Pentagonal cupola

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.

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

Properties

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 cannot be sliced by a plane without cutting within a face, so it is an elementary polyhedron.^[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] $${\displaystyle {\begin{aligned}h&={\sqrt {\frac {5-{\sqrt {5}}}{10}}}a&\approx 0.526a,\\R&={\frac {\sqrt {11+4{\sqrt {5}}}}{2}}a&\approx 2.233a,\\A&={\frac {20+5{\sqrt {3}}+{\sqrt {5\left(145+62{\sqrt {5}}\right)}}}{4}}a^{2}&\approx 16.580a^{2},\\V&={\frac {5+4{\sqrt {5}}}{6}}a^{3}&\approx 2.324a^{3}.\end{aligned}}}$$

3D model of a pentagonal cupola

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]

Related polyhedron

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 ${\displaystyle J_{74}}$
• trigyrate rhombicosidodecahedron ${\displaystyle J_{75}}$

Relatedly, a construction from polyhedra by removing one or more pentagonal cupolas is known as diminishment^[1]:

• 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 ${\displaystyle J_{82}}$
• tridiminished rhombicosidodecahedron ${\displaystyle J_{83}}$