# List of Johnson solids

In geometry, 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 (in which all vertices are symmetric to each other) from the definition; uniform polyhedra 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 non-uniform Johnson 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.
Seventeen 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 first six Johnson solids satisfy this criterion: the equilateral square pyramid, pentagonal pyramid, triangular cupola, square cupola, pentagonal cupola, and pentagonal rotunda. The criterion 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 360$^∘$/n {\textstyle {\frac {360^{\circ }}{n}}} 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 $2n$. 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 $C_{n\mathrm {v} }$ of order $2n$. The transformation that preserves a polyhedron's symmetry by reflecting it across a horizontal plane is known as the prismatic symmetry $D_{n\mathrm {h} }$ of order $4n$. The antiprismatic symmetry $D_{n\mathrm {d} }$ of order $4n$ preserves the symmetry by rotating its half bottom and reflection across the horizontal plane. The symmetry group $C_{n\mathrm {h} }$ of order $2n$ 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 $C_{1\mathrm {h} }$ 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.
The 92 Johnson solids :
| $J_{n}$ | Solid name | Vertices | Edges | Faces | Symmetry group and its order | Surface area and volume |
| 1 | Equilateral square pyramid | 5 | 8 | 5 | $C_{4v}$ of order 8 | ${\begin{aligned}A&=\left(1+{\sqrt {3}}\right)a^{2}\\&\approx 2.7321a^{2}\\V&={\frac {\sqrt {2}}{6}}a^{3}\\&\approx 0.2357a^{3}\end{aligned}}$ |
| 2 | Pentagonal pyramid | 6 | 10 | 6 | $C_{5v}$ of order 10 | ${\begin{aligned}A&={\frac {a^{2}}{2}}{\sqrt {{\frac {5}{2}}\left(10+{\sqrt {5}}+{\sqrt {75+30{\sqrt {5}}}}\right)}}\\&\approx 3.8855a^{2}\\V&=\left({\frac {5+{\sqrt {5}}}{24}}\right)a^{3}\\&\approx 0.3015a^{3}\end{aligned}}$ |
| 3 | Triangular cupola | 9 | 15 | 8 | $C_{3v}$ of order 6 | ${\begin{aligned}A&=\left(3+{\frac {5{\sqrt {3}}}{2}}\right)a^{2}\\&\approx 7.3301a^{2}\\V&=\left({\frac {5}{3{\sqrt {2}}}}\right)a^{3}\\&\approx 1.1785a^{3}\end{aligned}}$ |
| 4 | Square cupola | 12 | 20 | 10 | $C_{4v}$ of order 8 | ${\begin{aligned}A&=\left(7+2{\sqrt {2}}+{\sqrt {3}}\right)a^{2}\\&\approx 11.5605a^{2}\\V&=\left(1+{\frac {2{\sqrt {2}}}{3}}\right)a^{3}\\&\approx 1.9428a^{3}\end{aligned}}$ |
| 5 | Pentagonal cupola | 15 | 25 | 12 | $C_{5v}$ of order 10 | ${\begin{aligned}A&=\left({\frac {1}{4}}\left(20+5{\sqrt {3}}+{\sqrt {5\left(145+62{\sqrt {5}}\right)}}\right)\right)a^{2}\\&\approx 16.5798a^{2}\\V&=\left({\frac {1}{6}}\left(5+4{\sqrt {5}}\right)\right)a^{3}\\&\approx 2.3241a^{3}\end{aligned}}$ |
| 6 | Pentagonal rotunda | 20 | 35 | 17 | $C_{5v}$ of order 10 | ${\begin{aligned}A&=\left({\frac {1}{2}}\left(5{\sqrt {3}}+{\sqrt {10\left(65+29{\sqrt {5}}\right)}}\right)\right)a^{2}\\&\approx 22.3472a^{2}\\V&=\left({\frac {1}{12}}\left(45+17{\sqrt {5}}\right)\right)a^{3}\\&\approx 6.9178a^{3}\end{aligned}}$ |
| 7 | Elongated triangular pyramid | 7 | 12 | 7 | $C_{3v}$ of order 6 | ${\begin{aligned}A&=\left(3+{\sqrt {3}}\right)a^{2}\\&\approx 4.7321a^{2}\\V&=\left({\frac {1}{12}}\left({\sqrt {2}}+3{\sqrt {3}}\right)\right)a^{3}\\&\approx 0.5509a^{3}\end{aligned}}$ |
| 8 | Elongated square pyramid | 9 | 16 | 9 | $C_{4v}$ of order 8 | ${\begin{aligned}A&=\left(5+{\sqrt {3}}\right)a^{2}\\&\approx 6.7321a^{2}\\V&=\left(1+{\frac {\sqrt {2}}{6}}\right)a^{3}\\&\approx 1.2357a^{3}\end{aligned}}$ |
| 9 | Elongated pentagonal pyramid | 11 | 20 | 11 | $C_{5v}$ of order 10 | ${\begin{aligned}A&={\frac {20+5{\sqrt {3}}+{\sqrt {25+10{\sqrt {5}}}}}{4}}a^{2}\\&\approx 8.8855a^{2}\\V&=\left({\frac {5+{\sqrt {5}}+6{\sqrt {25+10{\sqrt {5}}}}}{24}}\right)a^{3}\\&\approx 2.022a^{3}\end{aligned}}$ |
| 10 | Gyroelongated square pyramid | 9 | 20 | 13 | $C_{4v}$ of order 8 | ${\begin{aligned}A&=(1+3{\sqrt {3}})a^{2}\\&\approx 6.1962a^{2}\\V&={\frac {1}{6}}\left({\sqrt {2}}+2{\sqrt {4+3{\sqrt {2}}}}\right)a^{3}\\&\approx 1.1927a^{3}\end{aligned}}$ |
| 11 | Gyroelongated pentagonal pyramid | 11 | 25 | 16 | $C_{5v}$ of order 10 | ${\begin{aligned}A&={\frac {1}{4}}\left(15{\sqrt {3}}+{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 8.2157a^{2}\\V&={\frac {1}{24}}\left(25+9{\sqrt {5}}\right)a^{3}\\&\approx 1.8802a^{3}\end{aligned}}$ |
| 12 | Triangular bipyramid | 5 | 9 | 6 | $D_{3h}$ of order 12 | ${\begin{aligned}A&={\frac {3{\sqrt {3}}}{2}}a^{2}\\&\approx 2.5981a^{2}\\V&={\frac {\sqrt {2}}{6}}a^{3}\\&\approx 0.2358a^{3}\end{aligned}}$ |
| 13 | Pentagonal bipyramid | 7 | 15 | 10 | $D_{5h}$ of order 20 | ${\begin{aligned}A&={\frac {5{\sqrt {3}}}{2}}a^{2}\\&\approx 4.3301a^{2}\\V&={\frac {1}{12}}\left(5+{\sqrt {5}}\right)a^{3}\\&\approx 0.603a^{3}\end{aligned}}$ |
| 14 | Elongated triangular bipyramid | 8 | 15 | 9 | $D_{3h}$ of order 12 | ${\begin{aligned}A&={\frac {3}{2}}\left(2+{\sqrt {3}}\right)a^{2}\\&\approx 5.5981a^{2}\\V&={\frac {1}{12}}\left(2{\sqrt {2}}+3{\sqrt {3}}\right)a^{3}\\&\approx 0.6687a^{3}\end{aligned}}$ |
| 15 | Elongated square bipyramid | 10 | 20 | 12 | $D_{4h}$ of order 16 | ${\begin{aligned}A&=2\left(2+{\sqrt {3}}\right)a^{2}\\&\approx 7.4641a^{2}\\V&={\frac {1}{3}}\left(3+{\sqrt {2}}\right)a^{3}\\&\approx 1.4714a^{3}\end{aligned}}$ |
| 16 | Elongated pentagonal bipyramid | 12 | 25 | 15 | $D_{5h}$ of order 20 | ${\begin{aligned}A&={\frac {5}{2}}\left(2+{\sqrt {3}}\right)a^{2}\\&\approx 9.3301a^{2}\\V&={\frac {1}{12}}\left(5+{\sqrt {5}}+3{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{3}\\&\approx 2.3235a^{3}\end{aligned}}$ |
| 17 | Gyroelongated square bipyramid | 10 | 24 | 16 | $D_{4d}$ of order 16 | ${\begin{aligned}A&=4{\sqrt {3}}a^{2}\\&\approx 6.9282a^{2}\\V&={\frac {1}{3}}\left({\sqrt {2}}+{\sqrt {4+3{\sqrt {2}}}}\right)a^{3}\\&\approx 1.4284a^{3}\end{aligned}}$ |
| 18 | Elongated triangular cupola | 15 | 27 | 14 | $C_{3v}$ of order 6 | ${\begin{aligned}A&={\frac {1}{2}}\left(18+5{\sqrt {3}}\right)a^{2}\\&\approx 13.3301a^{2}\\V&={\frac {1}{6}}\left(5{\sqrt {2}}+9{\sqrt {3}}\right)a^{3}\\&\approx 3.7766a^{3}\end{aligned}}$ |
| 19 | Elongated square cupola | 20 | 36 | 18 | $C_{4v}$ of order 8 | ${\begin{aligned}A&=(15+2{\sqrt {2}}+{\sqrt {3}})a^{2}\\&\approx 19.5605a^{2}\\V&=\left(3+{\frac {8{\sqrt {2}}}{3}}\right)a^{3}\\&\approx 6.7712a^{3}\end{aligned}}$ |
| 20 | Elongated pentagonal cupola | 25 | 45 | 22 | $C_{5v}$ of order 10 | ${\begin{aligned}A&={\frac {1}{4}}\left(60+5{\sqrt {3}}+10{\sqrt {5+2{\sqrt {5}}}}+{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 26.5798a^{2}\\V&={\frac {1}{6}}\left(5+4{\sqrt {5}}+15{\sqrt {5+2{\sqrt {5}}}}\right)a^{3}\\&\approx 10.0183a^{3}\end{aligned}}$ |
| 21 | Elongated pentagonal rotunda | 30 | 55 | 27 | $C_{5v}$ of order 10 | ${\begin{aligned}A&={\frac {1}{2}}a^{2}\left(20+5{\sqrt {3}}+5{\sqrt {5+2{\sqrt {5}}}}+3{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)\\&\approx 32.3472a^{2}\\V&={\frac {1}{12}}a^{3}\left(45+17{\sqrt {5}}+30{\sqrt {5+2{\sqrt {5}}}}\right)\\&\approx 14.612a^{3}\end{aligned}}$ |
| 22 | Gyroelongated triangular cupola | 15 | 33 | 20 | $C_{3v}$ of order 6 | ${\begin{aligned}A&={\frac {1}{2}}\left(6+11{\sqrt {3}}\right)a^{2}\\&\approx 12.5263a^{2}\\V&={\frac {1}{3}}{\sqrt {{\frac {61}{2}}+18{\sqrt {3}}+30{\sqrt {1+{\sqrt {3}}}}}}a^{3}\\&\approx 3.5161a^{3}\end{aligned}}$ |
| 23 | Gyroelongated square cupola | 20 | 44 | 26 | $C_{4v}$ of order 8 | ${\begin{aligned}A&=(7+2{\sqrt {2}}+5{\sqrt {3}})a^{2}\\&\approx 18.4887a^{2}\\V&=\left(1+{\frac {2}{3}}{\sqrt {2}}+{\frac {2}{3}}{\sqrt {4+2{\sqrt {2}}+2{\sqrt {146+103{\sqrt {2}}}}}}\right)a^{3}\\&\approx 6.2108a^{3}\end{aligned}}$ |
| 24 | Gyroelongated pentagonal cupola | 25 | 55 | 32 | $C_{5v}$ of order 10 | ${\begin{aligned}A&={\frac {1}{4}}\left(20+25{\sqrt {3}}+10{\sqrt {5+2{\sqrt {5}}}}+{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 25.2400a^{2}\\V&=\left({\frac {5}{6}}+{\frac {2}{3}}{\sqrt {5}}+{\frac {5}{6}}{\sqrt {2{\sqrt {650+290{\sqrt {5}}}}-2{\sqrt {5}}-2}}\right)a^{3}\\&\approx 9.0733a^{3}\end{aligned}}$ |
| 25 | Gyroelongated pentagonal rotunda | 30 | 65 | 37 | $C_{5v}$ of order 10 | ${\begin{aligned}A&={\frac {1}{2}}\left(15{\sqrt {3}}+\left(5+3{\sqrt {5}}\right){\sqrt {5+2{\sqrt {5}}}}\right)a^{2}\\&\approx 31.0075a^{2}\\V&=\left({\frac {45}{12}}+{\frac {17}{12}}{\sqrt {5}}+{\frac {5}{6}}{\sqrt {2{\sqrt {650+290{\sqrt {5}}}}-2{\sqrt {5}}-2}}\right)a^{3}\\&\approx 13.6671a^{3}\end{aligned}}$ |
| 26 | Gyrobifastigium | 8 | 14 | 8 | $D_{2d}$ of order 8 | ${\begin{aligned}A&=\left(4+{\sqrt {3}}\right)a^{2}\\&\approx 5.7321a^{2}\\V&=\left({\frac {\sqrt {3}}{2}}\right)a^{3}\\&\approx 0.866a^{3}\end{aligned}}$ |
| 27 | Triangular orthobicupola | 12 | 24 | 14 | $D_{3h}$ of order 12 | ${\begin{aligned}A&=2\left(3+{\sqrt {3}}\right)a^{2}\\&\approx 9.4641a^{2}\\V&={\frac {5{\sqrt {2}}}{3}}a^{3}\\&\approx 2.357a^{3}\end{aligned}}$ |
| 28 | Square orthobicupola | 16 | 32 | 18 | $D_{4h}$ of order 16 | ${\begin{aligned}A&=2(5+{\sqrt {3}})a^{2}\\&\approx 13.4641a^{2}\\V&=\left(2+{\frac {4{\sqrt {2}}}{3}}\right)a^{3}\\&\approx 3.8856a^{3}\end{aligned}}$ |
| 29 | Square gyrobicupola | 16 | 32 | 18 | $D_{4d}$ of order 16 | ${\begin{aligned}A&=2(5+{\sqrt {3}})a^{2}\\&\approx 13.4641a^{2}\\V&=\left(2+{\frac {4{\sqrt {2}}}{3}}\right)a^{3}\\&\approx 3.8856a^{3}\end{aligned}}$ |
| 30 | Pentagonal orthobicupola | 20 | 40 | 22 | $D_{5h}$ of order 20 | ${\begin{aligned}A&=\left(10+{\sqrt {{\frac {5}{2}}\left(10+{\sqrt {5}}+{\sqrt {75+30{\sqrt {5}}}}\right)}}\right)a^{2}\\&\approx 17.7711a^{2}\\V&={\frac {1}{3}}\left(5+4{\sqrt {5}}\right)a^{3}\\&\approx 4.6481a^{3}\end{aligned}}$ |
| 31 | Pentagonal gyrobicupola | 20 | 40 | 22 | $D_{5d}$ of order 20 | ${\begin{aligned}A&=\left(10+{\sqrt {{\frac {5}{2}}\left(10+{\sqrt {5}}+{\sqrt {75+30{\sqrt {5}}}}\right)}}\right)a^{2}\\&\approx 17.7711a^{2}\\V&={\frac {1}{3}}\left(5+4{\sqrt {5}}\right)a^{3}\\&\approx 4.6481a^{3}\end{aligned}}$ |
| 32 | Pentagonal orthocupolarotunda | 25 | 50 | 27 | $C_{5v}$ of order 10 | ${\begin{aligned}A&=\left(5+{\frac {1}{4}}{\sqrt {1900+490{\sqrt {5}}+210{\sqrt {75+30{\sqrt {5}}}}}}\right)a^{2}\\&\approx 23.5385a^{2}\\V&={\frac {5}{12}}\left(11+5{\sqrt {5}}\right)a^{3}\\&\approx 9.2418a^{3}\end{aligned}}$ |
| 33 | Pentagonal gyrocupolarotunda | 25 | 50 | 27 | $C_{5v}$ of order 10 | ${\begin{aligned}A&=\left(5+{\frac {15}{4}}{\sqrt {3}}+{\frac {7}{4}}{\sqrt {25+10{\sqrt {5}}}}\right)a^{2}\\&\approx 23.5385a^{2}\\V&={\frac {5}{12}}\left(11+5{\sqrt {5}}\right)a^{3}\\&\approx 9.2418a^{3}\end{aligned}}$ |
| 34 | Pentagonal orthobirotunda | 30 | 60 | 32 | $D_{5h}$ of order 20 | ${\begin{aligned}A&=\left((5{\sqrt {3}}+3{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 29.306a^{2}\\V&={\frac {1}{6}}(45+17{\sqrt {5}})a^{3}\\&\approx 13.8355a^{3}\end{aligned}}$ |
| 35 | Elongated triangular orthobicupola | 18 | 36 | 20 | $D_{3h}$ of order 12 | ${\begin{aligned}A&=2(6+{\sqrt {3}})a^{2}\\&\approx 15.4641a^{2}\\V&=\left({\frac {5{\sqrt {2}}}{3}}+{\frac {3{\sqrt {3}}}{2}}\right)a^{3}\\&\approx 4.9551a^{3}\end{aligned}}$ |
| 36 | Elongated triangular gyrobicupola | 18 | 36 | 20 | $D_{3d}$ of order 12 | ${\begin{aligned}A&=2(6+{\sqrt {3}})a^{2}\\&\approx 15.4641a^{2}\\V&=\left({\frac {5{\sqrt {2}}}{3}}+{\frac {3{\sqrt {3}}}{2}}\right)a^{3}\\&\approx 4.9551a^{3}\end{aligned}}$ |
| 37 | Elongated square gyrobicupola | 24 | 48 | 26 | $D_{4d}$ of order 16 | ${\begin{aligned}A&=2(9+{\sqrt {3}})a^{2}\\&\approx 21.4641a^{2}\\V&=\left(4+{\frac {10{\sqrt {2}}}{3}}\right)a^{3}\\&\approx 8.714a^{3}\end{aligned}}$ |
| 38 | Elongated pentagonal orthobicupola | 30 | 60 | 32 | $D_{5h}$ of order 20 | ${\begin{aligned}A&=\left(20+{\sqrt {{\frac {5}{2}}\left(10+{\sqrt {5}}+{\sqrt {75+30{\sqrt {5}}}}\right)}}\right)a^{2}\\&\approx 27.7711a^{2}\\V&={\frac {1}{6}}\left(10+8{\sqrt {5}}+15{\sqrt {5+2{\sqrt {5}}}}\right)a^{3}\\&\approx 12.3423a^{3}\end{aligned}}$ |
| 39 | Elongated pentagonal gyrobicupola | 30 | 60 | 32 | $D_{5d}$ of order 20 | ${\begin{aligned}A&=\left(20+{\sqrt {{\frac {5}{2}}\left(10+{\sqrt {5}}+{\sqrt {75+30{\sqrt {5}}}}\right)}}\right)a^{2}\\&\approx 27.7711a^{2}\\V&={\frac {1}{6}}\left(10+8{\sqrt {5}}+15{\sqrt {5+2{\sqrt {5}}}}\right)a^{3}\\&\approx 12.3423a^{3}\end{aligned}}$ |
| 40 | Elongated pentagonal orthocupolarotunda | 35 | 70 | 37 | $C_{5v}$ of order 10 | ${\begin{aligned}A&={\frac {1}{4}}\left(60+{\sqrt {10\left(190+49{\sqrt {5}}+21{\sqrt {75+30{\sqrt {5}}}}\right)}}\right)a^{2}\\&\approx 33.5385a^{2}\\V&={\frac {5}{12}}\left(11+5{\sqrt {5}}+6{\sqrt {5+2{\sqrt {5}}}}\right)a^{3}\\&\approx 16.936a^{3}\end{aligned}}$ |
| 41 | Elongated pentagonal gyrocupolarotunda | 35 | 70 | 37 | $C_{5v}$ of order 10 | ${\begin{aligned}A&={\frac {1}{4}}\left(60+{\sqrt {10\left(190+49{\sqrt {5}}+21{\sqrt {75+30{\sqrt {5}}}}\right)}}\right)a^{2}\\&\approx 33.5385a^{2}\\V&={\frac {5}{12}}\left(11+5{\sqrt {5}}+6{\sqrt {5+2{\sqrt {5}}}}\right)a^{3}\\&\approx 16.936a^{3}\end{aligned}}$ |
| 42 | Elongated pentagonal orthobirotunda | 40 | 80 | 42 | $D_{5h}$ of order 20 | ${\begin{aligned}A&=\left(10+{\sqrt {30\left(10+3{\sqrt {5}}+{\sqrt {75+30{\sqrt {5}}}}\right)}}\right)a^{2}\\&\approx 39.306a^{2}\\V&={\frac {1}{6}}\left(45+17{\sqrt {5}}+15{\sqrt {5+2{\sqrt {5}}}}\right)a^{3}\\&\approx 21.5297a^{3}\end{aligned}}$ |
| 43 | Elongated pentagonal gyrobirotunda | 40 | 80 | 42 | $D_{5d}$ of order 20 | ${\begin{aligned}A&=\left(10+{\sqrt {30\left(10+3{\sqrt {5}}+{\sqrt {75+30{\sqrt {5}}}}\right)}}\right)a^{2}\\&\approx 39.306a^{2}\\V&={\frac {1}{6}}\left(45+17{\sqrt {5}}+15{\sqrt {5+2{\sqrt {5}}}}\right)a^{3}\\&\approx 21.5297a^{3}\end{aligned}}$ |
| 44 | Gyroelongated triangular bicupola | 18 | 42 | 26 | $D_{3}$ of order 6 | ${\begin{aligned}A&=\left(6+5{\sqrt {3}}\right)a^{2}\\&\approx 14.6603a^{2}\\V&={\sqrt {2}}\left({\frac {5}{3}}+{\sqrt {1+{\sqrt {3}}}}\right)a^{3}\\&\approx 4.6946a^{3}\end{aligned}}$ |
| 45 | Gyroelongated square bicupola | 24 | 56 | 34 | $D_{4}$ of order 8 | ${\begin{aligned}A&=\left(10+6{\sqrt {3}}\right)a^{2}\\&\approx 20.3923a^{2}\\V&=\left(2+{\frac {4}{3}}{\sqrt {2}}+{\frac {2}{3}}{\sqrt {4+2{\sqrt {2}}+2{\sqrt {146+103{\sqrt {2}}}}}}\right)a^{3}\\&\approx 8.1536a^{3}\end{aligned}}$ |
| 46 | Gyroelongated pentagonal bicupola | 30 | 70 | 42 | $D_{5}$ of order 10 | ${\begin{aligned}A&={\frac {1}{2}}\left(20+15{\sqrt {3}}+{\sqrt {25+10{\sqrt {5}}}}\right)a^{2}\\&\approx 26.4313a^{2}\\V&=\left({\frac {5}{3}}+{\frac {4}{3}}{\sqrt {5}}+{\frac {5}{6}}{\sqrt {2{\sqrt {650+290{\sqrt {5}}}}-2{\sqrt {5}}-2}}\right)a^{3}\\&\approx 11.3974a^{3}\end{aligned}}$ |
| 47 | Gyroelongated pentagonal cupolarotunda | 35 | 80 | 47 | $C_{5}$ of order 5 | ${\begin{aligned}A&={\frac {1}{4}}\left(20+35{\sqrt {3}}+7{\sqrt {25+10{\sqrt {5}}}}\right)a^{2}\\&\approx 32.1988a^{2}\\V&=\left({\frac {55}{12}}+{\frac {25}{12}}{\sqrt {5}}+{\frac {5}{6}}{\sqrt {2{\sqrt {650+290{\sqrt {5}}}}-2{\sqrt {5}}-2}}\right)a^{3}\\&\approx 15.9911a^{3}\end{aligned}}$ |
| 48 | Gyroelongated pentagonal birotunda | 40 | 90 | 52 | $D_{5}$ of order 10 | ${\begin{aligned}A&=\left(10{\sqrt {3}}+3{\sqrt {25+10{\sqrt {5}}}}\right)a^{2}\\&\approx 37.9662a^{2}\\V&=\left({\frac {45}{6}}+{\frac {17}{6}}{\sqrt {5}}+{\frac {5}{6}}{\sqrt {2{\sqrt {650+290{\sqrt {5}}}}-2{\sqrt {5}}-2}}\right)a^{3}\\&\approx 20.5848a^{3}\end{aligned}}$ |
| 49 | Augmented triangular prism | 7 | 13 | 8 | $C_{2v}$ of order 4 | ${\begin{aligned}A&={\frac {1}{2}}(4+3{\sqrt {3}})a^{2}\\&\approx 4.5981a^{2}\\V&={\frac {1}{12}}(2{\sqrt {2}}+3{\sqrt {3}})a^{3}\\&\approx 0.6687a^{3}\end{aligned}}$ |
| 50 | Biaugmented triangular prism | 8 | 17 | 11 | $C_{2v}$ of order 4 | ${\begin{aligned}A&={\frac {1}{2}}(2+5{\sqrt {3}})a^{2}\\&\approx 5.3301a^{2}\\V&={\sqrt {{\frac {59}{144}}+{\frac {1}{\sqrt {6}}}}}a^{3}\\&\approx 0.9044a^{3}\end{aligned}}$ |
| 51 | Triaugmented triangular prism | 9 | 21 | 14 | $D_{3h}$ of order 12 | ${\begin{aligned}A&={\frac {7{\sqrt {3}}}{2}}a^{2}\\&\approx 6.0622a^{2}\\V&={\frac {2{\sqrt {2}}+{\sqrt {3}}}{4}}a^{3}\\&\approx 1.1401a^{3}\end{aligned}}$ |
| 52 | Augmented pentagonal prism | 11 | 19 | 10 | $C_{2v}$ of order 4 | ${\begin{aligned}A&={\frac {1}{2}}\left(8+2{\sqrt {3}}+{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 9.173a^{2}\\V&={\frac {1}{12}}{\sqrt {233+90{\sqrt {5}}+12{\sqrt {50+20{\sqrt {5}}}}}}a^{3}\\&\approx 1.9562a^{3}\end{aligned}}$ |
| 53 | Biaugmented pentagonal prism | 12 | 23 | 13 | $C_{2v}$ of order 4 | ${\begin{aligned}A&={\frac {1}{2}}a^{2}\left(6+4{\sqrt {3}}+{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)\\&\approx 9.9051a^{2}\\V&={\frac {1}{12}}a^{3}{\sqrt {257+90{\sqrt {5}}+24{\sqrt {50+20{\sqrt {5}}}}}}\\&\approx 2.1919a^{3}\end{aligned}}$ |
| 54 | Augmented hexagonal prism | 13 | 22 | 11 | $C_{2v}$ of order 4 | ${\begin{aligned}A&=(5+4{\sqrt {3}})a^{2}\\&\approx 11.9282a^{2}\\V&={\frac {1}{6}}\left({\sqrt {2}}+9{\sqrt {3}}\right)a^{3}\\&\approx 2.8338a^{3}\end{aligned}}$ |
| 55 | Parabiaugmented hexagonal prism | 14 | 26 | 14 | $D_{2h}$ of order 8 | ${\begin{aligned}A&=(4+5{\sqrt {3}})a^{2}\\&\approx 12.6603a^{2}\\V&={\frac {1}{6}}\left(2{\sqrt {2}}+9{\sqrt {3}}\right)a^{3}\\&\approx 3.0695a^{3}\end{aligned}}$ |
| 56 | Metabiaugmented hexagonal prism | 14 | 26 | 14 | $C_{2v}$ of order 4 | ${\begin{aligned}A&=(4+5{\sqrt {3}})a^{2}\\&\approx 12.6603a^{2}\\V&={\frac {1}{6}}\left(2{\sqrt {2}}+9{\sqrt {3}}\right)a^{3}\\&\approx 3.0695a^{3}\end{aligned}}$ |
| 57 | Triaugmented hexagonal prism | 15 | 30 | 17 | $D_{3h}$ of order 12 | ${\begin{aligned}A&=3\left(1+2{\sqrt {3}}\right)a^{2}\\&\approx 13.3923a^{2}\\V&=\left({\frac {1}{\sqrt {2}}}+{\frac {3{\sqrt {3}}}{2}}\right)a^{3}\\&\approx 3.3052a^{3}\end{aligned}}$ |
| 58 | Augmented dodecahedron | 21 | 35 | 16 | $C_{5v}$ of order 10 | ${\begin{aligned}A&={\frac {1}{4}}\left(5{\sqrt {3}}+11{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 21.0903a^{2}\\V&={\frac {1}{24}}\left(95+43{\sqrt {5}}\right)a^{3}\\&\approx 7.9646a^{3}\end{aligned}}$ |
| 59 | Parabiaugmented dodecahedron | 22 | 40 | 20 | $D_{5d}$ of order 20 | ${\begin{aligned}A&={\frac {5}{2}}\left({\sqrt {3}}+{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 21.5349a^{2}\\V&={\frac {1}{6}}\left(25+11{\sqrt {5}}\right)a^{3}\\&\approx 8.2661a^{3}\end{aligned}}$ |
| 60 | Metabiaugmented dodecahedron | 22 | 40 | 20 | $C_{2v}$ of order 4 | ${\begin{aligned}A&={\frac {5}{2}}\left({\sqrt {3}}+{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 21.5349a^{2}\\V&={\frac {1}{6}}\left(25+11{\sqrt {5}}\right)a^{3}\\&\approx 8.2661a^{3}\end{aligned}}$ |
| 61 | Triaugmented dodecahedron | 23 | 45 | 24 | $C_{3v}$ of order 6 | ${\begin{aligned}A&={\frac {3}{4}}\left(5{\sqrt {3}}+3{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 21.9795a^{2}\\V&={\frac {5}{8}}\left(7+3{\sqrt {5}}\right)a^{3}\\&\approx 8.5676a^{3}\end{aligned}}$ |
| 62 | Metabidiminished icosahedron | 10 | 20 | 12 | $C_{2v}$ of order 4 | ${\begin{aligned}A&={\frac {1}{2}}\left(5{\sqrt {3}}+{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 7.7711a^{2}\\V&={\frac {1}{6}}\left(5+2{\sqrt {5}}\right)a^{3}\\&\approx 1.5787a^{3}\end{aligned}}$ |
| 63 | Tridiminished icosahedron | 9 | 15 | 8 | $C_{3v}$ of order 6 | ${\begin{aligned}A&={\frac {1}{4}}\left(5{\sqrt {3}}+3{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 7.3265a^{2}\\V&=\left({\frac {5}{8}}+{\frac {7{\sqrt {5}}}{24}}\right)a^{3}\\&\approx 1.2772a^{3}\end{aligned}}$ |
| 64 | Augmented tridiminished icosahedron | 10 | 18 | 10 | $C_{3v}$ of order 6 | ${\begin{aligned}A&={\frac {1}{4}}\left(7{\sqrt {3}}+3{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 8.1925a^{2}\\V&={\frac {1}{24}}\left(15+2{\sqrt {2}}+7{\sqrt {5}}\right)a^{3}\\&\approx 1.395a^{3}\end{aligned}}$ |
| 65 | Augmented truncated tetrahedron | 15 | 27 | 14 | $C_{3v}$ of order 6 | ${\begin{aligned}A&={\frac {1}{2}}\left(6+13{\sqrt {3}}\right)a^{2}\\&\approx 14.2583a^{2}\\V&={\frac {11}{2{\sqrt {2}}}}a^{3}\\&\approx 3.8891a^{3}\end{aligned}}$ |
| 66 | Augmented truncated cube | 28 | 48 | 22 | $C_{4v}$ of order 8 | ${\begin{aligned}A&=(15+10{\sqrt {2}}+3{\sqrt {3}})a^{2}\\&\approx 34.3383a^{2}\\V&=\left(8+{\frac {16{\sqrt {2}}}{3}}\right)a^{3}\\&\approx 15.5425a^{3}\end{aligned}}$ |
| 67 | Biaugmented truncated cube | 32 | 60 | 30 | $D_{4h}$ of order 16 | ${\begin{aligned}A&=2\left(9+4{\sqrt {2}}+2{\sqrt {3}}\right)a^{2}\\&\approx 36.2419a^{2}\\V&=(9+6{\sqrt {2}})a^{3}\\&\approx 17.4853a^{3}\end{aligned}}$ |
| 68 | Augmented truncated dodecahedron | 65 | 105 | 42 | $C_{5v}$ of order 10 | ${\begin{aligned}A&={\frac {1}{4}}\left(20+25{\sqrt {3}}+110{\sqrt {5+2{\sqrt {5}}}}+{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 102.1821a^{2}\\V&=\left({\frac {505}{12}}+{\frac {81{\sqrt {5}}}{4}}\right)a^{3}\\&\approx 87.3637a^{3}\end{aligned}}$ |
| 69 | Parabiaugmented truncated dodecahedron | 70 | 120 | 52 | $D_{5d}$ of order 20 | ${\begin{aligned}A&={\frac {1}{2}}\left(20+15{\sqrt {3}}+50{\sqrt {5+2{\sqrt {5}}}}+{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 103.3734a^{2}\\V&={\frac {1}{12}}\left(515+251{\sqrt {5}}\right)a^{3}\\&\approx 89.6878a^{3}\end{aligned}}$ |
| 70 | Metabiaugmented truncated dodecahedron | 70 | 120 | 52 | $C_{2v}$ of order 4 | ${\begin{aligned}A&={\frac {1}{2}}\left(20+15{\sqrt {3}}+50{\sqrt {5+2{\sqrt {5}}}}+{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 103.3734a^{2}\\V&={\frac {1}{12}}\left(515+251{\sqrt {5}}\right)a^{3}\\&\approx 89.6878a^{3}\end{aligned}}$ |
| 71 | Triaugmented truncated dodecahedron | 75 | 135 | 62 | $C_{3v}$ of order 6 | ${\begin{aligned}A&={\frac {1}{4}}\left(60+35{\sqrt {3}}+90{\sqrt {5+2{\sqrt {5}}}}+3{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 104.5648a^{2}\\V&={\frac {7}{12}}\left(75+37{\sqrt {5}}\right)a^{3}\\&\approx 92.0118a^{3}\end{aligned}}$ |
| 72 | Gyrate rhombicosidodecahedron | 60 | 120 | 62 | $C_{5v}$ of order 10 | ${\begin{aligned}A&=\left(30+5{\sqrt {3}}+3{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 59.306a^{2}\\V&=\left(20+{\frac {29{\sqrt {5}}}{3}}\right)a^{3}\\&\approx 41.6153a^{3}\end{aligned}}$ |
| 73 | Parabigyrate rhombicosidodecahedron | 60 | 120 | 62 | $D_{5d}$ of order 20 | ${\begin{aligned}A&=\left(30+5{\sqrt {3}}+3{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 59.306a^{2}\\V&=\left(20+{\frac {29{\sqrt {5}}}{3}}\right)a^{3}\\&\approx 41.6153a^{3}\end{aligned}}$ |
| 74 | Metabigyrate rhombicosidodecahedron | 60 | 120 | 62 | $C_{2v}$ of order 4 | ${\begin{aligned}A&=\left(30+5{\sqrt {3}}+3{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 59.306a^{2}\\V&=\left(20+{\frac {29{\sqrt {5}}}{3}}\right)a^{3}\\&\approx 41.6153a^{3}\end{aligned}}$ |
| 75 | Trigyrate rhombicosidodecahedron | 60 | 120 | 62 | $C_{3v}$ of order 6 | ${\begin{aligned}A&=\left(30+5{\sqrt {3}}+3{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 59.306a^{2}\\V&=\left(20+{\frac {29{\sqrt {5}}}{3}}\right)a^{3}\\&\approx 41.6153a^{3}\end{aligned}}$ |
| 76 | Diminished rhombicosidodecahedron | 55 | 105 | 52 | $C_{5v}$ of order 10 | ${\begin{aligned}A&={\frac {1}{4}}\left(100+15{\sqrt {3}}+10{\sqrt {5+2{\sqrt {5}}}}+11{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 58.1147a^{2}\\V&=\left({\frac {115}{6}}+9{\sqrt {5}}\right)a^{3}\\&\approx 39.2913a^{3}\end{aligned}}$ |
| 77 | Paragyrate diminished rhombicosidodecahedron | 55 | 105 | 52 | $C_{5v}$ of order 10 | ${\begin{aligned}A&={\frac {1}{4}}\left(100+15{\sqrt {3}}+10{\sqrt {5+2{\sqrt {5}}}}+11{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 58.1147a^{2}\\V&=\left({\frac {115}{6}}+9{\sqrt {5}}\right)a^{3}\\&\approx 39.2913a^{3}\end{aligned}}$ |
| 78 | Metagyrate diminished rhombicosidodecahedron | 55 | 105 | 52 | $C_{s}$ of order 2 | ${\begin{aligned}A&={\frac {1}{4}}\left(100+15{\sqrt {3}}+10{\sqrt {5+2{\sqrt {5}}}}+11{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 58.1147a^{2}\\V&=\left({\frac {115}{6}}+9{\sqrt {5}}\right)a^{3}\\&\approx 39.2913a^{3}\end{aligned}}$ |
| 79 | Bigyrate diminished rhombicosidodecahedron | 55 | 105 | 52 | $C_{s}$ of order 2 | ${\begin{aligned}A&={\frac {1}{4}}\left(100+15{\sqrt {3}}+10{\sqrt {5+2{\sqrt {5}}}}+11{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 58.1147a^{2}\\V&=\left({\frac {115}{6}}+9{\sqrt {5}}\right)a^{3}\\&\approx 39.2913a^{3}\end{aligned}}$ |
| 80 | Parabidiminished rhombicosidodecahedron | 50 | 90 | 42 | $D_{5d}$ of order 20 | ${\begin{aligned}A&={\frac {5}{2}}\left(8+{\sqrt {3}}+2{\sqrt {5+2{\sqrt {5}}}}+{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 56.9233a^{2}\\V&={\frac {5}{3}}\left(11+5{\sqrt {5}}\right)a^{3}\\&\approx 36.9672a^{3}\end{aligned}}$ |
| 81 | Metabidiminished rhombicosidodecahedron | 50 | 90 | 42 | $C_{2v}$ of order 4 | ${\begin{aligned}A&={\frac {5}{2}}\left(8+{\sqrt {3}}+2{\sqrt {5+2{\sqrt {5}}}}+{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 56.9233a^{2}\\V&={\frac {5}{3}}\left(11+5{\sqrt {5}}\right)a^{3}\\&\approx 36.9672a^{3}\end{aligned}}$ |
| 82 | Gyrate bidiminished rhombicosidodecahedron | 50 | 90 | 42 | $C_{s}$ of order 2 | ${\begin{aligned}A&={\frac {5}{2}}\left(8+{\sqrt {3}}+2{\sqrt {5+2{\sqrt {5}}}}+{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 56.9233a^{2}\\V&={\frac {5}{3}}\left(11+5{\sqrt {5}}\right)a^{3}\\&\approx 36.9672a^{3}\end{aligned}}$ |
| 83 | Tridiminished rhombicosidodecahedron | 45 | 75 | 32 | $C_{3v}$ of order 6 | ${\begin{aligned}A&={\frac {1}{4}}\left(60+5{\sqrt {3}}+30{\sqrt {5+2{\sqrt {5}}}}+9{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 55.732a^{2}\\V&=\left({\frac {35}{2}}+{\frac {23{\sqrt {5}}}{3}}\right)a^{3}\\&\approx 34.6432a^{3}\end{aligned}}$ |
| 84 | Snub disphenoid | 8 | 18 | 12 | $D_{2d}$ of order 8 | ${\begin{aligned}A&=3{\sqrt {3}}a^{2}\\&\approx 5.1962a^{2}\\V&\approx 0.8595a^{3}\end{aligned}}$ |
| 85 | Snub square antiprism | 16 | 40 | 26 | $D_{4d}$ of order 16 | ${\begin{aligned}A&=2\left(1+3{\sqrt {3}}\right)a^{2}\\&\approx 12.3923a^{2}\\V&\approx 3.6012a^{3}\end{aligned}}$ |
| 86 | Sphenocorona | 10 | 22 | 14 | $C_{2v}$ of order 4 | ${\begin{aligned}A&=(2+3{\sqrt {3}})a^{2}\\&\approx 7.1962a^{2}\\V&={\frac {1}{2}}a^{3}{\sqrt {1+3{\sqrt {\frac {3}{2}}}+{\sqrt {13+3{\sqrt {6}}}}}}\\&\approx 1.5154a^{3}\end{aligned}}$ |
| 87 | Augmented sphenocorona | 11 | 26 | 17 | $C_{s}$ of order 2 | ${\begin{aligned}A&=(1+4{\sqrt {3}})a^{2}\\&\approx 7.9282a^{2}\\V&={\frac {1}{2}}a^{3}{\sqrt {1+3{\sqrt {\frac {3}{2}}}+{\sqrt {13+3{\sqrt {6}}}}}}+{\frac {1}{3{\sqrt {2}}}}\\&\approx 1.7511a^{3}\end{aligned}}$ |
| 88 | Sphenomegacorona | 12 | 28 | 18 | $C_{2v}$ of order 4 | ${\begin{aligned}A&=2\left(1+2{\sqrt {3}}\right)a^{2}\\&\approx 8.9282a^{2}\\V&\approx 1.9481a^{3}\end{aligned}}$ |
| 89 | Hebesphenomegacorona | 14 | 33 | 21 | $C_{2v}$ of order 4 | ${\begin{aligned}A&={\frac {3}{2}}\left(2+3{\sqrt {3}}\right)a^{2}\\&\approx 10.7942a^{2}\\V&\approx 2.9129a^{3}\end{aligned}}$ |
| 90 | Disphenocingulum | 16 | 38 | 24 | $D_{2d}$ of order 8 | ${\begin{aligned}A&=(4+5{\sqrt {3}})a^{2}\\&\approx 12.6603a^{2}\\V&\approx 3.7776a^{3}\end{aligned}}$ |
| 91 | Bilunabirotunda | 14 | 26 | 14 | $D_{2h}$ of order 8 | ${\begin{aligned}A&=\left(2+2{\sqrt {3}}+{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 12.346a^{2}\\V&={\frac {1}{12}}\left(17+9{\sqrt {5}}\right)a^{3}\\&\approx 3.0937a^{3}\end{aligned}}$ |
| 92 | Triangular hebesphenorotunda | 18 | 36 | 20 | $C_{3v}$ of order 6 | ${\begin{aligned}A&={\frac {1}{4}}\left(12+19{\sqrt {3}}+3{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 16.3887a^{2}\\V&=\left({\frac {5}{2}}+{\frac {7{\sqrt {5}}}{6}}\right)a^{3}\\&\approx 5.1087a^{3}\end{aligned}}$ |