View source for List of Johnson solids

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In geometry, a [[convex polyhedron]] whose faces are [[regular polygon]]s is known as a ''[[Johnson solid]]'', or sometimes as a ''Johnson–Zalgaller solid''.<ref>{{Cite journal |last=Araki |first=Yoshiaki |last2=Horiyama |first2=Takashi |last3=Uehara |first3=Ryuhei |date=2015 |editor-last=Rahman |editor-first=M. Sohel |editor2-last=Tomita |editor2-first=Etsuji |title=Common Unfolding of Regular Tetrahedron and Johnson-Zalgaller Solid |url=https://link.springer.com/chapter/10.1007/978-3-319-15612-5_26 |journal=WALCOM: Algorithms and Computation |language=en |location=Cham |publisher=Springer International Publishing |pages=294–305 |doi=10.1007/978-3-319-15612-5_26 |isbn=978-3-319-15612-5}}</ref> Some authors exclude [[uniform polyhedra]] (in which [[isogonal figure|all vertices are symmetric to each other]]) from the definition; uniform polyhedra include [[Platonic solid|Platonic]] and [[Archimedean solid]]s as well as [[Prism (geometry)|prism]]s and [[antiprism]]s.<ref>{{multiref
|{{harvp|Diudea|2018|p=[https://books.google.com/books?id=p_06DwAAQBAJ&pg=PA39 39]}}
|{{harvp|Todesco|2020|p=[https://books.google.com/books?id=wtIBEAAAQBAJ&pg=PA282 282]}}
|{{harvp|Williams|Monteleone|2021|p=[https://books.google.com/books?id=w5RBEAAAQBAJ&pg=PA23 23]}}
}}</ref>
The Johnson solids are named after American mathematician [[Norman Johnson (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.<ref>{{multiref
|{{harvp|Johnson|1966}}
|{{harvp|Zalgaller|1969}}
}}</ref>

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]].<ref>{{multiref
|{{harvp|Cromwell|1997|p=[https://archive.org/details/polyhedra0000crom/page/87/mode/1up 86–87]|loc=See the figure on p.[https://archive.org/details/polyhedra0000crom/page/89/mode/1up 89]}}
|{{harvp|Johnson|1966}}
}}</ref> 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 (geometry)|Augmentation]] involves attaching the Johnson solids onto one or more faces of polyhedra, while [[Elongation (geometry)|elongation]] or [[gyroelongation]] involve joining them onto the bases of a prism or antiprism, respectively. Some others are constructed by [[Diminishment (geometry)|diminishment]], the removal of one of the first six solids from one or more of a polyhedron's faces.<ref>{{multiref
|{{harvp|Rajwade|2001|p=[https://books.google.com/books?id=afJdDwAAQBAJ&pg=PA84 84–88]}}
|{{harvp|Slobodan|Obradović|Ðukanović|2015}}
|{{harvp|Berman|1971|p=350}}
}}</ref>

The following table contains the 92 Johnson solids, with edge length <math> a </math>. The table includes the solid's enumeration (denoted as <math> J_n </math>).{{sfnp|Uehara|2020|p=[https://books.google.com/books?id=51juDwAAQBAJ&pg=PA62 62]}} It also includes the number of vertices, edges, and faces of each solid, as well as its [[symmetry group]], surface area <math> A </math>, and volume <math> V </math>. Every polyhedron has its own [[Polyhedron#Characteristics|characteristics]], including [[Symmetry (geometry)|symmetry]] and measurement. An object is said to have symmetry if there is a [[Transformation (geometry)|transformation]] that maps it to itself. All of those transformations may be composed in a [[Group (mathematics)|group]], alongside the group's number of [[Element (mathematics)|elements]], known as the [[Order of a group|order]]. In two-dimensional space, these transformations include [[Rotational symmetry|rotating]] around the center of a polygon and [[reflection symmetry|reflecting an object]] around the [[Bisection#Perpendicular line segment bisector|perpendicular bisector]] of a polygon. A polygon that is rotated symmetrically by <math display="inline"> \frac{360^\circ}{n} </math> is denoted by <math> C_n </math>, a [[cyclic group]] of order <math> n </math>; combining this with the reflection symmetry results in the symmetry of [[dihedral group]] <math> D_n </math> of order <math> 2n </math>.<ref>{{multiref
|{{harvp|Powell|2010|p=[https://books.google.com/books?id=ojq5BQAAQBAJ&pg=PA27 27]}}
|{{harvp|Solomon|2003|p=[https://books.google.com/books?id=ouvZKQiykf4C&pg=PA40 40]}}
}}</ref> In [[Point groups in three dimensions|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 [[Axial symmetry|axis of symmetry]], and the reflection relative to perpendicular planes passing through the bisector of a base, which is known as the [[pyramidal symmetry]] <math> C_{n \mathrm{v}} </math> of order <math> 2n </math>. The transformation that preserves a polyhedron's symmetry by reflecting it across a horizontal plane is known as the [[prismatic symmetry]] <math> D_{n \mathrm{h}} </math> of order <math> 4n </math>. The [[antiprismatic symmetry]] <math> D_{n \mathrm{d}} </math> of order <math> 4n </math> preserves the symmetry by rotating its half bottom and reflection across the horizontal plane.{{sfnp|Flusser|Suk|Zitofa|2017|p=[https://books.google.com/books?id=jwKLDQAAQBAJ&pg=PA126 126]}} The symmetry group <math> C_{n \mathrm{h}} </math> of order <math> 2n </math> 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 <math> C_{1 \mathrm{h}} </math> of order 2, often denoted as <math> C_s </math>.<ref>{{multiref
|{{harvp|Flusser|Suk|Zitofa|2017|p=[https://books.google.com/books?id=jwKLDQAAQBAJ&pg=PA126 126]}}
|{{harvp|Hergert|Geilhufe|2018|p=[https://books.google.com/books?id=p6hWDwAAQBAJ&pg=PA56 56]}}
}}</ref> 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.{{sfnp|Walsh|2014|p=[https://books.google.com/books?id=ZhDdAwAAQBAJ&pg=PA284 284]}} A volume is a measurement of a region in three-dimensional space.{{sfnp|Parker|1997|p=[https://archive.org/details/mcgrawhilldictio00park_0/page/264 264]}} The volume of a polyhedron may be ascertained in different ways: either through its base and height (like for [[Pyramid (geometry)|pyramids]] and [[Prism (geometry)|prism]]s), by slicing it off into pieces and summing their individual volumes, or by finding the [[Zero of a function|root]] of a [[polynomial]] representing the polyhedron.<ref>{{multiref
|{{harvp|Cromwell|1997|p=[https://books.google.com/books?id=OJowej1QWpoC&&pg=PA36 36]}}
|{{harvp|Berman|1971}}
|{{harvp|Timofeenko|2009}}
}}</ref>
{{mw-datatable}}{{sticky-header}}{{sort-under}}
{| class="wikitable sortable mw-datatable sticky-header-multi sort-under plainrowheaders"
|+ The 92 Johnson solids
!scope=col | <math> J_n </math>
!scope=col | Solid name
!scope=col | Image
!scope=col | Vertices
!scope=col | Edges
!scope=col | Faces
!scope=col | [[List of spherical symmetry groups|Symmetry group]] and its [[Symmetry number|order]]{{sfnp|Johnson|1966}}
!scope=col | Surface area and volume{{sfnp|Berman|1971}}
|-
!scope=row | 1
| [[Equilateral square pyramid]]
| [[Image:Square pyramid.png|100px]]
| 5
| 8
| 5
| <math> C_{4v} </math> of order 8
| <math> \begin{align}
 A &= \left(1 + \sqrt{3}\right)a^2 \\
 &\approx 2.7321a^2 \\
 V &= \frac{\sqrt{2}}{6}a^3 \\
 &\approx 0.2357a^3 \end{align} </math>
|-
!scope=row | 2
| [[Pentagonal pyramid]]
| [[Image:Pentagonal pyramid.png|100px]]
| 6
| 10
| 6
| <math> C_{5v} </math> of order 10
| <math> \begin{align}
 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{align} </math>
|-
!scope=row | 3
| [[Triangular cupola]]
| [[Image:Triangular cupola.png|100px]]
| 9
| 15
| 8
| <math> C_{3v} </math> of order 6
| <math> \begin{align}
 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{align} </math>
|-
!scope=row | 4
| [[Square cupola]]
| [[Image:Square cupola.png|100px]]
| 12
| 20
| 10
| <math> C_{4v} </math> of order 8
| <math> \begin{align}
 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{align} </math>
|-
!scope=row | 5
| [[Pentagonal cupola]]
| [[Image:Pentagonal cupola.png|100px]]
| 15
| 25
| 12
| <math> C_{5v} </math> of order 10
| <math> \begin{align}
 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{align} </math>
|-
!scope=row | 6
| [[Pentagonal rotunda]]
| [[Image:Pentagonal rotunda.png|100px]]
| 20
| 35
| 17
| <math> C_{5v} </math> of order 10
| <math> \begin{align}
 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{align} </math>
|-
!scope=row | 7
| [[Elongated triangular pyramid]]
| [[Image:Elongated triangular pyramid.png|100px]]
| 7
| 12
| 7
| <math> C_{3v} </math> of order 6
| <math> \begin{align}
 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{align} </math>
|-
!scope=row | 8
| [[Elongated square pyramid]]
| [[Image:Elongated square pyramid.png|100px]]
| 9
| 16
| 9
| <math> C_{4v} </math> of order 8
| <math> \begin{align}
 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{align} </math>
|-
!scope=row | 9
| [[Elongated pentagonal pyramid]]
| [[Image:Elongated pentagonal pyramid.png|100px]]
| 11
| 20
| 11
| <math> C_{5v} </math> of order 10
| <math> \begin{align}
 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{align} </math>
|-
!scope=row | 10
| [[Gyroelongated square pyramid]]
| [[Image:Gyroelongated square pyramid.png|100px]]
| 9
| 20
| 13
| <math> C_{4v} </math> of order 8
| <math> \begin{align}
 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{align} </math>
|-
!scope=row | 11
| [[Gyroelongated pentagonal pyramid]]
| [[Image:Gyroelongated pentagonal pyramid.png|100px]]
| 11
| 25
| 16
| <math> C_{5v} </math> of order 10
| <math> \begin{align}
 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{align} </math>
|-
!scope=row | 12
| [[Triangular bipyramid]]
| [[Image:Triangular dipyramid.png|100px]]
| 5
| 9
| 6
| <math> D_{3h} </math> of order 12
| <math> \begin{align}
 A &= \frac{3\sqrt{3}}{2}a^2 \\
 &\approx 2.5981a^2 \\ 
 V &= \frac{\sqrt{2}}{6}a^3 \\
 &\approx 0.2358a^3
\end{align} </math>
|-
!scope=row | 13
| [[Pentagonal bipyramid]]
| [[Image:Pentagonal dipyramid.png|100px]]
| 7
| 15
| 10
| <math> D_{5h} </math> of order 20
| <math> \begin{align}
 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{align} </math>
|-
!scope=row | 14
| [[Elongated triangular bipyramid]]
| [[Image:Elongated triangular dipyramid.png|100px]]
| 8
| 15
| 9
| <math> D_{3h} </math> of order 12
| <math> \begin{align}
 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{align} </math>
|-
!scope=row | 15
| [[Elongated square bipyramid]]
| [[Image:Elongated square dipyramid.png|100px]]
| 10
| 20
| 12
| <math> D_{4h} </math> of order 16
| <math> \begin{align}
 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{align} </math>
|-
!scope=row | 16
| [[Elongated pentagonal bipyramid]]
| [[Image:Elongated pentagonal dipyramid.png|100px]]
| 12
| 25
| 15
| <math> D_{5h} </math> of order 20
| <math> \begin{align}
 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{align} </math>
|-
!scope=row | 17
| [[Gyroelongated square bipyramid]]
| [[Image:Gyroelongated square dipyramid.png|100px]]
| 10
| 24
| 16
| <math> D_{4d} </math> of order 16
| <math> \begin{align}
 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{align} </math>
|-
!scope=row | 18
| [[Elongated triangular cupola]]
| [[Image:Elongated triangular cupola.png|100px]]
| 15
| 27
| 14
| <math> C_{3v} </math> of order 6
| <math> \begin{align}
 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{align} </math>
|-
!scope=row | 19
| [[Elongated square cupola]]
| [[Image:Elongated square cupola.png|100px]]
| 20
| 36
| 18
| <math> C_{4v} </math> of order 8
| <math> \begin{align}
 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{align} </math>
|-
!scope=row | 20
| [[Elongated pentagonal cupola]]
| [[Image:Elongated pentagonal cupola.svg|100px]]
| 25
| 45
| 22
| <math> C_{5v} </math> of order 10
| <math> \begin{align}
 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{align} </math>
|-
!scope=row | 21
| [[Elongated pentagonal rotunda]]
| [[Image:Elongated pentagonal rotunda.png|100px]]
| 30
| 55
| 27
| <math> C_{5v} </math> of order 10
| <math> \begin{align}
 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{align} </math>
|-
!scope=row | 22
| [[Gyroelongated triangular cupola]]
| [[Image:Gyroelongated triangular cupola.png|100px]]
| 15
| 33
| 20
| <math> C_{3v} </math> of order 6
| <math> \begin{align}
 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{align} </math>
|-
!scope=row | 23
| [[Gyroelongated square cupola]]
| [[Image:Gyroelongated square cupola.png|100px]]
| 20
| 44
| 26
| <math> C_{4v} </math> of order 8
| <math> \begin{align}
 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{align} </math>
|-
!scope=row | 24
| [[Gyroelongated pentagonal cupola]]
| [[Image:Gyroelongated pentagonal cupola.png|100px]]
| 25
| 55
| 32
| <math> C_{5v} </math> of order 10
| <math> \begin{align}
 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{align} </math>
|-
!scope=row | 25
| [[Gyroelongated pentagonal rotunda]]
| [[Image:Gyroelongated pentagonal rotunda.png|100px]]
| 30
| 65
| 37
| <math> C_{5v} </math> of order 10
| <math> \begin{align}
 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{align} </math>
|-
!scope=row | 26
| [[Gyrobifastigium]]
| [[Image:Gyrobifastigium.png|100px]]
| 8
| 14
| 8
| <math> D_{2d} </math> of order 8
| <math> \begin{align}
 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{align} </math>
|-
!scope=row | 27
| [[Triangular orthobicupola]]
| [[Image:Triangular orthobicupola.png|100px]]
| 12
| 24
| 14
| <math> D_{3h} </math> of order 12
| <math> \begin{align}
 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{align} </math>
|-
!scope=row | 28
| [[Square orthobicupola]]
| [[Image:Square orthobicupola.png|100px]]
| 16
| 32
| 18
| <math> D_{4h} </math> of order 16
| <math> \begin{align}
 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{align} </math>
|-
!scope=row | 29
| [[Square gyrobicupola]]
| [[Image:Square gyrobicupola.png|100px]]
| 16
| 32
| 18
| <math> D_{4d} </math> of order 16
| <math> \begin{align}
 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{align} </math>
|-
!scope=row | 30
| [[Pentagonal orthobicupola]]
| [[Image:Pentagonal orthobicupola.png|100px]]
| 20
| 40
| 22
| <math> D_{5h} </math> of order 20
| <math> \begin{align}
 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{align} </math>
|-
!scope=row | 31
| [[Pentagonal gyrobicupola]]
| [[Image:Pentagonal gyrobicupola.png|100px]]
| 20
| 40
| 22
| <math> D_{5d} </math> of order 20
| <math> \begin{align}
 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{align} </math>
|-
!scope=row | 32
| [[Pentagonal orthocupolarotunda]]
| [[Image:Pentagonal orthocupolarotunda.png|100px]]
| 25
| 50
| 27
| <math> C_{5v} </math> of order 10
| <math> \begin{align}
 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{align} </math>
|-
!scope=row | 33
| [[Pentagonal gyrocupolarotunda]]
| [[Image:Pentagonal gyrocupolarotunda.png|100px]]
| 25
| 50
| 27
| <math> C_{5v} </math> of order 10
| <math> \begin{align}
 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{align} </math>
|-
!scope=row | 34
| [[Pentagonal orthobirotunda]]
| [[Image:Pentagonal orthobirotunda.png|100px]]
| 30
| 60
| 32
| <math> D_{5h} </math> of order 20
| <math> \begin{align}
 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{align} </math>
|-
!scope=row | 35
| [[Elongated triangular orthobicupola]]
| [[Image:Elongated triangular orthobicupola.png|100px]]
| 18
| 36
| 20
| <math> D_{3h} </math> of order 12
| <math> \begin{align}
 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{align} </math>
|-
!scope=row | 36
| [[Elongated triangular gyrobicupola]]
| [[Image:Elongated triangular gyrobicupola.png|100px]]
| 18
| 36
| 20
| <math> D_{3d} </math> of order 12
| <math> \begin{align}
 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{align} </math>
|-
!scope=row | 37
| [[Elongated square gyrobicupola]]
| [[Image:Elongated square gyrobicupola.png|100px]]
| 24
| 48
| 26
| <math> D_{4d} </math> of order 16
| <math> \begin{align}
 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{align} </math>
|-
!scope=row | 38
| [[Elongated pentagonal orthobicupola]]
| [[Image:Elongated pentagonal orthobicupola.png|100px]]
| 30
| 60
| 32
| <math> D_{5h} </math> of order 20
| <math> \begin{align}
 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{align} </math>
|-
!scope=row | 39
| [[Elongated pentagonal gyrobicupola]]
| [[Image:Elongated pentagonal gyrobicupola.png|100px]]
| 30
| 60
| 32
| <math> D_{5d} </math> of order 20
| <math> \begin{align}
 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{align} </math>
|-
!scope=row | 40
| [[Elongated pentagonal orthocupolarotunda]]
| [[Image:Elongated pentagonal orthocupolarotunda.png|100px]]
| 35
| 70
| 37
| <math> C_{5v} </math> of order 10
| <math> \begin{align}
 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{align} </math>
|-
!scope=row | 41
| [[Elongated pentagonal gyrocupolarotunda]]
| [[Image:Elongated pentagonal gyrocupolarotunda.png|100px]]
| 35
| 70
| 37
| <math> C_{5v} </math> of order 10
| <math> \begin{align}
 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{align} </math>
|-
!scope=row | 42
| [[Elongated pentagonal orthobirotunda]]
| [[Image:Elongated pentagonal orthobirotunda.png|100px]]
| 40
| 80
| 42
| <math> D_{5h} </math> of order 20
| <math> \begin{align}
 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{align} </math>
|-
!scope=row | 43
| [[Elongated pentagonal gyrobirotunda]]
| [[Image:Elongated pentagonal gyrobirotunda.png|100px]]
| 40
| 80
| 42
| <math> D_{5d} </math> of order 20
| <math> \begin{align}
 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{align} </math>
|-
!scope=row | 44
| [[Gyroelongated triangular bicupola]]
| [[Image:Gyroelongated triangular bicupola.png|100px]]
| 18
| 42
| 26
| <math> D_3 </math> of order 6
| <math> \begin{align}
 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{align} </math>
|-
!scope=row | 45
| [[Gyroelongated square bicupola]]
| [[Image:Gyroelongated square bicupola.png|100px]]
| 24
| 56
| 34
| <math> D_4 </math> of order 8
| <math> \begin{align}
 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{align} </math>
|-
!scope=row | 46
| [[Gyroelongated pentagonal bicupola]]
| [[Image:Gyroelongated pentagonal bicupola.png|100px]]
| 30
| 70
| 42
| <math> D_5 </math> of order 10
| <math> \begin{align}
 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{align} </math>
|-
!scope=row | 47
| [[Gyroelongated pentagonal cupolarotunda]]
| [[Image:Gyroelongated pentagonal cupolarotunda.png|100px]]
| 35
| 80
| 47
| <math> C_5 </math> of order 5
| <math> \begin{align}
 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{align} </math>
|-
!scope=row | 48
| [[Gyroelongated pentagonal birotunda]]
| [[Image:Gyroelongated pentagonal birotunda.png|100px]]
| 40
| 90
| 52
| <math> D_5 </math> of order 10
| <math> \begin{align}
 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{align} </math>
|-
!scope=row | 49
| [[Augmented triangular prism]]
| [[Image:Augmented triangular prism.png|100px]]
| 7
| 13
| 8
| <math> C_{2v} </math> of order 4
| <math> \begin{align}
 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{align} </math>
|-
!scope=row | 50
| [[Biaugmented triangular prism]]
| [[Image:Biaugmented triangular prism.png|100px]]
| 8
| 17
| 11
| <math> C_{2v} </math> of order 4
| <math> \begin{align}
 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{align} </math>
|-
!scope=row | 51
| [[Triaugmented triangular prism]]
| [[Image:Triaugmented triangular prism.png|100px]]
| 9
| 21
| 14
| <math> D_{3h} </math> of order 12
| <math> \begin{align}
 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{align} </math>
|-
!scope=row | 52
| [[Augmented pentagonal prism]]
| [[Image:Augmented pentagonal prism.png|100px]]
| 11
| 19
| 10
| <math> C_{2v} </math> of order 4
| <math> \begin{align}
 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{align} </math>
|-
!scope=row | 53
| [[Biaugmented pentagonal prism]]
| [[Image:Biaugmented pentagonal prism.png|100px]]
| 12
| 23
| 13
| <math> C_{2v} </math> of order 4
| <math> \begin{align}
 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{align} </math>
|-
!scope=row | 54
| [[Augmented hexagonal prism]]
| [[Image:Augmented hexagonal prism.png|100px]]
| 13
| 22
| 11
| <math> C_{2v} </math> of order 4
| <math> \begin{align}
 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{align} </math>
|-
!scope=row | 55
| [[Parabiaugmented hexagonal prism]]
| [[Image:Parabiaugmented hexagonal prism.png|100px]]
| 14
| 26
| 14
| <math> D_{2h} </math> of order 8
| <math> \begin{align}
 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{align} </math>
|-
!scope=row | 56
| [[Metabiaugmented hexagonal prism]]
| [[Image:Metabiaugmented hexagonal prism.png|100px]]
| 14
| 26
| 14
| <math> C_{2v} </math> of order 4
| <math> \begin{align}
 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{align} </math>
|-
!scope=row | 57
| [[Triaugmented hexagonal prism]]
| [[Image:Triaugmented hexagonal prism.png|100px]]
| 15
| 30
| 17
| <math> D_{3h} </math> of order 12
| <math> \begin{align}
 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{align} </math>
|-
!scope=row | 58
| [[Augmented dodecahedron]]
| [[Image:Augmented dodecahedron.png|100px]]
| 21
| 35
| 16
| <math> C_{5v} </math> of order 10
| <math> \begin{align}
 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{align} </math>
|-
!scope=row | 59
| [[Parabiaugmented dodecahedron]]
| [[Image:Parabiaugmented dodecahedron.png|100px]]
| 22
| 40
| 20
| <math> D_{5d} </math> of order 20
| <math> \begin{align}
 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{align} </math>
|-
!scope=row | 60
| [[Metabiaugmented dodecahedron]]
| [[Image:Metabiaugmented dodecahedron.png|100px]]
| 22
| 40
| 20
| <math> C_{2v} </math> of order 4
| <math> \begin{align}
 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{align} </math>
|-
!scope=row | 61
| [[Triaugmented dodecahedron]]
| [[Image:Triaugmented dodecahedron.png|100px]]
| 23
| 45
| 24
| <math> C_{3v} </math> of order 6
| <math> \begin{align}
 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{align} </math>
|-
!scope=row | 62
| [[Metabidiminished icosahedron]]
| [[Image:Metabidiminished icosahedron.png|100px]]
| 10
| 20
| 12
| <math> C_{2v} </math> of order 4
| <math> \begin{align}
 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{align} </math>
|-
!scope=row | 63
| [[Tridiminished icosahedron]]
| [[Image:Tridiminished icosahedron.png|100px]]
| 9
| 15
| 8
| <math> C_{3v} </math> of order 6
| <math> \begin{align}
 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{align} </math>
|-
!scope=row | 64
| [[Augmented tridiminished icosahedron]]
| [[Image:Augmented tridiminished icosahedron.png|100px]]
| 10
| 18
| 10
| <math> C_{3v} </math> of order 6
| <math> \begin{align}
 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{align} </math>
|-
!scope=row | 65
| [[Augmented truncated tetrahedron]]
| [[Image:Augmented truncated tetrahedron.png|100px]]
| 15
| 27
| 14
| <math> C_{3v} </math> of order 6
| <math> \begin{align}
 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{align} </math>
|-
!scope=row | 66
| [[Augmented truncated cube]]
| [[Image:Augmented truncated cube.png|100px]]
| 28
| 48
| 22
| <math> C_{4v} </math> of order 8
| <math> \begin{align}
 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{align} </math>
|-
!scope=row | 67
| [[Biaugmented truncated cube]]
| [[Image:Biaugmented truncated cube.png|100px]]
| 32
| 60
| 30
| <math> D_{4h} </math> of order 16
| <math> \begin{align}
 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{align} </math>
|-
!scope=row | 68
| [[Augmented truncated dodecahedron]]
| [[Image:Augmented truncated dodecahedron.png|100px]]
| 65
| 105
| 42
| <math> C_{5v} </math> of order 10
| <math> \begin{align}
 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{align} </math>
|-
!scope=row | 69
| [[Parabiaugmented truncated dodecahedron]]
| [[Image:Parabiaugmented truncated dodecahedron.png|100px]]
| 70
| 120
| 52
| <math> D_{5d} </math> of order 20
| <math> \begin{align}
 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{align} </math>
|-
!scope=row | 70
| [[Metabiaugmented truncated dodecahedron]]
| [[Image:Metabiaugmented truncated dodecahedron.png|100px]]
| 70
| 120
| 52
| <math> C_{2v} </math> of order 4
| <math> \begin{align}
 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{align} </math>
|-
!scope=row | 71
| [[Triaugmented truncated dodecahedron]]
| [[Image:Triaugmented truncated dodecahedron.png|100px]]
| 75
| 135
| 62
| <math> C_{3v} </math> of order 6
| <math> \begin{align}
 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{align} </math>
|-
!scope=row | 72
| [[Gyrate rhombicosidodecahedron]]
| [[Image:Gyrate rhombicosidodecahedron.png|100px]]
| 60
| 120
| 62
| <math> C_{5v} </math> of order 10
| <math> \begin{align}
 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{align} </math>
|-
!scope=row | 73
| [[Parabigyrate rhombicosidodecahedron]]
| [[Image:Parabigyrate rhombicosidodecahedron.png|100px]]
| 60
| 120
| 62
| <math> D_{5d} </math> of order 20
| <math> \begin{align}
 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{align} </math>
|-
!scope=row | 74
| [[Metabigyrate rhombicosidodecahedron]]
| [[Image:Metabigyrate rhombicosidodecahedron.png|100px]]
| 60
| 120
| 62
| <math> C_{2v} </math> of order 4
| <math> \begin{align}
 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{align} </math>
|-
!scope=row | 75
| [[Trigyrate rhombicosidodecahedron]]
| [[Image:Trigyrate rhombicosidodecahedron.png|100px]]
| 60
| 120
| 62
| <math> C_{3v} </math> of order 6
| <math> \begin{align}
 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{align} </math>
|-
!scope=row | 76
| [[Diminished rhombicosidodecahedron]]
| [[Image:Diminished rhombicosidodecahedron.png|100px]]
| 55
| 105
| 52
| <math> C_{5v} </math> of order 10
| <math> \begin{align}
 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{align} </math>
|-
!scope=row | 77
| [[Paragyrate diminished rhombicosidodecahedron]]
| [[Image:Paragyrate diminished rhombicosidodecahedron.png|100px]]
| 55
| 105
| 52
| <math> C_{5v} </math> of order 10
| <math> \begin{align}
 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{align} </math>
|-
!scope=row | 78
| [[Metagyrate diminished rhombicosidodecahedron]]
| [[Image:Metagyrate diminished rhombicosidodecahedron.png|100px]]
| 55
| 105
| 52
| <math> C_s </math> of order 2
| <math> \begin{align}
 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{align} </math>
|-
!scope=row | 79
| [[Bigyrate diminished rhombicosidodecahedron]]
| [[Image:Bigyrate diminished rhombicosidodecahedron.png|100px]]
| 55
| 105
| 52
| <math> C_s </math> of order 2
| <math> \begin{align}
 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{align} </math>
|-
!scope=row | 80
| [[Parabidiminished rhombicosidodecahedron]]
| [[Image:Parabidiminished rhombicosidodecahedron.png|100px]]
| 50
| 90
| 42
| <math> D_{5d} </math> of order 20
| <math> \begin{align}
 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{align} </math>
|-
!scope=row | 81
| [[Metabidiminished rhombicosidodecahedron]]
| [[Image:Metabidiminished rhombicosidodecahedron.png|100px]]
| 50
| 90
| 42
| <math> C_{2v} </math> of order 4
| <math> \begin{align}
 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{align} </math>
|-
!scope=row | 82
| [[Gyrate bidiminished rhombicosidodecahedron]]
| [[Image:Gyrate bidiminished rhombicosidodecahedron.png|100px]]
| 50
| 90
| 42
| <math> C_s </math> of order 2
| <math> \begin{align}
 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{align} </math>
|-
!scope=row | 83
| [[Tridiminished rhombicosidodecahedron]]
| [[Image:Tridiminished rhombicosidodecahedron.png|100px]]
| 45
| 75
| 32
| <math> C_{3v} </math> of order 6
| <math> \begin{align}
 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{align} </math>
|-
!scope=row | 84
| [[Snub disphenoid]]
| [[Image:Snub disphenoid.png|100px]]
| 8
| 18
| 12
| <math> D_{2d} </math> of order 8
| <math> \begin{align}
 A &= 3 \sqrt{3}a^2 \\
 &\approx 5.1962a^2 \\
 V &\approx 0.8595a^3
\end{align} </math>
|-
!scope=row | 85
| [[Snub square antiprism]]
| [[Image:Snub square antiprism.png|100px]]
| 16
| 40
| 26
| <math> D_{4d} </math> of order 16
| <math> \begin{align}
 A &= 2 \left(1+3 \sqrt{3}\right)a^2 \\
 &\approx 12.3923a^2 \\
 V &\approx 3.6012a^3
\end{align} </math>
|-
!scope=row | 86
| [[Sphenocorona]]
| [[Image:Sphenocorona.png|100px]]
| 10
| 22
| 14
| <math> C_{2v} </math> of order 4
| <math> \begin{align}
 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{align} </math>
|-
!scope=row | 87
| [[Augmented sphenocorona]]
| [[Image:Augmented sphenocorona.png|100px]]
| 11
| 26
| 17
| <math> C_s </math> of order 2
| <math> \begin{align}
 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{align} </math>
|-
!scope=row | 88
| [[Sphenomegacorona]]
| [[Image:Sphenomegacorona.png|100px]]
| 12
| 28
| 18
| <math> C_{2v} </math> of order 4
| <math> \begin{align}
 A &= 2 \left(1+2 \sqrt{3}\right)a^2 \\
 &\approx 8.9282a^2 \\
 V &\approx 1.9481a^3
\end{align} </math>
|-
!scope=row | 89
| [[Hebesphenomegacorona]]
| [[Image:Hebesphenomegacorona.png|100px]]
| 14
| 33
| 21
| <math> C_{2v} </math> of order 4
| <math> \begin{align}
 A &= \frac{3}{2} \left(2+3 \sqrt{3}\right)a^2 \\
 &\approx 10.7942a^2 \\
 V &\approx 2.9129a^3
\end{align} </math>
|-
!scope=row | 90
| [[Disphenocingulum]]
| [[Image:Disphenocingulum.png|100px]]
| 16
| 38
| 24
| <math> D_{2d} </math> of order 8
| <math> \begin{align}
 A &= (4+5 \sqrt{3})a^2 \\
 &\approx 12.6603a^2 \\
 V &\approx 3.7776a^3
\end{align} </math>
|-
!scope=row | 91
| [[Bilunabirotunda]]
| [[Image:Bilunabirotunda.png|100px]]
| 14
| 26
| 14
| <math> D_{2h} </math> of order 8
| <math> \begin{align}
 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{align} </math>
|-
!scope=row | 92
| [[Triangular hebesphenorotunda]]
| [[Image:Triangular hebesphenorotunda.png|100px]]
| 18
| 36
| 20
| <math> C_{3v} </math> of order 6
| <math> \begin{align}
 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{align} </math>
|}

== References ==
{{reflist}}

== Bibliography ==
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{{Refend}}

== External links==
* {{Cite web
 | last = Hart | first = George W. | author-link = George W. Hart
 | title = Johnson Solids
 | url = http://www.georgehart.com/virtual-polyhedra/johnson-info.html
}}
* {{Cite web
 | url = http://bulatov.org/polyhedra/johnson/
 | title = Johnson solids
 | last = Bulatov | first = Vladimir
}}{{snd}}VRML models of Johnson solids
* {{Cite journal
 | last = Gagnon | first = Sylvain
 | url = https://upcommons.upc.edu/bitstream/handle/2099/890/st6-11-a7.pdf
 | title = Les polyèdres convexes aux faces régulières
 | trans-title = Convex polyhedra with regular faces
 | journal = Topologie Structurale
 | trans-journal = Structural Topology
 | issue = 6
 | year = 1982
 | pages = 83–95
 | language = fr
}}

{{Johnson solids navigator}}

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[[Category:Johnson solids|*]]
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