List of Johnson solids explained

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. A polyhedron is considered to be convex if:

• 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 convex polyhedron whose faces are regular polygons is known as a Johnson solid, or sometimes as a Johnson–Zalgaller solid. Some authors exclude uniform polyhedra from the definition. A uniform polyhedron is a polyhedron in which the faces are regular and they are isogonal; examples include Platonic and Archimedean solids as well as prisms and antiprisms.The Johnson solids are named after American mathematician Norman Johnson (1930–2017), who published a list of 92 such polyhedra in 1966. His conjecture that the list was complete and no other examples existed was proven by Russian-Israeli mathematician Victor Zalgaller (1920–2020) in 1969.

Some of the Johnson solids may be categorized as elementary polyhedra, meaning they cannot be separated by a plane to create two small convex polyhedra with regular faces. The Johnson solids satisfying this criteria are the first six—equilateral square pyramid, pentagonal pyramid, triangular cupola, square cupola, pentagonal cupola, and pentagonal rotunda. The criteria is also satisfied by eleven other Johnson solids, specifically the tridiminished icosahedron, parabidiminished rhombicosidodecahedron, tridiminished rhombicosidodecahedron, snub disphenoid, snub square antiprism, sphenocorona, sphenomegacorona, hebesphenomegacorona, disphenocingulum, bilunabirotunda, and triangular hebesphenorotunda. The rest of the Johnson solids are not elementary, and they are constructed using the first six Johnson solids together with Platonic and Archimedean solids in various processes. Augmentation involves attaching the Johnson solids onto one or more faces of polyhedra, while elongation or gyroelongation involve joining them onto the bases of a prism or antiprism, respectively. Some others are constructed by diminishment, the removal of one of the first six solids from one or more of a polyhedron's faces.

The following table contains the 92 Johnson solids, with edge length

a . The table includes the solid's enumeration (denoted as J _{n} ). It also includes the number of vertices, edges, and faces of each solid, as well as its symmetry group, surface area A , and volume V . Every polyhedron has its own characteristics, including symmetry and measurement. An object is said to have symmetry if there is a transformation that maps it to itself. All of those transformations may be composed in a group, alongside the group's number of elements, known as the 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 by $\backslash frac$ is denoted by C _{n} , a cyclic group of order n ; combining this with the reflection symmetry results in the symmetry of dihedral group D _{n} of order 2 n . In three-dimensional symmetry point groups, the transformations preserving a polyhedron's symmetry include the rotation around the line passing through the base center, known as the axis of symmetry, and the reflection relative to perpendicular planes passing through the bisector of a base, which is known as the pyramidal symmetry C _{n} of order 2 n . The transformation that preserves a polyhedron's symmetry by reflecting it across a horizontal plane is known as the prismatic symmetry D _{n} of order 4 n . The antiprismatic symmetry D _{n} of order 4 n preserves the symmetry by rotating its half bottom and reflection across the horizontal plane. The symmetry group C _{n} of order 2 n preserves the symmetry by rotation around the axis of symmetry and reflection on the horizontal plane; the specific case preserving the symmetry by one full rotation is C _{1} of order 2, often denoted as C _{s} . 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; for a polyhedron, the surface area is the sum of the areas of all of its faces. A volume is a measurement of a region in three-dimensional space. The volume of a polyhedron may be ascertained in different ways: either through its base and height (like for pyramids and prisms), by slicing it off into pieces and summing their individual volumes, or by finding the root of a polynomial representing the polyhedron.

J _{n} !scope=col

Solid name	scope=col	Image	scope=col	Vertices	scope=col	Edges	scope=col	Faces	scope=col	Symmetry group and its order	scope=col	Surface area and volume
scope=row	1	Equilateral square pyramid		5	8	5	C _{4v} of order 8	\begin{align} A & = \left( 1 + \sqrt{ 3 }\right)a ^{2} \\ & ≈ 2 . 7321 a ^{2} \\ V & = (\sqrt{ 2)/() }a^3 \\ &\approx 0.2357a^3 \end
scope=row	2	Pentagonal pyramid		6	10	6	C _{5v} of order 10	\begin{align} A & = (a ^{2})/(\sqrt{ 2) (5)/(2) \left( 10+ \sqrt{ 5 } + \sqrt{ 75+30 \sqrt{ 5 }}\right)} \\ & ≈ 3 . 8855 a ^{2} \\ V & = \left( (5+ \sqrt{ 5)/() }\right)a^3 \\ &\approx 0.3015a^3 \end
scope=row	3	Triangular cupola		9	15	8	C _{3v} of order 6	\begin{align} A & = \left( 3+ (5 \sqrt{ 3)/() } \right) a^2 \\ &\approx 7.3301a^2 \\ V &= \left(\frac\right) a^3 \\ &\approx 1.1785a^3 \end
scope=row	4	Square cupola		12	20	10	C _{4v} of order 8	\begin{align} A & = \left( 7+2 \sqrt{ 2 } + \sqrt{ 3 }\right)a ^{2} \\ & ≈ 11 . 5605 a ^{2} \\ V & = \left( 1+ (2 \sqrt{ 2)/() }\right)a^3 \\ &\approx 1.9428a^3 \end
scope=row	5	Pentagonal cupola		15	25	12	C _{5v} of order 10	\begin{align} A & = \left( (1)/(4) \left( 20+5 \sqrt{ 3 } + \sqrt{ 5 \left( 145+62 \sqrt{ 5 }\right)}\right)\right)a ^{2} \\ & ≈ 16 . 5798 a ^{2} \\ V & = \left( (1)/(6) \left( 5+4 \sqrt{ 5 }\right)\right)a ^{3} \\ & ≈ 2 . 3241 a ^{3} \end{align}
scope=row	6	Pentagonal rotunda		20	35	17	C _{5v} of order 10	\begin{align} A & = \left( (1)/(2) \left( 5 \sqrt{ 3 } + \sqrt{ 10 \left( 65+29 \sqrt{ 5 }\right)}\right)\right)a ^{2} \\ & ≈ 22 . 3472 a ^{2} \\ V & = \left( (1)/(12) \left( 45+17 \sqrt{ 5 }\right)\right)a ^{3} \\ & ≈ 6 . 9178 a ^{3} \end{align}
scope=row	7	Elongated triangular pyramid		7	12	7	C _{3v} of order 6	\begin{align} A & = \left( 3+ \sqrt{ 3 }\right)a ^{2} \\ & ≈ 4 . 7321 a ^{2} \\ V & = \left( (1)/(12) \left(\sqrt{ 2 } +3 \sqrt{ 3 }\right)\right)a ^{3} \\ & ≈ 0 . 5509 a ^{3 \end{align}}
scope=row	8	Elongated square pyramid		9	16	9	C _{4v} of order 8	\begin{align} A & = \left( 5 + \sqrt{ 3 } \right)a ^{2} \\ & ≈ 6 . 7321 a ^{2} \\ V & = \left( 1 + (\sqrt{ 2)/() }\right)a^3 \\ &\approx 1.2357a^3 \end
scope=row	9	Elongated pentagonal pyramid		11	20	11	C _{5v} of order 10	\begin{align} A & = (20+5 \sqrt{ 3)/(+) \sqrt{ 25 + 10 \sqrt{ 5 }}}{ 4 }a ^{2} \\ & ≈ 8 . 8855 a ^{2} \\ V & = \left( (5+ \sqrt{ 5)/(+) 6 \sqrt{ 25 + 10 \sqrt{ 5 }}}{ 24 } \right)a ^{3} \\ & ≈ 2 . 022 a ^{3} \end{align}
scope=row	10	Gyroelongated square pyramid		9	20	13	C _{4v} of order 8	\begin{align} A & = ( 1 + 3 \sqrt{ 3 })a ^{2} \\ & ≈ 6 . 1962 a ^{2} \\ V & = (1)/(6) \left(\sqrt{ 2 } +2 \sqrt{ 4+3 \sqrt{ 2 }}\right)a ^{3} \\ & ≈ 1 . 1927 a ^{3 \end{align}}
scope=row	11	Gyroelongated pentagonal pyramid		11	25	16	C _{5v} of order 10	\begin{align} A & = (1)/(4) \left( 15 \sqrt{ 3 } + \sqrt{ 5 \left( 5+2 \sqrt{ 5 }\right)}\right)a ^{2} \\ & ≈ 8 . 2157 a ^{2} \\ V & = (1)/(24) \left( 25+9 \sqrt{ 5 }\right)a ^{3} \\ & ≈ 1 . 8802 a ^{3 \end{align}}
scope=row	12	Triangular bipyramid		5	9	6	D _{3h} of order 12	\begin{align} A & = (3 \sqrt{ 3)/() }a^2 \\ &\approx 2.5981a^2 \\ V &= \fraca^3 \\ &\approx 0.2358a^3\end
scope=row	13	Pentagonal bipyramid		7	15	10	D _{5h} of order 20	\begin{align} A & = (5 \sqrt{ 3)/() }a^2 \\ &\approx 4.3301a^2 \\ V &= \frac \left(5+\sqrt\right)a^3 \\ &\approx 0.603a^3\end
scope=row	14	Elongated triangular bipyramid		8	15	9	D _{3h} of order 12	\begin{align} A & = (3)/(2) \left( 2+ \sqrt{ 3 }\right)a ^{2} \\ & ≈ 5 . 5981 a ^{2} \\ V & = (1)/(12) \left( 2 \sqrt{ 2 } +3 \sqrt{ 3 }\right)a ^{3} \\ & ≈ 0 . 6687 a ^{3 \end{align}}
scope=row	15	Elongated square bipyramid		10	20	12	D _{4h} of order 16	\begin{align} A & = 2 \left( 2+ \sqrt{ 3 }\right)a ^{2} \\ & ≈ 7 . 4641 a ^{2} \\ V & = (1)/(3) \left( 3+ \sqrt{ 2 }\right)a ^{3} \\ & ≈ 1 . 4714 a ^{3 \end{align}}
scope=row	16	Elongated pentagonal bipyramid		12	25	15	D _{5h} of order 20	\begin{align} A & = (5)/(2) \left( 2+ \sqrt{ 3 }\right)a ^{2} \\ & ≈ 9 . 3301 a ^{2} \\ V & = (1)/(12) \left( 5+ \sqrt{ 5 } +3 \sqrt{ 5 \left( 5+2 \sqrt{ 5 }\right)}\right)a ^{3} \\ & ≈ 2 . 3235 a ^{3 \end{align}}
scope=row	17	Gyroelongated square bipyramid		10	24	16	D _{4d} of order 16	\begin{align} A & = 4 \sqrt{ 3 }a ^{2} \\ & ≈ 6 . 9282 a ^{2} \\ V & = (1)/(3) \left(\sqrt{ 2 } + \sqrt{ 4 + 3 \sqrt{ 2 }}\right)a ^{3} \\ & ≈ 1 . 4284 a ^{3 \end{align}}
scope=row	18	Elongated triangular cupola		15	27	14	C _{3v} of order 6	\begin{align} A & = (1)/(2) \left( 18+5 \sqrt{ 3 }\right)a ^{2} \\ & ≈ 13 . 3301 a ^{2} \\ V & = (1)/(6) \left( 5 \sqrt{ 2 } +9 \sqrt{ 3 }\right)a ^{3} \\ & ≈ 3 . 7766 a ^{3 \end{align}}
scope=row	19	Elongated square cupola		20	36	18	C _{4v} of order 8	\begin{align} A & = ( 15+2 \sqrt{ 2 } + \sqrt{ 3 })a ^{2\\} & ≈ 19 . 5605 a ^{2} \\ V & = \left( 3+ (8 \sqrt{ 2)/() }\right)a^3 \\ &\approx 6.7712a^3\end
scope=row	20	Elongated pentagonal cupola		25	45	22	C _{5v} of order 10	\begin{align} A & = (1)/(4) \left( 60+5 \sqrt{ 3 } +10 \sqrt{ 5+2 \sqrt{ 5 }} + \sqrt{ 5 \left( 5+2 \sqrt{ 5 }\right)}\right)a ^{2} \\ & ≈ 26 . 5798 a ^{2} \\ V & = (1)/(6) \left( 5+4 \sqrt{ 5 } +15 \sqrt{ 5+2 \sqrt{ 5 }}\right)a ^{3} \\ & ≈ 10 . 0183 a ^{3 \end{align}}
scope=row	21	Elongated pentagonal rotunda		30	55	27	C _{5v} of order 10	\begin{align} A & = (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) \\ & ≈ 32 . 3472 a ^{2} \\ V & = (1)/(12) a ^{3} \left( 45+17 \sqrt{ 5 } +30 \sqrt{ 5+2 \sqrt{ 5 }}\right) \\ & ≈ 14 . 612 a ^{3 \end{align}}
scope=row	22	Gyroelongated triangular cupola		15	33	20	C _{3v} of order 6	\begin{align} A & = (1)/(2) \left( 6+11 \sqrt{ 3 }\right)a ^{2} \\ & ≈ 12 . 5263 a ^{2} \\ V & = (1)/(\sqrt{ 3) (61)/(2) +18 \sqrt{ 3 } +30 \sqrt{ 1+ \sqrt{ 3 }}}a ^{3} \\ & ≈ 3 . 5161 a ^{3 \end{align}}
scope=row	23	Gyroelongated square cupola		20	44	26	C _{4v} of order 8	\begin{align} A & = ( 7+2 \sqrt{ 2 } +5 \sqrt{ 3 })a ^{2} \\ & ≈ 18 . 4887 a ^{2} \\ V & = \left( 1+ (2)/(3) \sqrt{ 2 } + (2)/(3) \sqrt{ 4+2 \sqrt{ 2 } +2 \sqrt{ 146+103 \sqrt{ 2 }}}\right)a ^{3} \\ & ≈ 6 . 2108 a ^{3 \end{align}}
scope=row	24	Gyroelongated pentagonal cupola		25	55	32	C _{5v} of order 10	\begin{align} A & = (1)/(4) \left( 20+25 \sqrt{ 3 } +10 \sqrt{ 5+2 \sqrt{ 5 }} + \sqrt{ 5 \left( 5+2 \sqrt{ 5 }\right)}\right)a ^{2} \\ & ≈ 25 . 2400 a ^{2} \\ V & = \left( (5)/(+ 6) (2)/(3) \sqrt{ 5 } + (5)/(6) \sqrt{ 2 \sqrt{ 650+290 \sqrt{ 5 }} -2 \sqrt{ 5 } -2 }\right) a ^{3} \\ & ≈ 9 . 0733 a ^{3 \end{align}}
scope=row	25	Gyroelongated pentagonal rotunda		30	65	37	C _{5v} of order 10	\begin{align} A & = (1)/(2) \left( 15 \sqrt{ 3 } + \left( 5+3 \sqrt{ 5 }\right)\sqrt{ 5+2 \sqrt{ 5 }}\right)a ^{2} \\ & ≈ 31 . 0075 a ^{2} \\ V & = \left( (45)/(+ 12) (17)/(12) \sqrt{ 5 } + (5)/(6) \sqrt{ 2 \sqrt{ 650+290 \sqrt{ 5 }} -2 \sqrt{ 5 } -2 }\right) a ^{3} \\ & ≈ 13 . 6671 a ^{3 \end{align}}
scope=row	26	Gyrobifastigium		8	14	8	D _{2d} of order 8	\begin{align} A & = \left( 4+ \sqrt{ 3 }\right)a ^{2} \\ & ≈ 5 . 7321 a ^{2} \\ V & = \left( (\sqrt{ 3)/() }\right)a^3 \\ &\approx 0.866a^3\end
scope=row	27	Triangular orthobicupola		12	24	14	D _{3h} of order 12	\begin{align} A & = 2 \left( 3+ \sqrt{ 3 }\right)a ^{2} \\ & ≈ 9 . 4641 a ^{2} \\ V & = (5 \sqrt{ 2)/() }a^3 \\ &\approx 2.357a^3\end
scope=row	28	Square orthobicupola		16	32	18	D _{4h} of order 16	\begin{align} A & = 2 ( 5 + \sqrt{ 3 })a ^{2} \\ & ≈ 13 . 4641 a ^{2} \\ V & = \left( 2 + (4 \sqrt{ 2)/() }\right)a^3 \\ &\approx 3.8856a^3\end
scope=row	29	Square gyrobicupola		16	32	18	D _{4d} of order 16	\begin{align} A & = 2 ( 5 + \sqrt{ 3 })a ^{2} \\ & ≈ 13 . 4641 a ^{2} \\ V & = \left( 2 + (4 \sqrt{ 2)/() }\right)a^3 \\ &\approx 3.8856a^3\end
scope=row	30	Pentagonal orthobicupola		20	40	22	D _{5h} of order 20	\begin{align} A & = \left( 10+ \sqrt{ (5)/(2) \left( 10+ \sqrt{ 5 } + \sqrt{ 75+30 \sqrt{ 5 }}\right)}\right)a ^{2} \\ & ≈ 17 . 7711 a ^{2} \\ V & = (1)/(3) \left( 5+4 \sqrt{ 5 }\right)a ^{3} \\ & ≈ 4 . 6481 a ^{3 \end{align}}
scope=row	31	Pentagonal gyrobicupola		20	40	22	D _{5d} of order 20	\begin{align} A & = \left( 10+ \sqrt{ (5)/(2) \left( 10+ \sqrt{ 5 } + \sqrt{ 75+30 \sqrt{ 5 }}\right)}\right)a ^{2} \\ & ≈ 17 . 7711 a ^{2} \\ V & = (1)/(3) \left( 5+4 \sqrt{ 5 }\right)a ^{3} \\ & ≈ 4 . 6481 a ^{3 \end{align}}
scope=row	32	Pentagonal orthocupolarotunda		25	50	27	C _{5v} of order 10	\begin{align} A & = \left( 5+ (1)/(4) \sqrt{ 1900+490 \sqrt{ 5 } +210 \sqrt{ 75+30 \sqrt{ 5 }}}\right)a ^{2} \\ & ≈ 23 . 5385 a ^{2} \\ V & = (5)/(12) \left( 11+5 \sqrt{ 5 }\right)a ^{3} \\ & ≈ 9 . 2418 a ^{3 \end{align}}
scope=row	33	Pentagonal gyrocupolarotunda		25	50	27	C _{5v} of order 10	\begin{align} A & = \left( 5+ (15)/(\sqrt{ 3 } + 4) (7)/(4) \sqrt{ 25+10 \sqrt{ 5 }}\right)a ^{2} \\ & ≈ 23 . 5385 a ^{2} \\ V & = (5)/(12) \left( 11+5 \sqrt{ 5 }\right)a ^{3} \\ & ≈ 9 . 2418 a ^{3 \end{align}}
scope=row	34	Pentagonal orthobirotunda		30	60	32	D _{5h} of order 20	\begin{align} A & = \left(( 5 \sqrt{ 3 } +3 \sqrt{ 5 \left( 5+2 \sqrt{ 5 }\right)}\right)a ^{2} \\ & ≈ 29 . 306 a ^{2} \\ V & = (1)/(6) ( 45 + 17 \sqrt{ 5 })a ^{3} \\ & ≈ 13 . 8355 a ^{3 \end{align}}
scope=row	35	Elongated triangular orthobicupola		18	36	20	D _{3h} of order 12	\begin{align} A & = 2 ( 6 + \sqrt{ 3 })a ^{2} \\ & ≈ 15 . 4641 a ^{2} \\ V & = \left( (5 \sqrt{ 2)/() } + \frac\right)a^3 \\ &\approx 4.9551a^3\end
scope=row	36	Elongated triangular gyrobicupola		18	36	20	D _{3d} of order 12	\begin{align} A & = 2 ( 6 + \sqrt{ 3 })a ^{2} \\ & ≈ 15 . 4641 a ^{2} \\ V & = \left( (5 \sqrt{ 2)/() } + \frac\right)a^3 \\ &\approx 4.9551a^3\end
scope=row	37	Elongated square gyrobicupola		24	48	26	D _{4d} of order 16	\begin{align} A & = 2 ( 9 + \sqrt{ 3 })a ^{2} \\ & ≈ 21 . 4641 a ^{2} \\ V & = \left( 4 + (10 \sqrt{ 2)/() }\right)a^3 \\ &\approx 8.714a^3\end
scope=row	38	Elongated pentagonal orthobicupola		30	60	32	D _{5h} of order 20	\begin{align} A & = \left( 20+ \sqrt{ (5)/(2) \left( 10+ \sqrt{ 5 } + \sqrt{ 75+30 \sqrt{ 5 }}\right)}\right)a ^{2} \\ & ≈ 27 . 7711 a ^{2} \\ V & = (1)/(6) \left( 10+8 \sqrt{ 5 } +15 \sqrt{ 5+2 \sqrt{ 5 }}\right)a ^{3} \\ & ≈ 12 . 3423 a ^{3 \end{align}}
scope=row	39	Elongated pentagonal gyrobicupola		30	60	32	D _{5d} of order 20	\begin{align} A & = \left( 20+ \sqrt{ (5)/(2) \left( 10+ \sqrt{ 5 } + \sqrt{ 75+30 \sqrt{ 5 }}\right)}\right)a ^{2} \\ & ≈ 27 . 7711 a ^{2} \\ V & = (1)/(6) \left( 10+8 \sqrt{ 5 } +15 \sqrt{ 5+2 \sqrt{ 5 }}\right)a ^{3} \\ & ≈ 12 . 3423 a ^{3 \end{align}}
scope=row	40	Elongated pentagonal orthocupolarotunda		35	70	37	C _{5v} of order 10	\begin{align} A & = (1)/(4) \left( 60+ \sqrt{ 10 \left( 190+49 \sqrt{ 5 } +21 \sqrt{ 75+30 \sqrt{ 5 }}\right)}\right)a ^{2} \\ & ≈ 33 . 5385 a ^{2} \\ V & = (5)/(12) \left( 11+5 \sqrt{ 5 } +6 \sqrt{ 5+2 \sqrt{ 5 }}\right)a ^{3} \\ & ≈ 16 . 936 a ^{3 \end{align}}
scope=row	41	Elongated pentagonal gyrocupolarotunda		35	70	37	C _{5v} of order 10	\begin{align} A & = (1)/(4) \left( 60+ \sqrt{ 10 \left( 190+49 \sqrt{ 5 } +21 \sqrt{ 75+30 \sqrt{ 5 }}\right)}\right)a ^{2} \\ & ≈ 33 . 5385 a ^{2} \\ V & = (5)/(12) \left( 11+5 \sqrt{ 5 } +6 \sqrt{ 5+2 \sqrt{ 5 }}\right)a ^{3} \\ & ≈ 16 . 936 a ^{3 \end{align}}
scope=row	42	Elongated pentagonal orthobirotunda		40	80	42	D _{5h} of order 20	\begin{align} A & = \left( 10+ \sqrt{ 30 \left( 10+3 \sqrt{ 5 } + \sqrt{ 75+30 \sqrt{ 5 }}\right)}\right)a ^{2} \\ & ≈ 39 . 306 a ^{2} \\ V & = (1)/(6) \left( 45+17 \sqrt{ 5 } +15 \sqrt{ 5+2 \sqrt{ 5 }}\right)a ^{3} \\ & ≈ 21 . 5297 a ^{3 \end{align}}
scope=row	43	Elongated pentagonal gyrobirotunda		40	80	42	D _{5d} of order 20	\begin{align} A & = \left( 10+ \sqrt{ 30 \left( 10+3 \sqrt{ 5 } + \sqrt{ 75+30 \sqrt{ 5 }}\right)}\right)a ^{2} \\ & ≈ 39 . 306 a ^{2} \\ V & = (1)/(6) \left( 45+17 \sqrt{ 5 } +15 \sqrt{ 5+2 \sqrt{ 5 }}\right)a ^{3} \\ & ≈ 21 . 5297 a ^{3 \end{align}}
scope=row	44	Gyroelongated triangular bicupola		18	42	26	D _{3} of order 6	\begin{align} A & = \left( 6+5 \sqrt{ 3 }\right)a ^{2} \\ & ≈ 14 . 6603 a ^{2} \\ V & = \sqrt{ 2 } \left( (5)/(3) + \sqrt{ 1+ \sqrt{ 3 }}\right) a ^{3} \\ & ≈ 4 . 6946 a ^{3 \end{align}}
scope=row	45	Gyroelongated square bicupola		24	56	34	D _{4} of order 8	\begin{align} A & = \left( 10+6 \sqrt{ 3 }\right) a ^{2} \\ & ≈ 20 . 3923 a ^{2} \\ V & = \left( 2+ (4)/(3) \sqrt{ 2 } + (2)/(3) \sqrt{ 4+2 \sqrt{ 2 } +2 \sqrt{ 146+103 \sqrt{ 2 }}}\right) a ^{3} \\ & ≈ 8 . 1536 a ^{3 \end{align}}
scope=row	46	Gyroelongated pentagonal bicupola		30	70	42	D _{5} of order 10	\begin{align} A & = (1)/(2) \left( 20+15 \sqrt{ 3 } + \sqrt{ 25+10 \sqrt{ 5 }}\right)a ^{2} \\ & ≈ 26 . 4313 a ^{2} \\ V & = \left( (5)/(+ 3) (4)/(3) \sqrt{ 5 } + (5)/(6) \sqrt{ 2 \sqrt{ 650+290 \sqrt{ 5 }} -2 \sqrt{ 5 } -2 }\right) a ^{3} \\ & ≈ 11 . 3974 a ^{3 \end{align}}
scope=row	47	Gyroelongated pentagonal cupolarotunda		35	80	47	C _{5} of order 5	\begin{align} A & = (1)/(4) \left( 20+35 \sqrt{ 3 } +7 \sqrt{ 25+10 \sqrt{ 5 }}\right)a ^{2} \\ & ≈ 32 . 1988 a ^{2} \\ V & = \left( (55)/(+ 12) (25)/(12) \sqrt{ 5 } + (5)/(6) \sqrt{ 2 \sqrt{ 650+290 \sqrt{ 5 }} -2 \sqrt{ 5 } -2 }\right) a ^{3} \\ & ≈ 15 . 9911 a ^{3 \end{align}}
scope=row	48	Gyroelongated pentagonal birotunda		40	90	52	D _{5} of order 10	\begin{align} A & = \left( 10 \sqrt{ 3 } + 3 \sqrt{ 25+10 \sqrt{ 5 }}\right) a ^{2} \\ & ≈ 37 . 9662 a ^{2} \\ V & = \left( (45)/(+ 6) (17)/(6) \sqrt{ 5 } + (5)/(6) \sqrt{ 2 \sqrt{ 650+290 \sqrt{ 5 }} -2 \sqrt{ 5 } -2 }\right) a ^{3} \\ & ≈ 20 . 5848 a ^{3 \end{align}}
scope=row	49	Augmented triangular prism		7	13	8	C _{2v} of order 4	\begin{align} A & = (1)/(2) ( 4 + 3 \sqrt{ 3 })a ^{2} \\ & ≈ 4 . 5981 a ^{2} \\ V & = (1)/(12) ( 2 \sqrt{ 2 } + 3 \sqrt{ 3 })a ^{3} \\ & ≈ 0 . 6687 a ^{3 \end{align}}
scope=row	50	Biaugmented triangular prism		8	17	11	C _{2v} of order 4	\begin{align} A & = (1)/(2) ( 2 + 5 \sqrt{ 3 })a ^{2} \\ & ≈ 5 . 3301 a ^{2} \\ V & = \sqrt{ (59)/(144) + (1)/(\sqrt{ 6) }}a^3 \\ &\approx 0.9044a^3\end
scope=row	51	Triaugmented triangular prism		9	21	14	D _{3h} of order 12	\begin{align} A & = (7 \sqrt{ 3)/() }a^2 \\ &\approx 6.0622a^2 \\ V &= \fraca^3 \\ &\approx 1.1401a^3\end
scope=row	52	Augmented pentagonal prism		11	19	10	C _{2v} of order 4	\begin{align} A & = (1)/(2) \left( 8+2 \sqrt{ 3 } + \sqrt{ 5 \left( 5+2 \sqrt{ 5 }\right)}\right)a ^{2} \\ & ≈ 9 . 173 a ^{2} \\ V & = (1)/(12) \sqrt{ 233+90 \sqrt{ 5 } +12 \sqrt{ 50+20 \sqrt{ 5 }}}a ^{3} \\ & ≈ 1 . 9562 a ^{3 \end{align}}
scope=row	53	Biaugmented pentagonal prism		12	23	13	C _{2v} of order 4	\begin{align} A & = (1)/(2) a ^{2} \left( 6+4 \sqrt{ 3 } + \sqrt{ 5 \left( 5+2 \sqrt{ 5 }\right)}\right) \\ & ≈ 9 . 9051 a ^{2} \\ V & = (1)/(12) a ^{3} \sqrt{ 257+90 \sqrt{ 5 } +24 \sqrt{ 50+20 \sqrt{ 5 }}} \\ & ≈ 2 . 1919 a ^{3 \end{align}}
scope=row	54	Augmented hexagonal prism		13	22	11	C _{2v} of order 4	\begin{align} A & = ( 5+4 \sqrt{ 3 })a ^{2} \\ & ≈ 11 . 9282 a ^{2} \\ V & = (1)/(6) \left(\sqrt{ 2 } +9 \sqrt{ 3 }\right)a ^{3} \\ & ≈ 2 . 8338 a ^{3 \end{align}}
scope=row	55	Parabiaugmented hexagonal prism		14	26	14	D _{2h} of order 8	\begin{align} A & = ( 4+5 \sqrt{ 3 })a ^{2} \\ & ≈ 12 . 6603 a ^{2} \\ V & = (1)/(6) \left( 2 \sqrt{ 2 } +9 \sqrt{ 3 }\right)a ^{3} \\ & ≈ 3 . 0695 a ^{3 \end{align}}
scope=row	56	Metabiaugmented hexagonal prism		14	26	14	C _{2v} of order 4	\begin{align} A & = ( 4+5 \sqrt{ 3 })a ^{2} \\ & ≈ 12 . 6603 a ^{2} \\ V & = (1)/(6) \left( 2 \sqrt{ 2 } +9 \sqrt{ 3 }\right)a ^{3} \\ & ≈ 3 . 0695 a ^{3 \end{align}}
scope=row	57	Triaugmented hexagonal prism		15	30	17	D _{3h} of order 12	\begin{align} A & = 3 \left( 1+2 \sqrt{ 3 }\right)a ^{2} \\ & ≈ 13 . 3923 a ^{2} \\ V & = \left( (1)/(\sqrt{ 2) }+\frac\right)a^3 \\ &\approx 3.3052a^3\end
scope=row	58	Augmented dodecahedron		21	35	16	C _{5v} of order 10	\begin{align} A & = (1)/(4) \left( 5 \sqrt{ 3 } +11 \sqrt{ 5 \left( 5+2 \sqrt{ 5 }\right)}\right)a ^{2} \\ & ≈ 21 . 0903 a ^{2} \\ V & = (1)/(24) \left( 95+43 \sqrt{ 5 }\right)a ^{3} \\ & ≈ 7 . 9646 a ^{3 \end{align}}
scope=row	59	Parabiaugmented dodecahedron		22	40	20	D _{5d} of order 20	\begin{align} A & = (5)/(2) \left(\sqrt{ 3 } + \sqrt{ 5 \left( 5+2 \sqrt{ 5 }\right)}\right)a ^{2} \\ & ≈ 21 . 5349 a ^{2} \\ V & = (1)/(6) \left( 25+11 \sqrt{ 5 }\right)a ^{3} \\ & ≈ 8 . 2661 a ^{3 \end{align}}
scope=row	60	Metabiaugmented dodecahedron		22	40	20	C _{2v} of order 4	\begin{align} A & = (5)/(2) \left(\sqrt{ 3 } + \sqrt{ 5 \left( 5+2 \sqrt{ 5 }\right)}\right)a ^{2} \\ & ≈ 21 . 5349 a ^{2} \\ V & = (1)/(6) \left( 25+11 \sqrt{ 5 }\right)a ^{3} \\ & ≈ 8 . 2661 a ^{3 \end{align}}
scope=row	61	Triaugmented dodecahedron		23	45	24	C _{3v} of order 6	\begin{align} A & = (3)/(4) \left( 5 \sqrt{ 3 } +3 \sqrt{ 5 \left( 5+2 \sqrt{ 5 }\right)}\right)a ^{2} \\ & ≈ 21 . 9795 a ^{2} \\ V & = (5)/(8) \left( 7+3 \sqrt{ 5 }\right)a ^{3} \\ & ≈ 8 . 5676 a ^{3 \end{align}}
scope=row	62	Metabidiminished icosahedron		10	20	12	C _{2v} of order 4	\begin{align} A & = (1)/(2) \left( 5 \sqrt{ 3 } + \sqrt{ 5 \left( 5+2 \sqrt{ 5 }\right)}\right)a ^{2} \\ & ≈ 7 . 7711 a ^{2} \\ V & = (1)/(6) \left( 5+2 \sqrt{ 5 }\right)a ^{3} \\ & ≈ 1 . 5787 a ^{3 \end{align}}
scope=row	63	Tridiminished icosahedron		9	15	8	C _{3v} of order 6	\begin{align} A & = (1)/(4) \left( 5 \sqrt{ 3 } +3 \sqrt{ 5 \left( 5+2 \sqrt{ 5 }\right)}\right)a ^{3} \\ & ≈ 7 . 3265 a ^{3} \\ V & = \left( (5)/(+ 8) (7 \sqrt{ 5)/() }\right)a^3 \\ &\approx 1.2772a^3\end
scope=row	64	Augmented tridiminished icosahedron		10	18	10	C _{3v} of order 6	\begin{align} A & = (1)/(4) \left( 7 \sqrt{ 3 } +3 \sqrt{ 5 \left( 5+2 \sqrt{ 5 }\right)}\right)a ^{2} \\ & ≈ 8 . 1925 a ^{2} \\ V & = (1)/(24) \left( 15+2 \sqrt{ 2 } +7 \sqrt{ 5 }\right)a ^{3} \\ & ≈ 1 . 395 a ^{3 \end{align}}
scope=row	65	Augmented truncated tetrahedron		15	27	14	C _{3v} of order 6	\begin{align} A & = (1)/(2) \left( 6+13 \sqrt{ 3 }\right)a ^{2} \\ & ≈ 14 . 2583 a ^{2} \\ V & = (11)/(2 \sqrt{ 2) }a^3 \\ &\approx 3.8891a^3\end
scope=row	66	Augmented truncated cube		28	48	22	C _{4v} of order 8	\begin{align} A & = ( 15+10 \sqrt{ 2 } +3 \sqrt{ 3 })a ^{2} \\ & ≈ 34 . 3383 a ^{2} \\ V & = \left( 8+ (16 \sqrt{ 2)/() }\right)a^3 \\ &\approx 15.5425a^3\end
scope=row	67	Biaugmented truncated cube		32	60	30	D _{4h} of order 16	\begin{align} A & = 2 \left( 9+4 \sqrt{ 2 } +2 \sqrt{ 3 }\right)a ^{2} \\ & ≈ 36 . 2419 a ^{2} \\ V & = ( 9+6 \sqrt{ 2 })a ^{3} \\ & ≈ 17 . 4853 a ^{3 \end{align}}
scope=row	68	Augmented truncated dodecahedron		65	105	42	C _{5v} of order 10	\begin{align} A & = (1)/(4) \left( 20+25 \sqrt{ 3 } +110 \sqrt{ 5+2 \sqrt{ 5 }} + \sqrt{ 5 \left( 5+2 \sqrt{ 5 }\right)}\right)a ^{2} \\ & ≈ 102 . 1821 a ^{2} \\ V & = \left( (505)/(+ 12) (81 \sqrt{ 5)/() }\right)a^3 \\ &\approx 87.3637a^3\end
scope=row	69	Parabiaugmented truncated dodecahedron		70	120	52	D _{5d} of order 20	\begin{align} A & = (1)/(2) \left( 20+15 \sqrt{ 3 } +50 \sqrt{ 5+2 \sqrt{ 5 }} + \sqrt{ 5 \left( 5+2 \sqrt{ 5 }\right)}\right)a ^{2} \\ & ≈ 103 . 3734 a ^{2} \\ V & = (1)/(12) \left( 515+251 \sqrt{ 5 }\right)a ^{3} \\ & ≈ 89 . 6878 a ^{3 \end{align}}
scope=row	70	Metabiaugmented truncated dodecahedron		70	120	52	C _{2v} of order 4	\begin{align} A & = (1)/(2) \left( 20+15 \sqrt{ 3 } +50 \sqrt{ 5+2 \sqrt{ 5 }} + \sqrt{ 5 \left( 5+2 \sqrt{ 5 }\right)}\right)a ^{2} \\ & ≈ 103 . 3734 a ^{2} \\ V & = (1)/(12) \left( 515+251 \sqrt{ 5 }\right)a ^{3} \\ & ≈ 89 . 6878 a ^{3 \end{align}}
scope=row	71	Triaugmented truncated dodecahedron		75	135	62	C _{3v} of order 6	\begin{align} A & = (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} \\ & ≈ 104 . 5648 a ^{2} \\ V & = (7)/(12) \left( 75+37 \sqrt{ 5 }\right)a ^{3} \\ & ≈ 92 . 0118 a ^{3 \end{align}}
scope=row	72	Gyrate rhombicosidodecahedron		60	120	62	C _{5v} of order 10	\begin{align} A & = \left( 30+5 \sqrt{ 3 } +3 \sqrt{ 5 \left( 5+2 \sqrt{ 5 }\right)}\right)a ^{2} \\ & ≈ 59 . 306 a ^{2} \\ V & = \left( 20+ (29 \sqrt{ 5)/() }\right)a^3 \\ &\approx 41.6153a^3\end
scope=row	73	Parabigyrate rhombicosidodecahedron		60	120	62	D _{5d} of order 20	\begin{align} A & = \left( 30+5 \sqrt{ 3 } +3 \sqrt{ 5 \left( 5+2 \sqrt{ 5 }\right)}\right)a ^{2} \\ & ≈ 59 . 306 a ^{2} \\ V & = \left( 20+ (29 \sqrt{ 5)/() }\right)a^3 \\ &\approx 41.6153a^3\end
scope=row	74	Metabigyrate rhombicosidodecahedron		60	120	62	C _{2v} of order 4	\begin{align} A & = \left( 30+5 \sqrt{ 3 } +3 \sqrt{ 5 \left( 5+2 \sqrt{ 5 }\right)}\right)a ^{2} \\ & ≈ 59 . 306 a ^{2} \\ V & = \left( 20+ (29 \sqrt{ 5)/() }\right)a^3 \\ &\approx 41.6153a^3\end
scope=row	75	Trigyrate rhombicosidodecahedron		60	120	62	C _{3v} of order 6	\begin{align} A & = \left( 30+5 \sqrt{ 3 } +3 \sqrt{ 5 \left( 5+2 \sqrt{ 5 }\right)}\right)a ^{2} \\ & ≈ 59 . 306 a ^{2} \\ V & = \left( 20+ (29 \sqrt{ 5)/() }\right)a^3 \\ &\approx 41.6153a^3\end
scope=row	76	Diminished rhombicosidodecahedron		55	105	52	C _{5v} of order 10	\begin{align} A & = (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} \\ & ≈ 58 . 1147 a ^{2} \\ V & = \left( (115)/(6) +9 \sqrt{ 5 }\right)a ^{3} \\ & ≈ 39 . 2913 a ^{3 \end{align}}
scope=row	77	Paragyrate diminished rhombicosidodecahedron		55	105	52	C _{5v} of order 10	\begin{align} A & = (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} \\ & ≈ 58 . 1147 a ^{2} \\ V & = \left( (115)/(6) +9 \sqrt{ 5 }\right)a ^{3} \\ & ≈ 39 . 2913 a ^{3 \end{align}}
scope=row	78	Metagyrate diminished rhombicosidodecahedron		55	105	52	C _{s} of order 2	\begin{align} A & = (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} \\ & ≈ 58 . 1147 a ^{2} \\ V & = \left( (115)/(6) +9 \sqrt{ 5 }\right)a ^{3} \\ & ≈ 39 . 2913 a ^{3 \end{align}}
scope=row	79	Bigyrate diminished rhombicosidodecahedron		55	105	52	C _{s} of order 2	\begin{align} A & = (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} \\ & ≈ 58 . 1147 a ^{2} \\ V & = \left( (115)/(6) +9 \sqrt{ 5 }\right)a ^{3} \\ & ≈ 39 . 2913 a ^{3 \end{align}}
scope=row	80	Parabidiminished rhombicosidodecahedron		50	90	42	D _{5d} of order 20	\begin{align} A & = (5)/(2) \left( 8+ \sqrt{ 3 } +2 \sqrt{ 5+2 \sqrt{ 5 }} + \sqrt{ 5 \left( 5+2 \sqrt{ 5 }\right)}\right)a ^{2} \\ & ≈ 56 . 9233 a ^{2} \\ V & = (5)/(3) \left( 11+5 \sqrt{ 5 }\right)a ^{3} \\ & ≈ 36 . 9672 a ^{3 \end{align}}
scope=row	81	Metabidiminished rhombicosidodecahedron		50	90	42	C _{2v} of order 4	\begin{align} A & = (5)/(2) \left( 8+ \sqrt{ 3 } +2 \sqrt{ 5+2 \sqrt{ 5 }} + \sqrt{ 5 \left( 5+2 \sqrt{ 5 }\right)}\right)a ^{2} \\ & ≈ 56 . 9233 a ^{2} \\ V & = (5)/(3) \left( 11+5 \sqrt{ 5 }\right)a ^{3} \\ & ≈ 36 . 9672 a ^{3 \end{align}}
scope=row	82	Gyrate bidiminished rhombicosidodecahedron		50	90	42	C _{s} of order 2	\begin{align} A & = (5)/(2) \left( 8+ \sqrt{ 3 } +2 \sqrt{ 5+2 \sqrt{ 5 }} + \sqrt{ 5 \left( 5+2 \sqrt{ 5 }\right)}\right)a ^{2} \\ & ≈ 56 . 9233 a ^{2} \\ V & = (5)/(3) \left( 11+5 \sqrt{ 5 }\right)a ^{3} \\ & ≈ 36 . 9672 a ^{3 \end{align}}
scope=row	83	Tridiminished rhombicosidodecahedron		45	75	32	C _{3v} of order 6	\begin{align} A & = (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} \\ & ≈ 55 . 732 a ^{2} \\ V & = \left( (35)/(+ 2) (23 \sqrt{ 5)/() }\right)a^3 \\ &\approx 34.6432a^3\end
scope=row	84	Snub disphenoid		8	18	12	D _{2d} of order 8	\begin{align} A & = 3 \sqrt{ 3 }a ^{2} \\ & ≈ 5 . 1962 a ^{2} \\ V & ≈ 0 . 8595 a ^{3 \end{align}}
scope=row	85	Snub square antiprism		16	40	26	D _{4d} of order 16	\begin{align} A & = 2 \left( 1+3 \sqrt{ 3 }\right)a ^{2} \\ & ≈ 12 . 3923 a ^{2} \\ V & ≈ 3 . 6012 a ^{3 \end{align}}
scope=row	86	Sphenocorona		10	22	14	C _{2v} of order 4	\begin{align} A & = ( 2+3 \sqrt{ 3 })a ^{2} \\ & ≈ 7 . 1962 a ^{2} \\ V & = (1)/(2) a ^{3} \sqrt{ 1+3 \sqrt{ (3)/(2) }+\sqrt} \\ &\approx 1.5154a^3\end
scope=row	87	Augmented sphenocorona		11	26	17	C _{s} of order 2	\begin{align} A & = ( 1+4 \sqrt{ 3 })a ^{2} \\ & ≈ 7 . 9282 a ^{2} \\ V & = (1)/(2) a ^{3\sqrt{1} + 3 \sqrt{ (3)/(2) } + \sqrt}+\frac \\ &\approx 1.7511a^3\end
scope=row	88	Sphenomegacorona		12	28	18	C _{2v} of order 4	\begin{align} A & = 2 \left( 1+2 \sqrt{ 3 }\right)a ^{2} \\ & ≈ 8 . 9282 a ^{2} \\ V & ≈ 1 . 9481 a ^{3 \end{align}}
scope=row	89	Hebesphenomegacorona		14	33	21	C _{2v} of order 4	\begin{align} A & = (3)/(2) \left( 2+3 \sqrt{ 3 }\right)a ^{2} \\ & ≈ 10 . 7942 a ^{2} \\ V & ≈ 2 . 9129 a ^{3 \end{align}}
scope=row	90	Disphenocingulum		16	38	24	D _{2d} of order 8	\begin{align} A & = ( 4+5 \sqrt{ 3 })a ^{2} \\ & ≈ 12 . 6603 a ^{2} \\ V & ≈ 3 . 7776 a ^{3 \end{align}}
scope=row	91	Bilunabirotunda		14	26	14	D _{2h} of order 8	\begin{align} A & = \left( 2+2 \sqrt{ 3 } + \sqrt{ 5 \left( 5+2 \sqrt{ 5 }\right)}\right)a ^{2} \\ & ≈ 12 . 346 a ^{2} \\ V & = (1)/(12) \left( 17+9 \sqrt{ 5 }\right)a ^{3} \\ & ≈ 3 . 0937 a ^{3 \end{align}}
scope=row	92	Triangular hebesphenorotunda		18	36	20	C _{3v} of order 6	\begin{align} A & = (1)/(4) \left( 12+19 \sqrt{ 3 } +3 \sqrt{ 5 \left( 5+2 \sqrt{ 5 }\right)}\right)a ^{2} \\ & ≈ 16 . 3887 a ^{2} \\ V & = \left( (5)/(+ 2) (7 \sqrt{ 5)/() }\right)a^3 \\ &\approx 5.1087a^3\end

Bibliography

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External links

• Web site: Hart . George W. . George W. Hart . Johnson Solids .
• Web site: Johnson Polyhedra . Images of all 92 categorised Johnson solids
• Web site: Johnson Solids. Visualisations of all 92 Johnson solids
• Web site: Johnson solids . Bulatov . Vladimir. VRML models of Johnson solids
• Gagnon . Sylvain . Les polyèdres convexes aux faces régulières . Convex polyhedra with regular faces . Topologie Structurale . Structural Topology . 6 . 1982 . 83–95 . fr.