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

. 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

, and volume

. 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

\frac

is denoted by

C_n

, a cyclic group of order

; combining this with the reflection symmetry results in the symmetry of dihedral group

D_n

of order

. 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

. 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

. The antiprismatic symmetry

D_n

of order

preserves the symmetry by rotating its half bottom and reflection across the horizontal plane. The symmetry group

C_n

of order

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

C₁

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

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Faces

scope=col

Symmetry group and its order

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Surface area and volume

scope=row

Equilateral square pyramid

C_4v

of order 8

\begin{align} A&=\left(1+\sqrt{3}\right)a²\\ & ≈ 2.7321a²\\ V&=

	\sqrt{2

}a^3 \\ &\approx 0.2357a^3 \end

scope=row

Pentagonal pyramid

C_5v

of order 10

\begin{align} A&=

	a²	\sqrt{
	2

	5
	2

\left(10+\sqrt{5}+\sqrt{75+30\sqrt{5}}\right)}\\ & ≈ 3.8855a²\\ V&=\left(

	5+\sqrt{5

}\right)a^3 \\ &\approx 0.3015a^3 \end

scope=row

Triangular cupola

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

Square cupola

C_4v

of order 8

\begin{align} A&=\left(7+2\sqrt{2}+\sqrt{3}\right)a²\\ & ≈ 11.5605a²\\ V&=\left(1+

	2\sqrt{2

}\right)a^3 \\ &\approx 1.9428a^3 \end

scope=row

Pentagonal cupola

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)a²\\ & ≈ 16.5798a²\\ V&=\left(

	1
	6

\left(5+4\sqrt{5}\right)\right)a³\\ & ≈ 2.3241a³\end{align}

scope=row

Pentagonal rotunda

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)a²\\ & ≈ 22.3472a²\\ V&=\left(

	1
	12

\left(45+17\sqrt{5}\right)\right)a³\\ & ≈ 6.9178a³\end{align}

scope=row

Elongated triangular pyramid

C_3v

of order 6

\begin{align} A&=\left(3+\sqrt{3}\right)a²\\ & ≈ 4.7321a²\\ V&=\left(

	1
	12

\left(\sqrt{2}+3\sqrt{3}\right)\right)a³\\ & ≈ 0.5509a^{3
\end{align}}

scope=row

Elongated square pyramid

C_4v

of order 8

\begin{align} A&=\left(5+\sqrt{3}\right)a²\\ & ≈ 6.7321a²\\ V&=\left(1+

	\sqrt{2

}\right)a^3 \\ &\approx 1.2357a^3 \end

scope=row

Elongated pentagonal pyramid

C_5v

of order 10

\begin{align} A&=

	20+5\sqrt{3
	+

\sqrt{25+10\sqrt{5}}}{4}a²\\ & ≈ 8.8855a²\\ V&=\left(

	5+\sqrt{5
	+

6\sqrt{25+10\sqrt{5}}}{24}\right)a³\\ & ≈ 2.022a³\end{align}

scope=row

Gyroelongated square pyramid

C_4v

of order 8

\begin{align} A&=(1+3\sqrt{3})a²\\ & ≈ 6.1962a²\\ V&=

	1
	6

\left(\sqrt{2}+2\sqrt{4+3\sqrt{2}}\right)a³\\ & ≈ 1.1927a^{3
\end{align}}

scope=row

Gyroelongated pentagonal pyramid

C_5v

of order 10

\begin{align} A&=

	1
	4

\left(15\sqrt{3}+\sqrt{5\left(5+2\sqrt{5}\right)}\right)a²\\ & ≈ 8.2157a²\\ V&=

	1
	24

\left(25+9\sqrt{5}\right)a³\\ & ≈ 1.8802a^{3
\end{align}}

scope=row

Triangular bipyramid

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

Pentagonal bipyramid

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

Elongated triangular bipyramid

D_3h

of order 12

\begin{align} A&=

	3
	2

\left(2+\sqrt{3}\right)a²\\ & ≈ 5.5981a²\\ V&=

	1
	12

\left(2\sqrt{2}+3\sqrt{3}\right)a³\\ & ≈ 0.6687a^{3
\end{align}}

scope=row

Elongated square bipyramid

D_4h

of order 16

\begin{align} A&=2\left(2+\sqrt{3}\right)a²\\ & ≈ 7.4641a²\\ V&=

	1
	3

\left(3+\sqrt{2}\right)a³\\ & ≈ 1.4714a^{3
\end{align}}

scope=row

Elongated pentagonal bipyramid

D_5h

of order 20

\begin{align} A&=

	5
	2

\left(2+\sqrt{3}\right)a²\\ & ≈ 9.3301a²\\ V&=

	1
	12

\left(5+\sqrt{5}+3\sqrt{5\left(5+2\sqrt{5}\right)}\right)a³\\ & ≈ 2.3235a^{3
\end{align}}

scope=row

Gyroelongated square bipyramid

D_4d

of order 16

\begin{align} A&=4\sqrt{3}a²\\ & ≈ 6.9282a²\\ V&=

	1
	3

\left(\sqrt{2}+\sqrt{4+3\sqrt{2}}\right)a³\\ & ≈ 1.4284a^{3
\end{align}}

scope=row

Elongated triangular cupola

C_3v

of order 6

\begin{align} A&=

	1
	2

\left(18+5\sqrt{3}\right)a²\\ & ≈ 13.3301a²\\ V&=

	1
	6

\left(5\sqrt{2}+9\sqrt{3}\right)a³\\ & ≈ 3.7766a^{3
\end{align}}

scope=row

Elongated square cupola

C_4v

of order 8

\begin{align} A&=(15+2\sqrt{2}+\sqrt{3})a^2\\& ≈ 19.5605a²\\ V&=\left(3+

	8\sqrt{2

}\right)a^3 \\ &\approx 6.7712a^3\end

scope=row

Elongated pentagonal cupola

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)a²\\ & ≈ 26.5798a²\\ V&=

	1
	6

\left(5+4\sqrt{5}+15\sqrt{5+2\sqrt{5}}\right)a³\\ & ≈ 10.0183a^{3
\end{align}}

scope=row

Elongated pentagonal rotunda

C_5v

of order 10

\begin{align} A&=

	1
	2

a²\left(20+5\sqrt{3}+5\sqrt{5+2\sqrt{5}}+3\sqrt{5\left(5+2\sqrt{5}\right)}\right)\\ & ≈ 32.3472a²\\ V&=

	1
	12

a³\left(45+17\sqrt{5}+30\sqrt{5+2\sqrt{5}}\right)\\ & ≈ 14.612a^{3
\end{align}}

scope=row

Gyroelongated triangular cupola

C_3v

of order 6

\begin{align} A&=

	1
	2

\left(6+11\sqrt{3}\right)a²\\ & ≈ 12.5263a²\\ V&=

	1	\sqrt{
	3

	61
	2

+18\sqrt{3}+30\sqrt{1+\sqrt{3}}}a³\\ & ≈ 3.5161a^{3
\end{align}}

scope=row

Gyroelongated square cupola

C_4v

of order 8

\begin{align} A&=(7+2\sqrt{2}+5\sqrt{3})a²\\ & ≈ 18.4887a²\\ V&=\left(1+

	2
	3

\sqrt{2}+

	2
	3

\sqrt{4+2\sqrt{2}+2\sqrt{146+103\sqrt{2}}}\right)a³\\ & ≈ 6.2108a^{3
\end{align}}

scope=row

Gyroelongated pentagonal cupola

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)a²\\ & ≈ 25.2400a²\\ V&=\left(

	5	+
	6

	2
	3

\sqrt{5}+

	5
	6

\sqrt{2\sqrt{650+290\sqrt{5}}-2\sqrt{5}-2}\right)a³\\ & ≈ 9.0733a^{3
\end{align}}

scope=row

Gyroelongated pentagonal rotunda

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)a²\\ & ≈ 31.0075a²\\ V&=\left(

	45	+
	12

	17
	12

\sqrt{5}+

	5
	6

\sqrt{2\sqrt{650+290\sqrt{5}}-2\sqrt{5}-2}\right)a³\\ & ≈ 13.6671a^{3
\end{align}}

scope=row

Gyrobifastigium

D_2d

of order 8

\begin{align} A&=\left(4+\sqrt{3}\right)a²\\ & ≈ 5.7321a²\\ V&=\left(

	\sqrt{3

}\right)a^3 \\ &\approx 0.866a^3\end

scope=row

Triangular orthobicupola

D_3h

of order 12

\begin{align} A&=2\left(3+\sqrt{3}\right)a²\\ & ≈ 9.4641a²\\ V&=

	5\sqrt{2

}a^3 \\ &\approx 2.357a^3\end

scope=row

Square orthobicupola

D_4h

of order 16

\begin{align} A&=2(5+\sqrt{3})a²\\ & ≈ 13.4641a²\\ V&=\left(2+

	4\sqrt{2

}\right)a^3 \\ &\approx 3.8856a^3\end

scope=row

Square gyrobicupola

D_4d

of order 16

\begin{align} A&=2(5+\sqrt{3})a²\\ & ≈ 13.4641a²\\ V&=\left(2+

	4\sqrt{2

}\right)a^3 \\ &\approx 3.8856a^3\end

scope=row

Pentagonal orthobicupola

D_5h

of order 20

\begin{align} A&=\left(10+\sqrt{

	5
	2

\left(10+\sqrt{5}+\sqrt{75+30\sqrt{5}}\right)}\right)a²\\ & ≈ 17.7711a²\\ V&=

	1
	3

\left(5+4\sqrt{5}\right)a³\\ & ≈ 4.6481a^{3
\end{align}}

scope=row

Pentagonal gyrobicupola

D_5d

of order 20

\begin{align} A&=\left(10+\sqrt{

	5
	2

\left(10+\sqrt{5}+\sqrt{75+30\sqrt{5}}\right)}\right)a²\\ & ≈ 17.7711a²\\ V&=

	1
	3

\left(5+4\sqrt{5}\right)a³\\ & ≈ 4.6481a^{3
\end{align}}

scope=row

Pentagonal orthocupolarotunda

C_5v

of order 10

\begin{align} A&=\left(5+

	1
	4

\sqrt{1900+490\sqrt{5}+210\sqrt{75+30\sqrt{5}}}\right)a²\\ & ≈ 23.5385a²\\ V&=

	5
	12

\left(11+5\sqrt{5}\right)a³\\ & ≈ 9.2418a^{3
\end{align}}

scope=row

Pentagonal gyrocupolarotunda

C_5v

of order 10

\begin{align} A&=\left(5+

	15	\sqrt{3}+
	4

	7
	4

\sqrt{25+10\sqrt{5}}\right)a²\\ & ≈ 23.5385a²\\ V&=

	5
	12

\left(11+5\sqrt{5}\right)a³\\ & ≈ 9.2418a^{3
\end{align}}

scope=row

Pentagonal orthobirotunda

D_5h

of order 20

\begin{align} A&=\left((5\sqrt{3}+3\sqrt{5\left(5+2\sqrt{5}\right)}\right)a²\\ & ≈ 29.306a²\\ V&=

	1
	6

(45+17\sqrt{5})a³\\ & ≈ 13.8355a^{3
\end{align}}

scope=row

Elongated triangular orthobicupola

D_3h

of order 12

\begin{align} A&=2(6+\sqrt{3})a²\\ & ≈ 15.4641a²\\ V&=\left(

	5\sqrt{2

} + \frac\right)a^3 \\ &\approx 4.9551a^3\end

scope=row

Elongated triangular gyrobicupola

D_3d

of order 12

\begin{align} A&=2(6+\sqrt{3})a²\\ & ≈ 15.4641a²\\ V&=\left(

	5\sqrt{2

} + \frac\right)a^3 \\ &\approx 4.9551a^3\end

scope=row

Elongated square gyrobicupola

D_4d

of order 16

\begin{align} A&=2(9+\sqrt{3})a²\\ & ≈ 21.4641a²\\ V&=\left(4+

	10\sqrt{2

}\right)a^3 \\ &\approx 8.714a^3\end

scope=row

Elongated pentagonal orthobicupola

D_5h

of order 20

\begin{align} A&=\left(20+\sqrt{

	5
	2

\left(10+\sqrt{5}+\sqrt{75+30\sqrt{5}}\right)}\right)a²\\ & ≈ 27.7711a²\\ V&=

	1
	6

\left(10+8\sqrt{5}+15\sqrt{5+2\sqrt{5}}\right)a³\\ & ≈ 12.3423a^{3
\end{align}}

scope=row

Elongated pentagonal gyrobicupola

D_5d

of order 20

\begin{align} A&=\left(20+\sqrt{

	5
	2

\left(10+\sqrt{5}+\sqrt{75+30\sqrt{5}}\right)}\right)a²\\ & ≈ 27.7711a²\\ V&=

	1
	6

\left(10+8\sqrt{5}+15\sqrt{5+2\sqrt{5}}\right)a³\\ & ≈ 12.3423a^{3
\end{align}}

scope=row

Elongated pentagonal orthocupolarotunda

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)a²\\ & ≈ 33.5385a²\\ V&=

	5
	12

\left(11+5\sqrt{5}+6\sqrt{5+2\sqrt{5}}\right)a³\\ & ≈ 16.936a^{3
\end{align}}

scope=row

Elongated pentagonal gyrocupolarotunda

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)a²\\ & ≈ 33.5385a²\\ V&=

	5
	12

\left(11+5\sqrt{5}+6\sqrt{5+2\sqrt{5}}\right)a³\\ & ≈ 16.936a^{3
\end{align}}

scope=row

Elongated pentagonal orthobirotunda

D_5h

of order 20

\begin{align} A&=\left(10+\sqrt{30\left(10+3\sqrt{5}+\sqrt{75+30\sqrt{5}}\right)}\right)a²\\ & ≈ 39.306a²\\ V&=

	1
	6

\left(45+17\sqrt{5}+15\sqrt{5+2\sqrt{5}}\right)a³\\ & ≈ 21.5297a^{3
\end{align}}

scope=row

Elongated pentagonal gyrobirotunda

D_5d

of order 20

\begin{align} A&=\left(10+\sqrt{30\left(10+3\sqrt{5}+\sqrt{75+30\sqrt{5}}\right)}\right)a²\\ & ≈ 39.306a²\\ V&=

	1
	6

\left(45+17\sqrt{5}+15\sqrt{5+2\sqrt{5}}\right)a³\\ & ≈ 21.5297a^{3
\end{align}}

scope=row

Gyroelongated triangular bicupola

D₃

of order 6

\begin{align} A&=\left(6+5\sqrt{3}\right)a²\\ & ≈ 14.6603a²\\ V&=\sqrt{2}\left(

	5
	3

+\sqrt{1+\sqrt{3}}\right)a³\\ & ≈ 4.6946a^{3
\end{align}}

scope=row

Gyroelongated square bicupola

D₄

of order 8

\begin{align} A&=\left(10+6\sqrt{3}\right)a²\\ & ≈ 20.3923a²\\ V&=\left(2+

	4
	3

\sqrt{2}+

	2
	3

\sqrt{4+2\sqrt{2}+2\sqrt{146+103\sqrt{2}}}\right)a³\\ & ≈ 8.1536a^{3
\end{align}}

scope=row

Gyroelongated pentagonal bicupola

D₅

of order 10

\begin{align} A&=

	1
	2

\left(20+15\sqrt{3}+\sqrt{25+10\sqrt{5}}\right)a²\\ & ≈ 26.4313a²\\ V&=\left(

	5	+
	3

	4
	3

\sqrt{5}+

	5
	6

\sqrt{2\sqrt{650+290\sqrt{5}}-2\sqrt{5}-2}\right)a³\\ & ≈ 11.3974a^{3
\end{align}}

scope=row

Gyroelongated pentagonal cupolarotunda

C₅

of order 5

\begin{align} A&=

	1
	4

\left(20+35\sqrt{3}+7\sqrt{25+10\sqrt{5}}\right)a²\\ & ≈ 32.1988a²\\ V&=\left(

	55	+
	12

	25
	12

\sqrt{5}+

	5
	6

\sqrt{2\sqrt{650+290\sqrt{5}}-2\sqrt{5}-2}\right)a³\\ & ≈ 15.9911a^{3
\end{align}}

scope=row

Gyroelongated pentagonal birotunda

D₅

of order 10

\begin{align} A&=\left(10\sqrt{3}+3\sqrt{25+10\sqrt{5}}\right)a²\\ & ≈ 37.9662a²\\ V&=\left(

	45	+
	6

	17
	6

\sqrt{5}+

	5
	6

\sqrt{2\sqrt{650+290\sqrt{5}}-2\sqrt{5}-2}\right)a³\\ & ≈ 20.5848a^{3
\end{align}}

scope=row

Augmented triangular prism

C_2v

of order 4

\begin{align} A&=

	1
	2

(4+3\sqrt{3})a²\\ & ≈ 4.5981a²\\ V&=

	1
	12

(2\sqrt{2}+3\sqrt{3})a³\\ & ≈ 0.6687a^{3
\end{align}}

scope=row

Biaugmented triangular prism

C_2v

of order 4

\begin{align} A&=

	1
	2

(2+5\sqrt{3})a²\\ & ≈ 5.3301a²\\ V&=\sqrt{

	59
	144

	1
	\sqrt{6

}}a^3 \\ &\approx 0.9044a^3\end

scope=row

Triaugmented triangular prism

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

Augmented pentagonal prism

C_2v

of order 4

\begin{align} A&=

	1
	2

\left(8+2\sqrt{3}+\sqrt{5\left(5+2\sqrt{5}\right)}\right)a²\\ & ≈ 9.173a²\\ V&=

	1
	12

\sqrt{233+90\sqrt{5}+12\sqrt{50+20\sqrt{5}}}a³\\ & ≈ 1.9562a^{3
\end{align}}

scope=row

Biaugmented pentagonal prism

C_2v

of order 4

\begin{align} A&=

	1
	2

a²\left(6+4\sqrt{3}+\sqrt{5\left(5+2\sqrt{5}\right)}\right)\\ & ≈ 9.9051a²\\ V&=

	1
	12

a³\sqrt{257+90\sqrt{5}+24\sqrt{50+20\sqrt{5}}}\\ & ≈ 2.1919a^{3
\end{align}}

scope=row

Augmented hexagonal prism

C_2v

of order 4

\begin{align} A&=(5+4\sqrt{3})a²\\ & ≈ 11.9282a²\\ V&=

	1
	6

\left(\sqrt{2}+9\sqrt{3}\right)a³\\ & ≈ 2.8338a^{3
\end{align}}

scope=row

Parabiaugmented hexagonal prism

D_2h

of order 8

\begin{align} A&=(4+5\sqrt{3})a²\\ & ≈ 12.6603a²\\ V&=

	1
	6

\left(2\sqrt{2}+9\sqrt{3}\right)a³\\ & ≈ 3.0695a^{3
\end{align}}

scope=row

Metabiaugmented hexagonal prism

C_2v

of order 4

\begin{align} A&=(4+5\sqrt{3})a²\\ & ≈ 12.6603a²\\ V&=

	1
	6

\left(2\sqrt{2}+9\sqrt{3}\right)a³\\ & ≈ 3.0695a^{3
\end{align}}

scope=row

Triaugmented hexagonal prism

D_3h

of order 12

\begin{align} A&=3\left(1+2\sqrt{3}\right)a²\\ & ≈ 13.3923a²\\ V&=\left(

	1
	\sqrt{2

}+\frac\right)a^3 \\ &\approx 3.3052a^3\end

scope=row

Augmented dodecahedron

C_5v

of order 10

\begin{align} A&=

	1
	4

\left(5\sqrt{3}+11\sqrt{5\left(5+2\sqrt{5}\right)}\right)a²\\ & ≈ 21.0903a²\\ V&=

	1
	24

\left(95+43\sqrt{5}\right)a³\\ & ≈ 7.9646a^{3
\end{align}}

scope=row

Parabiaugmented dodecahedron

D_5d

of order 20

\begin{align} A&=

	5
	2

\left(\sqrt{3}+\sqrt{5\left(5+2\sqrt{5}\right)}\right)a²\\ & ≈ 21.5349a²\\ V&=

	1
	6

\left(25+11\sqrt{5}\right)a³\\ & ≈ 8.2661a^{3
\end{align}}

scope=row

Metabiaugmented dodecahedron

C_2v

of order 4

\begin{align} A&=

	5
	2

\left(\sqrt{3}+\sqrt{5\left(5+2\sqrt{5}\right)}\right)a²\\ & ≈ 21.5349a²\\ V&=

	1
	6

\left(25+11\sqrt{5}\right)a³\\ & ≈ 8.2661a^{3
\end{align}}

scope=row

Triaugmented dodecahedron

C_3v

of order 6

\begin{align} A&=

	3
	4

\left(5\sqrt{3}+3\sqrt{5\left(5+2\sqrt{5}\right)}\right)a²\\ & ≈ 21.9795a²\\ V&=

	5
	8

\left(7+3\sqrt{5}\right)a³\\ & ≈ 8.5676a^{3
\end{align}}

scope=row

Metabidiminished icosahedron

C_2v

of order 4

\begin{align} A&=

	1
	2

\left(5\sqrt{3}+\sqrt{5\left(5+2\sqrt{5}\right)}\right)a²\\ & ≈ 7.7711a²\\ V&=

	1
	6

\left(5+2\sqrt{5}\right)a³\\ & ≈ 1.5787a^{3
\end{align}}

scope=row

Tridiminished icosahedron

C_3v

of order 6

\begin{align} A&=

	1
	4

\left(5\sqrt{3}+3\sqrt{5\left(5+2\sqrt{5}\right)}\right)a²\\ & ≈ 7.3265a²\\ V&=\left(

	5	+
	8

	7\sqrt{5

}\right)a^3 \\ &\approx 1.2772a^3\end

scope=row

Augmented tridiminished icosahedron

C_3v

of order 6

\begin{align} A&=

	1
	4

\left(7\sqrt{3}+3\sqrt{5\left(5+2\sqrt{5}\right)}\right)a²\\ & ≈ 8.1925a²\\ V&=

	1
	24

\left(15+2\sqrt{2}+7\sqrt{5}\right)a³\\ & ≈ 1.395a^{3
\end{align}}

scope=row

Augmented truncated tetrahedron

C_3v

of order 6

\begin{align} A&=

	1
	2

\left(6+13\sqrt{3}\right)a²\\ & ≈ 14.2583a²\\ V&=

	11
	2\sqrt{2

}a^3 \\ &\approx 3.8891a^3\end

scope=row

Augmented truncated cube

C_4v

of order 8

\begin{align} A&=(15+10\sqrt{2}+3\sqrt{3})a²\\ & ≈ 34.3383a²\\ V&=\left(8+

	16\sqrt{2

}\right)a^3 \\ &\approx 15.5425a^3\end

scope=row

Biaugmented truncated cube

D_4h

of order 16

\begin{align} A&=2\left(9+4\sqrt{2}+2\sqrt{3}\right)a²\\ & ≈ 36.2419a²\\ V&=(9+6\sqrt{2})a³\\ & ≈ 17.4853a^{3
\end{align}}

scope=row

Augmented truncated dodecahedron

105

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)a²\\ & ≈ 102.1821a²\\ V&=\left(

	505	+
	12

	81\sqrt{5

}\right)a^3 \\ &\approx 87.3637a^3\end

scope=row

Parabiaugmented truncated dodecahedron

120

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)a²\\ & ≈ 103.3734a²\\ V&=

	1
	12

\left(515+251\sqrt{5}\right)a³\\ & ≈ 89.6878a^{3
\end{align}}

scope=row

Metabiaugmented truncated dodecahedron

120

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)a²\\ & ≈ 103.3734a²\\ V&=

	1
	12

\left(515+251\sqrt{5}\right)a³\\ & ≈ 89.6878a^{3
\end{align}}

scope=row

Triaugmented truncated dodecahedron

135

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)a²\\ & ≈ 104.5648a²\\ V&=

	7
	12

\left(75+37\sqrt{5}\right)a³\\ & ≈ 92.0118a^{3
\end{align}}

scope=row

Gyrate rhombicosidodecahedron

120

C_5v

of order 10

\begin{align} A&=\left(30+5\sqrt{3}+3\sqrt{5\left(5+2\sqrt{5}\right)}\right)a²\\ & ≈ 59.306a²\\ V&=\left(20+

	29\sqrt{5

}\right)a^3 \\ &\approx 41.6153a^3\end

scope=row

Parabigyrate rhombicosidodecahedron

120

D_5d

of order 20

\begin{align} A&=\left(30+5\sqrt{3}+3\sqrt{5\left(5+2\sqrt{5}\right)}\right)a²\\ & ≈ 59.306a²\\ V&=\left(20+

	29\sqrt{5

}\right)a^3 \\ &\approx 41.6153a^3\end

scope=row

Metabigyrate rhombicosidodecahedron

120

C_2v

of order 4

\begin{align} A&=\left(30+5\sqrt{3}+3\sqrt{5\left(5+2\sqrt{5}\right)}\right)a²\\ & ≈ 59.306a²\\ V&=\left(20+

	29\sqrt{5

}\right)a^3 \\ &\approx 41.6153a^3\end

scope=row

Trigyrate rhombicosidodecahedron

120

C_3v

of order 6

\begin{align} A&=\left(30+5\sqrt{3}+3\sqrt{5\left(5+2\sqrt{5}\right)}\right)a²\\ & ≈ 59.306a²\\ V&=\left(20+

	29\sqrt{5

}\right)a^3 \\ &\approx 41.6153a^3\end

scope=row

Diminished rhombicosidodecahedron

105

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)a²\\ & ≈ 58.1147a²\\ V&=\left(

	115
	6

+9\sqrt{5}\right)a³\\ & ≈ 39.2913a^{3
\end{align}}

scope=row

Paragyrate diminished rhombicosidodecahedron

105

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)a²\\ & ≈ 58.1147a²\\ V&=\left(

	115
	6

+9\sqrt{5}\right)a³\\ & ≈ 39.2913a^{3
\end{align}}

scope=row

Metagyrate diminished rhombicosidodecahedron

105

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)a²\\ & ≈ 58.1147a²\\ V&=\left(

	115
	6

+9\sqrt{5}\right)a³\\ & ≈ 39.2913a^{3
\end{align}}

scope=row

Bigyrate diminished rhombicosidodecahedron

105

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)a²\\ & ≈ 58.1147a²\\ V&=\left(

	115
	6

+9\sqrt{5}\right)a³\\ & ≈ 39.2913a^{3
\end{align}}

scope=row

Parabidiminished rhombicosidodecahedron

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)a²\\ & ≈ 56.9233a²\\ V&=

	5
	3

\left(11+5\sqrt{5}\right)a³\\ & ≈ 36.9672a^{3
\end{align}}

scope=row

Metabidiminished rhombicosidodecahedron

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)a²\\ & ≈ 56.9233a²\\ V&=

	5
	3

\left(11+5\sqrt{5}\right)a³\\ & ≈ 36.9672a^{3
\end{align}}

scope=row

Gyrate bidiminished rhombicosidodecahedron

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)a²\\ & ≈ 56.9233a²\\ V&=

	5
	3

\left(11+5\sqrt{5}\right)a³\\ & ≈ 36.9672a^{3
\end{align}}

scope=row

Tridiminished rhombicosidodecahedron

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)a²\\ & ≈ 55.732a²\\ V&=\left(

	35	+
	2

	23\sqrt{5

}\right)a^3 \\ &\approx 34.6432a^3\end

scope=row

Snub disphenoid

D_2d

of order 8

\begin{align} A&=3\sqrt{3}a²\\ & ≈ 5.1962a²\\ V& ≈ 0.8595a^{3
\end{align}}

scope=row

Snub square antiprism

D_4d

of order 16

\begin{align} A&=2\left(1+3\sqrt{3}\right)a²\\ & ≈ 12.3923a²\\ V& ≈ 3.6012a^{3
\end{align}}

scope=row

Sphenocorona

C_2v

of order 4

\begin{align} A&=(2+3\sqrt{3})a²\\ & ≈ 7.1962a²\\ V&=

	1
	2

a³\sqrt{1+3\sqrt{

	3
	2

}+\sqrt} \\ &\approx 1.5154a^3\end

scope=row

Augmented sphenocorona

C_s

of order 2

\begin{align} A&=(1+4\sqrt{3})a²\\ & ≈ 7.9282a²\\ V&=

	1
	2

a^3\sqrt{1+3\sqrt{

	3
	2

} + \sqrt}+\frac \\ &\approx 1.7511a^3\end

scope=row

Sphenomegacorona

C_2v

of order 4

\begin{align} A&=2\left(1+2\sqrt{3}\right)a²\\ & ≈ 8.9282a²\\ V& ≈ 1.9481a^{3
\end{align}}

scope=row

Hebesphenomegacorona

C_2v

of order 4

\begin{align} A&=

	3
	2

\left(2+3\sqrt{3}\right)a²\\ & ≈ 10.7942a²\\ V& ≈ 2.9129a^{3
\end{align}}

scope=row

Disphenocingulum

D_2d

of order 8

\begin{align} A&=(4+5\sqrt{3})a²\\ & ≈ 12.6603a²\\ V& ≈ 3.7776a^{3
\end{align}}

scope=row

Bilunabirotunda

D_2h

of order 8

\begin{align} A&=\left(2+2\sqrt{3}+\sqrt{5\left(5+2\sqrt{5}\right)}\right)a²\\ & ≈ 12.346a²\\ V&=

	1
	12

\left(17+9\sqrt{5}\right)a³\\ & ≈ 3.0937a^{3
\end{align}}

scope=row

Triangular hebesphenorotunda

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)a²\\ & ≈ 16.3887a²\\ V&=\left(

	5	+
	2

	7\sqrt{5

}\right)a^3 \\ &\approx 5.1087a^3\end

Bibliography

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Book: Cromwell, P. R. . Polyhedra . 1997 . . 978-0-521-66405-9.
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External links

Web site: Hart . George W. . George W. Hart . 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.