Pentagonal cupola

Pentagonal cupola
Type	Johnson J4 – J5 – J6
Faces	5 triangles 5 squares 1 pentagon 1 decagon
Edges	25
Vertices	15
Vertex configuration	${\displaystyle 10\times (3\times 4\times 10)}$ ${\displaystyle 5\times (3\times 4\times 5\times 4)}$
Symmetry group	${\displaystyle C_{\mathrm {v} }}$
Properties	convex, elementary
Net

In geometry, the pentagonal cupola is one of the Johnson solids (J₅). It can be obtained as a slice of the rhombicosidodecahedron. The pentagonal cupola consists of 5 equilateral triangles, 5 squares, 1 pentagon, and 1 decagon.

The pentagonal cupola's faces are five equilateral triangles, five squares, one regular pentagon, and one regular decagon.^[1] It has the property of convexity and regular polygonal faces, from which it is classified as the fifth Johnson solid.^[2] This cupola produces two or more regular polyhedrons by slicing it with a plane, an elementary polyhedron's example.^[3]

The following formulae for circumradius ${\displaystyle R}$, and height ${\displaystyle h}$, surface area ${\displaystyle A}$, and volume ${\displaystyle V}$ may be applied if all faces are regular with edge length ${\displaystyle a}$:^[4] $${\displaystyle {\begin{aligned}h&={\sqrt {\frac {5-{\sqrt {5}}}{10}}}a&\approx 0.526a,\\R&={\frac {\sqrt {11+4{\sqrt {5}}}}{2}}a&\approx 2.233a,\\A&={\frac {20+5{\sqrt {3}}+{\sqrt {5\left(145+62{\sqrt {5}}\right)}}}{4}}a^{2}&\approx 16.580a^{2},\\V&={\frac {5+4{\sqrt {5}}}{6}}a^{3}&\approx 2.324a^{3}.\end{aligned}}}$$

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 ${\displaystyle C_{5\mathrm {v} }}$ of order ten.^[3]

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.^[5][6] Some of the Johnson solids with such constructions are: elongated pentagonal cupola ${\displaystyle J_{20}}$, gyroelongated pentagonal cupola ${\displaystyle J_{24}}$, pentagonal orthobicupola ${\displaystyle J_{30}}$, pentagonal gyrobicupola ${\displaystyle J_{31}}$, pentagonal orthocupolarotunda ${\displaystyle J_{32}}$, pentagonal gyrocupolarotunda ${\displaystyle J_{33}}$, elongated pentagonal orthobicupola ${\displaystyle J_{38}}$, elongated pentagonal gyrobicupola ${\displaystyle J_{39}}$, elongated pentagonal orthocupolarotunda ${\displaystyle J_{40}}$, gyroelongated pentagonal bicupola ${\displaystyle J_{46}}$, gyroelongated pentagonal cupolarotunda ${\displaystyle J_{47}}$, augmented truncated dodecahedron ${\displaystyle J_{68}}$, parabiaugmented truncated dodecahedron ${\displaystyle J_{69}}$, metabiaugmented truncated dodecahedron ${\displaystyle J_{70}}$, triaugmented truncated dodecahedron ${\displaystyle J_{71}}$, gyrate rhombicosidodecahedron ${\displaystyle J_{72}}$, parabigyrate rhombicosidodecahedron ${\displaystyle J_{73}}$, metabigyrate rhombicosidodecahedron ${\displaystyle J_{74}}$, and trigyrate rhombicosidodecahedron ${\displaystyle J_{75}}$. Relatedly, a construction from polyhedra by removing one or more pentagonal cupolas is known as diminishment: diminished rhombicosidodecahedron ${\displaystyle J_{76}}$, paragyrate diminished rhombicosidodecahedron ${\displaystyle J_{77}}$, metagyrate diminished rhombicosidodecahedron ${\displaystyle J_{78}}$, bigyrate diminished rhombicosidodecahedron ${\displaystyle J_{79}}$, parabidiminished rhombicosidodecahedron ${\displaystyle J_{80}}$, metabidiminished rhombicosidodecahedron ${\displaystyle J_{81}}$, gyrate bidiminished rhombicosidodecahedron ${\displaystyle J_{82}}$, and tridiminished rhombicosidodecahedron ${\displaystyle J_{83}}$.^[1]

• Weisstein, Eric W., "Pentagonal cupola" ("Johnson solid") at MathWorld.