pentagonal cupola

The pentagonal cupola's faces are five equilateral triangles, five squares, one regular pentagon, and one regular decagon.{{r|berman}} It has the property of convexity and regular polygonal faces, from which it is classified as the fifth Johnson solid.{{r|uehara}} This cupola produces two or more regular polyhedrons by slicing it with a plane, an elementary polyhedron's example.{{r|johnson}}

The following formulae for circumradius $R$, and height $h$, surface area $A$, and volume $V$ may be applied if all faces are regular with edge length $a$:{{r|bcp}}

$$\backslash begin\{align\}$$

h &= \sqrt{\frac{5 - \sqrt{5}}{10}}a &\approx 0.526a, \\

R &= \frac{\sqrt{11+4\sqrt{5}}}{2}a &\approx 2.233a, \\

A &= \frac{20+5\sqrt{3}+\sqrt{5\left(145+62\sqrt{5}\right)}}{4}a^2 &\approx 16.580a^2, \\

V &= \frac{5+4\sqrt{5}}{6}a^3 &\approx 2.324a^3.

\end{align}

File:Cupula pentagonal 3D.stl

It has an axis of symmetry passing through the center of both top and base, which is symmetrical by rotating around it at one-, two-, three-, and four-fifth of a full-turn angle. It is also mirror-symmetric relative to any perpendicular plane passing through a bisector of the hexagonal base. Therefore, it has pyramidal symmetry, the cyclic group $C\_\{5\backslash mathrm\{v\}\}$ of order ten.{{r|johnson}}

The pentagonal cupola can be applied to construct a polyhedron. A construction that involves the attachment of its base to another polyhedron is known as augmentation; attaching it to prisms or antiprisms is known as elongation or gyroelongation.{{r|demey|slobodan}} Some of the Johnson solids with such constructions are: elongated pentagonal cupola $J\_\{20\}$, gyroelongated pentagonal cupola $J\_\{24\}$, pentagonal orthobicupola $J\_\{30\}$, pentagonal gyrobicupola $J\_\{31\}$, pentagonal orthocupolarotunda $J\_\{32\}$, pentagonal gyrocupolarotunda $J\_\{33\}$, elongated pentagonal orthobicupola $J\_\{38\}$, elongated pentagonal gyrobicupola $J\_\{39\}$, elongated pentagonal orthocupolarotunda $J\_\{40\}$, gyroelongated pentagonal bicupola $J\_\{46\}$, gyroelongated pentagonal cupolarotunda $J\_\{47\}$, augmented truncated dodecahedron $J\_\{68\}$, parabiaugmented truncated dodecahedron $J\_\{69\}$, metabiaugmented truncated dodecahedron $J\_\{70\}$, triaugmented truncated dodecahedron $J\_\{71\}$, gyrate rhombicosidodecahedron $J\_\{72\}$, parabigyrate rhombicosidodecahedron $J\_\{73\}$, metabigyrate rhombicosidodecahedron $J\_\{74\}$, and trigyrate rhombicosidodecahedron $J\_\{75\}$. Relatedly, a construction from polyhedra by removing one or more pentagonal cupolas is known as diminishment: diminished rhombicosidodecahedron $J\_\{76\}$, paragyrate diminished rhombicosidodecahedron $J\_\{77\}$, metagyrate diminished rhombicosidodecahedron $J\_\{78\}$, bigyrate diminished rhombicosidodecahedron $J\_\{79\}$, parabidiminished rhombicosidodecahedron $J\_\{80\}$, metabidiminished rhombicosidodecahedron $J\_\{81\}$, gyrate bidiminished rhombicosidodecahedron $J\_\{82\}$, and tridiminished rhombicosidodecahedron $J\_\{83\}$.{{r|berman}}

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• {{mathworld2 | urlname2 = JohnsonSolid | title2 = Johnson solid| urlname =PentagonalCupola| title =Pentagonal cupola}}

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