)$
71	Triaugmented truncated dodecahedron	104.564756354	${\frac {1}{4}}\left(60+35{\sqrt {3}}+90{\sqrt {5+2{\sqrt {5}}}}+3{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)$
72	Gyrate rhombicosidodecahedron	59.305982845	$30+5{\sqrt {3}}+3{\sqrt {5\left(5+2{\sqrt {5}}\right)}}$
73	Parabigyrate rhombicosidodecahedron	59.305982845	$30+5{\sqrt {3}}+3{\sqrt {5\left(5+2{\sqrt {5}}\right)}}$
74	Metabigyrate rhombicosidodecahedron	59.305982845	$30+5{\sqrt {3}}+3{\sqrt {5\left(5+2{\sqrt {5}}\right)}}$
75	Trigyrate rhombicosidodecahedron	59.305982845	$30+5{\sqrt {3}}+3{\sqrt {5\left(5+2{\sqrt {5}}\right)}}$
76	Diminished rhombicosidodecahedron	58.114650778	${\frac {1}{4}}\left(100+15{\sqrt {3}}+10{\sqrt {5+2{\sqrt {5}}}}+11{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)$
77	Paragyrate diminished rhombicosidodecahedron	58.114650778	${\frac {1}{4}}\left(100+15{\sqrt {3}}+10{\sqrt {5+2{\sqrt {5}}}}+11{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)$
78	Metagyrate diminished rhombicosidodecahedron	58.114650778	${\frac {1}{4}}\left(100+15{\sqrt {3}}+10{\sqrt {5+2{\sqrt {5}}}}+11{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)$
79	Bigyrate diminished rhombicosidodecahedron	58.114650778	${\frac {1}{4}}\left(100+15{\sqrt {3}}+10{\sqrt {5+2{\sqrt {5}}}}+11{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)$
80	Parabidiminished rhombicosidodecahedron	56.923318711	${\frac {5}{2}}\left(8+{\sqrt {3}}+2{\sqrt {5+2{\sqrt {5}}}}+{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)$
81	Metabidiminished rhombicosidodecahedron	56.923318711	${\frac {5}{2}}\left(8+{\sqrt {3}}+2{\sqrt {5+2{\sqrt {5}}}}+{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)$
82	Gyrate bidiminished rhombicosidodecahedron	56.923318711	${\frac {5}{2}}\left(8+{\sqrt {3}}+2{\sqrt {5+2{\sqrt {5}}}}+{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)$
83	Tridiminished rhombicosidodecahedron	55.731986644	${\frac {1}{4}}\left(60+5{\sqrt {3}}+30{\sqrt {5+2{\sqrt {5}}}}+9{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)$
84	Snub disphenoid	5.196152423	$3{\sqrt {3}}$
85	Snub square antiprism	12.392304845	$2\left(1+3{\sqrt {3}}\right)$
86	Sphenocorona	7.196152423	$2+3{\sqrt {3}}$
87	Augmented sphenocorona	7.928203230	$1+4{\sqrt {3}}$
88	Sphenomegacorona	8.928203230	$2\left(1+2{\sqrt {3}}\right)$
89	Hebesphenomegacorona	10.794228634	${\frac {3}{2}}\left(2+3{\sqrt {3}}\right)$
90	Disphenocingulum	12.660254038	$4+5{\sqrt {3}}$
91	Bilunabirotunda	12.346011217	$2+2{\sqrt {3}}+{\sqrt {5\left(5+2{\sqrt {5}}\right)}}$
92	Triangular hebesphenorotunda	16.388673538	${\frac {1}{4}}\left(12+19{\sqrt {3}}+3{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)$

For a fixed edge length, the triangular dipyramid $J 12$ has the smallest surface area and the triaugmented truncated dodecahedron $J 71$ has the largest, more than 40 times larger.

Volume

The following table lists the volume of each Johnson solid. Here $V$ is the volume (not the number of vertices, as in the first table) and $a$ is the edge length.

The source for this table is the PolyhedronData[..., "Volume"] command in Wolfram Research's Mathematica.

These volumes can be calculated from a set of vertex coordinates; such coordinates are known for all 92 Johnson solids. A conceptually simple approach is to triangulate the surface of the solid (for example, by adding an extra point in the center of each non-triangular face) and choose some interior point as an "origin" so that the interior can be subdivided into irregular tetrahedra. Each tetrahedron has one vertex at the origin inside and three vertices on the surface. The volume of the solid is then the sum of the volumes of these tetrahedra. There is a simple formula for the volume of an irregular tetrahedron.

J_n	Solid name	V/a³ (approx.)	V/a³ (exact)
1	Square pyramid	0.235702260	${\frac {1}{3{\sqrt {2}}}}$
2	Pentagonal pyramid	0.301502832	${\frac {1}{24}}\left(5+{\sqrt {5}}\right)$
3	Triangular cupola	1.178511302	${\frac {5}{3{\sqrt {2}}}}$
4	Square cupola	1.942809042	$1+{\frac {2{\sqrt {2}}}{3}}$
5	Pentagonal cupola	2.324045318	${\frac {1}{6}}\left(5+4{\sqrt {5}}\right)$
6	Pentagonal rotunda	6.917762968	${\frac {1}{12}}\left(45+17{\sqrt {5}}\right)$
7	Elongated triangular pyramid	0.550863832	${\frac {1}{12}}\left({\sqrt {2}}+3{\sqrt {3}}\right)$
8	Elongated square pyramid	1.235702260	${\frac {1}{6}}\left(6+{\sqrt {2}}\right)$
9	Elongated pentagonal pyramid	2.021980233	${\frac {1}{24}}\left(5+{\sqrt {5}}+6{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)$
10	Gyroelongated square pyramid	1.192702242	${\frac {1}{6}}\left({\sqrt {2}}+2{\sqrt {4+3{\sqrt {2}}}}\right)$
11	Gyroelongated pentagonal pyramid	1.880192158	${\frac {1}{24}}\left(25+9{\sqrt {5}}\right)$
12	Triangular dipyramid	0.235702260	${\frac {1}{3{\sqrt {2}}}}$
13	Pentagonal dipyramid	0.603005665	${\frac {1}{12}}\left(5+{\sqrt {5}}\right)$
14	Elongated triangular dipyramid	0.668714962	${\frac {1}{12}}\left(2{\sqrt {2}}+3{\sqrt {3}}\right)$
15	Elongated square dipyramid	1.471404521	${\frac {1}{3}}\left(3+{\sqrt {2}}\right)$
16	Elongated pentagonal dipyramid	2.323483065	${\frac {1}{12}}\left(5+{\sqrt {5}}+3{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)$
17	Gyroelongated square dipyramid	1.428404503	${\frac {1}{3}}\left({\sqrt {2}}+{\sqrt {4+3{\sqrt {2}}}}\right)$
18	Elongated triangular cupola	3.776587513	${\frac {1}{6}}\left(5{\sqrt {2}}+9{\sqrt {3}}\right)$
19	Elongated square cupola	6.771236166	$3+{\frac {8{\sqrt {2}}}{3}}$
20	Elongated pentagonal cupola	10.018254161	${\frac {1}{6}}\left(5+4{\sqrt {5}}+15{\sqrt {5+2{\sqrt {5}}}}\right)$
21	Elongated pentagonal rotunda	14.611971811	${\frac {1}{12}}\left(45+17{\sqrt {5}}+30{\sqrt {5+2{\sqrt {5}}}}\right)$
22	Gyroelongated triangular cupola	3.516053091	${\frac {1}{3}}{\sqrt {{\frac {61}{2}}+18{\sqrt {3}}+30{\sqrt {1+{\sqrt {3}}}}}}$
23	Gyroelongated square cupola	6.210765792	root of polynomial of degree 8
24	Gyroelongated pentagonal cupola	9.073333194	root of polynomial of degree 8
25	Gyroelongated pentagonal rotunda	13.667050844	root of polynomial of degree 8
26	Gyrobifastigium	0.866025404	${\frac {\sqrt {3}}{2}}$
27	Triangular orthobicupola	2.357022604	${\frac {5{\sqrt {2}}}{3}}$
28	Square orthobicupola	3.885618083	$2+{\frac {4{\sqrt {2}}}{3}}$
29	Square gyrobicupola	3.885618083	$2+{\frac {4{\sqrt {2}}}{3}}$
30	Pentagonal orthobicupola	4.648090637	${\frac {1}{3}}\left(5+4{\sqrt {5}}\right)$
31	Pentagonal gyrobicupola	4.648090637	${\frac {1}{3}}\left(5+4{\sqrt {5}}\right)$
32	Pentagonal orthocupolarotunda	9.241808286	${\frac {5}{12}}\left(11+5{\sqrt {5}}\right)$
33	Pentagonal gyrocupolarotunda	9.241808286	${\frac {5}{12}}\left(11+5{\sqrt {5}}\right)$
34	Pentagonal orthobirotunda	13.835525936	${\frac {1}{6}}\left(45+17{\sqrt {5}}\right)$
35	Elongated triangular orthobicupola	4.955098815	${\frac {5{\sqrt {2}}}{3}}+{\frac {3{\sqrt {3}}}{2}}$
36	Elongated triangular gyrobicupola	4.955098815	${\frac {5{\sqrt {2}}}{3}}+{\frac {3{\sqrt {3}}}{2}}$
37	Elongated square gyrobicupola	8.714045208	$4+{\frac {10{\sqrt {2}}}{3}}$
38	Elongated pentagonal orthobicupola	12.342299480	${\frac {1}{6}}\left(10+8{\sqrt {5}}+15{\sqrt {5+2{\sqrt {5}}}}\right)$
39	Elongated pentagonal gyrobicupola	12.342299480	${\frac {1}{6}}\left(10+8{\sqrt {5}}+15{\sqrt {5+2{\sqrt {5}}}}\right)$
40	Elongated pentagonal orthocupolarotunda	16.936017129	${\frac {5}{12}}\left(11+5{\sqrt {5}}+6{\sqrt {5+2{\sqrt {5}}}}\right)$
41	Elongated pentagonal gyrocupolarotunda	16.936017129	${\frac {5}{12}}\left(11+5{\sqrt {5}}+6{\sqrt {5+2{\sqrt {5}}}}\right)$
42	Elongated pentagonal orthobirotunda	21.529734779	${\frac {1}{6}}\left(45+17{\sqrt {5}}+15{\sqrt {5+2{\sqrt {5}}}}\right)$
43	Elongated pentagonal gyrobirotunda	21.529734779	${\frac {1}{6}}\left(45+17{\sqrt {5}}+15{\sqrt {5+2{\sqrt {5}}}}\right)$
44	Gyroelongated triangular bicupola	4.694564393	${\frac {1}{3}}{\sqrt {68+18{\sqrt {3}}+60{\sqrt {1+{\sqrt {3}}}}}}$
45	Gyroelongated square bicupola	8.153574834	root of polynomial of degree 8
46	Gyroelongated pentagonal bicupola	11.397378512	root of polynomial of degree 8
47	Gyroelongated pentagonal cupolarotunda	15.991096162	root of polynomial of degree 8
48	Gyroelongated pentagonal birotunda	20.584813812	root of polynomial of degree 8
49	Augmented triangular prism	0.668714962	${\frac {1}{12}}\left(2{\sqrt {2}}+3{\sqrt {3}}\right)$
50	Biaugmented triangular prism	0.904417223	${\sqrt {{\frac {59}{144}}+{\frac {1}{\sqrt {6}}}}}$
51	Triaugmented triangular prism	1.140119483	${\frac {1}{\sqrt {2}}}+{\frac {\sqrt {3}}{4}}$
52	Augmented pentagonal prism	1.956179661	${\frac {1}{12}}{\sqrt {233+90{\sqrt {5}}+12{\sqrt {50+20{\sqrt {5}}}}}}$
53	Biaugmented pentagonal prism	2.191881921	${\frac {1}{12}}{\sqrt {257+90{\sqrt {5}}+24{\sqrt {50+20{\sqrt {5}}}}}}$
54	Augmented hexagonal prism	2.833778472	${\frac {1}{6}}\left({\sqrt {2}}+9{\sqrt {3}}\right)$
55	Parabiaugmented hexagonal prism	3.069480732	${\frac {1}{6}}\left(2{\sqrt {2}}+9{\sqrt {3}}\right)$
56	Metabiaugmented hexagonal prism	3.069480732	${\frac {1}{6}}\left(2{\sqrt {2}}+9{\sqrt {3}}\right)$
57	Triaugmented hexagonal prism	3.305182993	${\frac {1}{\sqrt {2}}}+{\frac {3{\sqrt {3}}}{2}}$
58	Augmented dodecahedron	7.964621793	${\frac {1}{24}}\left(95+43{\sqrt {5}}\right)$
59	Parabiaugmented dodecahedron	8.266124625	${\frac {1}{6}}\left(25+11{\sqrt {5}}\right)$
60	Metabiaugmented dodecahedron	8.266124625	${\frac {1}{6}}\left(25+11{\sqrt {5}}\right)$
61	Triaugmented dodecahedron	8.567627458	${\frac {5}{8}}\left(7+3{\sqrt {5}}\right)$
62	Metabidiminished icosahedron	1.578689326	${\frac {1}{6}}\left(5+2{\sqrt {5}}\right)$
63	Tridiminished icosahedron	1.277186493	${\frac {5}{8}}+{\frac {7{\sqrt {5}}}{24}}$
64	Augmented tridiminished icosahedron	1.395037624	${\frac {1}{24}}\left(15+2{\sqrt {2}}+7{\sqrt {5}}\right)$
65	Augmented truncated tetrahedron	3.889087297	${\frac {11}{2{\sqrt {2}}}}$
66	Augmented truncated cube	15.542472333	$8+{\frac {16{\sqrt {2}}}{3}}$
67	Biaugmented truncated cube	17.485281374	$9+6{\sqrt {2}}$
68	Augmented truncated dodecahedron	87.363709878	${\frac {505}{12}}+{\frac {81{\sqrt {5}}}{4}}$
69	Parabiaugmented truncated dodecahedron	89.687755196	${\frac {1}{12}}\left(515+251{\sqrt {5}}\right)$
70	Metabiaugmented truncated dodecahedron	89.687755196	${\frac {1}{12}}\left(515+251{\sqrt {5}}\right)$
71	Triaugmented truncated dodecahedron	92.011800514	${\frac {7}{12}}\left(75+37{\sqrt {5}}\right)$
72	Gyrate rhombicosidodecahedron	41.615323782	$20+{\frac {29{\sqrt {5}}}{3}}$
73	Parabigyrate rhombicosidodecahedron	41.615323782	$20+{\frac {29{\sqrt {5}}}{3}}$
74	Metabigyrate rhombicosidodecahedron	41.615323782	$20+{\frac {29{\sqrt {5}}}{3}}$
75	Trigyrate rhombicosidodecahedron	41.615323782	$20+{\frac {29{\sqrt {5}}}{3}}$
76	Diminished rhombicosidodecahedron	39.291278464	${\frac {115}{6}}+9{\sqrt {5}}$
77	Paragyrate diminished rhombicosidodecahedron	39.291278464	${\frac {115}{6}}+9{\sqrt {5}}$
78	Metagyrate diminished rhombicosidodecahedron	39.291278464	${\frac {115}{6}}+9{\sqrt {5}}$
79	Bigyrate diminished rhombicosidodecahedron	39.291278464	${\frac {115}{6}}+9{\sqrt {5}}$
80	Parabidiminished rhombicosidodecahedron	36.967233146	${\frac {5}{3}}\left(11+5{\sqrt {5}}\right)$
81	Metabidiminished rhombicosidodecahedron	36.967233146	${\frac {5}{3}}\left(11+5{\sqrt {5}}\right)$
82	Gyrate bidiminished rhombicosidodecahedron	36.967233146	${\frac {5}{3}}\left(11+5{\sqrt {5}}\right)$
83	Tridiminished rhombicosidodecahedron	34.643187827	${\frac {35}{2}}+{\frac {23{\sqrt {5}}}{3}}$
84	Snub disphenoid	0.859493646	root of polynomial of degree 6
85	Snub square antiprism	3.601222010	root of polynomial of degree 12
86	Sphenocorona	1.515351640	${\frac {1}{2}}{\sqrt {1+3{\sqrt {\frac {3}{2}}}+{\sqrt {13+3{\sqrt {6}}}}}}$
87	Augmented sphenocorona	1.751053900	root of polynomial of degree 16
88	Sphenomegacorona	1.948108229	root of polynomial of degree 32
89	Hebesphenomegacorona	2.912910415	root of polynomial of degree 20
90	Disphenocingulum	3.777645342	root of polynomial of degree 24
91	Bilunabirotunda	3.093717650	${\frac {1}{12}}\left(17+9{\sqrt {5}}\right)$
92	Triangular hebesphenorotunda	5.108745974	${\frac {5}{2}}+{\frac {7{\sqrt {5}}}{6}}$

For a fixed edge length, the square pyramid $J 1$ and the triangular dipyramid $J 12$ have the smallest volume and the triaugmented truncated dodecahedron $J 71$ has the largest, more than 390 times larger.

Thirteen of the 92 Johnson solids have volumes for which $V / a 3$ is not a number expressible using radicals. These values are the greatest real root of the following polynomials.

J_n	Polynomial
23	$6561 x 8 - 52488 x 7 + 113724 x 6 - 9720 x 5 - 1616922 x 4 + 396360 x 3 + 1537020 x 2 - 178632 x - 3391$
24	$1679616 x 8 - 11197440 x 7 + 27060480 x 6 + 35769600 x 5 - 4456749600 x 4 - 10714248000 x 3 + 3828402000 x 2 + 13859430000 x + 5340175625$
25	$1679616 x 8 - 50388480 x 7 + 603262080 x 6 - 3520972800 x 5 + 5215460400 x 4 + 4128624000 x 3 - 8894943000 x 2 + 3881385000 x - 424924375$
45	$6561 x 8 - 104976 x 7 + 594864 x 6 - 1384128 x 5 - 552096 x 4 + 1569024 x 3 + 246528 x 2 - 119808 x + 4352$
46	$6561 x 8 - 87480 x 7 + 313470 x 6 + 753300 x 5 - 22424850 x 4 �� 84591000 x 3 - 85909500 x 2 + 8715000 x + 35547500$
47	$1679616 x 8 - 61585920 x 7 + 851472000 x 6 - 5108832000 x 5 + 4745790000 x 4 + 21346200000 x 3 - 29019375000 x 2 - 4576875000 x - 405859375$
48	$6561 x 8 - 393660 x 7 + 9316620 x 6 - 108207900 x 5 + 601832025 x 4 - 1417189500 x 3 + 965841750 x 2 + 597667500 x - 668786875$
84	$5832 x 6 - 1377 x 4 - 2160 x 2 - 4$
85	$531441 x 12 - 85726026 x 8 - 48347280 x 6 + 11588832 x 4 + 4759488 x 2 - 892448$
87	$45137758519296 x 16 - 110336743047168 x 14 - 191069246324736 x 12 + 209269081571328 x 10 + 364547659290624 x 8 - 58793017190400 x 6 + 3306865979520 x 4 - 1275399855936 x 2 + 1439671249$
88	$521578814501447328359509917696 x 32 - 985204427391622731345740955648 x 30 - 16645447351681991898880656015360 x 28 + 79710816694053483249372512649216 x 26 - 152195045391070538203422101864448 x 24 + 156280253448056209478031589244928 x 22 - 96188116617075838858708654227456 x 20 + 30636368373570166303441645731840 x 18 + 5828527077458909552923002273792 x 16 - 8060049780765551057159394951168 x 14 + 1018074792115156107372011716608 x 12 + 35220131544370794950945931264 x 10 + 327511698517355918956755959808 x 8 - 116978732884218191486738706432 x 6 + 10231563774949176791703149568 x 4 - 366323949299263261553952192 x 2 + 3071435678740442112675625$
89	$47330370277129322496 x 20 - 722445512980071186432 x 18 + 3596480447590271287296 x 16 - 8432333285523990773760 x 14 + 8973584611317745975296 x 12 - 3065290664181478981632 x 10 + 366229890219212144640 x 8 - 8337259437908852736 x 6 - 22211277300912896 x 4 + 132615435213216 x 2 + 2693461945329$
90	$1213025622610333925376 x 24 + 54451372392730545094656 x 22 - 796837093078664749252608 x 20 - 4133410366404688544268288 x 18 + 20902529024429842816303104 x 16 - 133907540390420673677230080 x 14 + 246234688242991598853881856 x 12 - 63327534106871321714442240 x 10 + 14389309497459555704164608 x 8 + 48042947402464500749392128 x 6 - 5891096640600351061013664 x 4 - 3212114716816853362953264 x 2 + 479556973248657693884401$

Inradius, midradius, and circumradius

The following table lists the radius $R i$ of the insphere, the radius $R m$ of the midsphere, and the radius $R c$ of the circumsphere, each divided by the edge length $a$ , when these spheres exist.

A polyhedron does not necessarily have an insphere, or a midsphere, or a circumsphere. For example, it does not have a circumsphere unless all its vertices lie on some sphere. The Johnson solids, having less symmetry than, say, the Platonic solids, lack many of these spheres. Only $J 1$ and $J 2$ possess all three of these spheres.

The source for this table is the PolyhedronData[..., "Inradius"], PolyhedronData[..., "Midradius"], and PolyhedronData[..., "Circumradius"] commands in Wolfram Research's Mathematica. The output has been simplified to a consistent form in terms of radicals.

J_n	R_i/a (approx.)	R_i/a (exact)	R_m/a (approx.)	R_m/a (exact)	R_c/a (approx.)	R_c/a (exact)
1	0.258819045	${\frac {\sqrt {2-{\sqrt {3}}}}{2}}$	0.500000000	${\frac {1}{2}}$	0.707106781	${\frac {1}{\sqrt {2}}}$
2	0.232788309	${\frac {1}{\sqrt {25-7{\sqrt {5}}+{\sqrt {30\left(5-{\sqrt {5}}\right)}}}}}$	0.809016994	${\frac {1}{4}}\left(1+{\sqrt {5}}\right)$	0.951056516	${\frac {1}{2}}{\sqrt {{\frac {1}{2}}\left(5+{\sqrt {5}}\right)}}$
3	-	-	0.866025404	${\frac {\sqrt {3}}{2}}$	1.000000000	$1$
4	-	-	1.306562965	${\sqrt {1+{\frac {1}{\sqrt {2}}}}}$	1.398966326	${\sqrt {{\frac {5}{4}}+{\frac {1}{\sqrt {2}}}}}$
5	-	-	2.176250899	${\sqrt {{\frac {5}{2}}+{\sqrt {5}}}}$	2.232950509	${\frac {1}{2}}{\sqrt {11+4{\sqrt {5}}}}$
6	-	-	1.538841769	${\frac {1}{2}}{\sqrt {5+2{\sqrt {5}}}}$	1.618033989	${\frac {1}{2}}\left(1+{\sqrt {5}}\right)$
7	-	-	-	-	-	-
8	-	-	-	-	-	-
9	-	-	-	-	-	-
10	-	-	-	-	-	-
11	-	-	0.809016994	${\frac {1}{4}}\left(1+{\sqrt {5}}\right)$	0.951056516	${\frac {1}{2}}{\sqrt {{\frac {1}{2}}\left(5+{\sqrt {5}}\right)}}$
12	0.272165527	${\frac {\sqrt {\frac {2}{3}}}{3}}$	-	-	-	-
13	0.417774579	${\sqrt {{\frac {1}{30}}\left(3+{\sqrt {5}}\right)}}$	-	-	-	-
14	-	-	-	-	-	-
15	-	-	-	-	-	-
16	-	-	-	-	-	-
17	-	-	-	-	-	-
18	-	-	-	-	-	-
19	-	-	1.306562965	${\sqrt {1+{\frac {1}{\sqrt {2}}}}}$	1.398966326	${\sqrt {{\frac {5}{4}}+{\frac {1}{\sqrt {2}}}}}$
20	-	-	-	-	-	-
21	-	-	-	-	-	-
22	-	-	-	-	-	-
23	-	-	-	-	-	-
24	-	-	-	-	-	-
25	-	-	-	-	-	-
26	-	-	-	-	-	-
27	-	-	0.866025404	${\frac {\sqrt {3}}{2}}$	1.000000000	$1$
28	-	-	-	-	-	-
29	-	-	-	-	-	-
30	-	-	-	-	-	-
31	-	-	-	-	-	-
32	-	-	-	-	-	-
33	-	-	-	-	-	-
34	-	-	1.538841769	${\frac {1}{2}}{\sqrt {5+2{\sqrt {5}}}}$	1.618033989	${\frac {1}{2}}\left(1+{\sqrt {5}}\right)$
35	-	-	-	-	-	-
36	-	-	-	-	-	-
37	-	-	1.306562965	${\sqrt {1+{\frac {1}{\sqrt {2}}}}}$	1.398966326	${\sqrt {{\frac {5}{4}}+{\frac {1}{\sqrt {2}}}}}$
38	-	-	-	-	-	-
39	-	-	-	-	-	-
40	-	-	-	-	-	-
41	-	-	-	-	-	-
42	-	-	-	-	-	-
43	-	-	-	-	-	-
44	-	-	-	-	-	-
45	-	-	-	-	-	-
46	-	-	-	-	-	-
47	-	-	-	-	-	-
48	-	-	-	-	-	-
49	-	-	-	-	-	-
50	-	-	-	-	-	-
51	-	-	-	-	-	-
52	-	-	-	-	-	-
53	-	-	-	-	-	-
54	-	-	-	-	-	-
55	-	-	-	-	-	-
56	-	-	-	-	-	-
57	-	-	-	-	-	-
58	-	-	-	-	-	-
59	-	-	-	-	-	-
60	-	-	-	-	-	-
61	-	-	-	-	-	-
62	-	-	0.809016994	${\frac {1}{4}}\left(1+{\sqrt {5}}\right)$	0.951056516	${\frac {1}{2}}{\sqrt {{\frac {1}{2}}\left(5+{\sqrt {5}}\right)}}$
63	-	-	0.809016994	${\frac {1}{4}}\left(1+{\sqrt {5}}\right)$	0.951056516	${\frac {1}{2}}{\sqrt {{\frac {1}{2}}\left(5+{\sqrt {5}}\right)}}$
64	-	-	-	-	-	-
65	-	-	-	-	-	-
66	-	-	-	-	-	-
67	-	-	-	-	-	-
68	-	-	-	-	-	-
69	-	-	-	-	-	-
70	-	-	-	-	-	-
71	-	-	-	-	-	-
72	-	-	2.176250899	${\sqrt {{\frac {5}{2}}+{\sqrt {5}}}}$	2.232950509	${\frac {1}{2}}{\sqrt {11+4{\sqrt {5}}}}$
73	-	-	2.176250899	${\sqrt {{\frac {5}{2}}+{\sqrt {5}}}}$	2.232950509	${\frac {1}{2}}{\sqrt {11+4{\sqrt {5}}}}$
74	-	-	2.176250899	${\sqrt {{\frac {5}{2}}+{\sqrt {5}}}}$	2.232950509	${\frac {1}{2}}{\sqrt {11+4{\sqrt {5}}}}$
75	-	-	2.176250899	${\sqrt {{\frac {5}{2}}+{\sqrt {5}}}}$	2.232950509	${\frac {1}{2}}{\sqrt {11+4{\sqrt {5}}}}$
76	-	-	2.176250899	${\sqrt {{\frac {5}{2}}+{\sqrt {5}}}}$	2.232950509	${\frac {1}{2}}{\sqrt {11+4{\sqrt {5}}}}$
77	-	-	2.176250899	${\sqrt {{\frac {5}{2}}+{\sqrt {5}}}}$	2.232950509	${\frac {1}{2}}{\sqrt {11+4{\sqrt {5}}}}$
78	-	-	2.176250899	${\sqrt {{\frac {5}{2}}+{\sqrt {5}}}}$	2.232950509	${\frac {1}{2}}{\sqrt {11+4{\sqrt {5}}}}$
79	-	-	2.176250899	${\sqrt {{\frac {5}{2}}+{\sqrt {5}}}}$	2.232950509	${\frac {1}{2}}{\sqrt {11+4{\sqrt {5}}}}$
80	-	-	2.176250899	${\sqrt {{\frac {5}{2}}+{\sqrt {5}}}}$	2.232950509	${\frac {1}{2}}{\sqrt {11+4{\sqrt {5}}}}$
81	-	-	2.176250899	${\sqrt {{\frac {5}{2}}+{\sqrt {5}}}}$	2.232950509	${\frac {1}{2}}{\sqrt {11+4{\sqrt {5}}}}$
82	-	-	2.176250899	${\sqrt {{\frac {5}{2}}+{\sqrt {5}}}}$	2.232950509	${\frac {1}{2}}{\sqrt {11+4{\sqrt {5}}}}$
83	-	-	2.176250899	${\sqrt {{\frac {5}{2}}+{\sqrt {5}}}}$	2.232950509	${\frac {1}{2}}{\sqrt {11+4{\sqrt {5}}}}$
84	-	-	-	-	-	-
85	-	-	-	-	-	-
86	-	-	-	-	-	-
87	-	-	-	-	-	-
88	-	-	-	-	-	-
89	-	-	-	-	-	-
90	-	-	-	-	-	-
91	-	-	-	-	-	-
92	-	-	-	-	-	-

References

Norman W. Johnson, "Convex Solids with Regular Faces", Canadian Journal of Mathematics, 18, 1966, pages 169–200. Contains the original enumeration of the 92 solids and the conjecture that there are no others.
Victor A. Zalgaller (1969). Convex Polyhedra with Regular Faces. Consultants Bureau. No ISBN. The first proof that there are only 92 Johnson solids.

External links

Sylvain Gagnon, "Convex polyhedra with regular faces^{[permanent dead link]}", Structural Topology, No. 6, 1982, 83-95.
Johnson Solids by George W. Hart.
Images of all 92 solids, categorized, on one page
Weisstein, Eric W. "Johnson Solid". MathWorld.
VRML models of Johnson Solids by Jim McNeill
VRML models of Johnson Solids by Vladimir Bulatov

v t Johnson solids
Pyramids, cupolae and rotundae	square pyramid pentagonal pyramid triangular cupola square cupola pentagonal cupola pentagonal rotunda
Modified pyramids	elongated triangular pyramid elongated square pyramid elongated pentagonal pyramid gyroelongated square pyramid gyroelongated pentagonal pyramid triangular bipyramid pentagonal bipyramid elongated triangular bipyramid elongated square bipyramid elongated pentagonal bipyramid gyroelongated square bipyramid
Modified cupolae and rotundae	elongated triangular cupola elongated square cupola elongated pentagonal cupola elongated pentagonal rotunda gyroelongated triangular cupola gyroelongated square cupola gyroelongated pentagonal cupola gyroelongated pentagonal rotunda gyrobifastigium triangular orthobicupola square orthobicupola square gyrobicupola pentagonal orthobicupola pentagonal gyrobicupola pentagonal orthocupolarotunda pentagonal gyrocupolarotunda pentagonal orthobirotunda elongated triangular orthobicupola elongated triangular gyrobicupola elongated square gyrobicupola elongated pentagonal orthobicupola elongated pentagonal gyrobicupola elongated pentagonal orthocupolarotunda elongated pentagonal gyrocupolarotunda elongated pentagonal orthobirotunda elongated pentagonal gyrobirotunda gyroelongated triangular bicupola gyroelongated square bicupola gyroelongated pentagonal bicupola gyroelongated pentagonal cupolarotunda gyroelongated pentagonal birotunda
Augmented prisms	augmented triangular prism biaugmented triangular prism triaugmented triangular prism augmented pentagonal prism biaugmented pentagonal prism augmented hexagonal prism parabiaugmented hexagonal prism metabiaugmented hexagonal prism triaugmented hexagonal prism
Modified Platonic solids	augmented dodecahedron parabiaugmented dodecahedron metabiaugmented dodecahedron triaugmented dodecahedron metabidiminished icosahedron tridiminished icosahedron augmented tridiminished icosahedron
Modified Archimedean solids	augmented truncated tetrahedron augmented truncated cube biaugmented truncated cube augmented truncated dodecahedron parabiaugmented truncated dodecahedron metabiaugmented truncated dodecahedron triaugmented truncated dodecahedron gyrate rhombicosidodecahedron parabigyrate rhombicosidodecahedron metabigyrate rhombicosidodecahedron trigyrate rhombicosidodecahedron diminished rhombicosidodecahedron paragyrate diminished rhombicosidodecahedron metagyrate diminished rhombicosidodecahedron bigyrate diminished rhombicosidodecahedron parabidiminished rhombicosidodecahedron metabidiminished rhombicosidodecahedron gyrate bidiminished rhombicosidodecahedron tridiminished rhombicosidodecahedron
Elementary solids	snub disphenoid snub square antiprism sphenocorona augmented sphenocorona sphenomegacorona hebesphenomegacorona disphenocingulum bilunabirotunda triangular hebesphenorotunda
(See also List of Johnson solids, a sortable table)