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.

Properties

The pentagonal cupola's faces are five equilateral triangles, five squares, one regular pentagon, and one regular decagon.Template:R It has the property of convexity and regular polygonal faces, from which it is classified as the fifth Johnson solid.Template:R This cupola cannot be sliced by a plane without cutting within a face, so it is an elementary polyhedron.Template:R

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 $a$:Template:R $$\begin{array}{rlr}h& =\sqrt{\frac{5-\sqrt{5}}{10}}a& \approx 0.526a,\\ R& =\frac{\sqrt{11+4\sqrt{5}}}{2}a& \approx 2.233a,\\ A& =\frac{20+5\sqrt{3}+\sqrt{5(145+62\sqrt{5})}}{4}{a}^2& \approx 16.580a^2,\\ V& =\frac{5+4\sqrt{5}}{6}{a}^3& \approx 2.324a^3.\end{array}$$

File:Cupula pentagonal 3D.stl

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 $C_{5\mathrm{v}}$ of order ten.Template:R

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.Template:R 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 $J_{74}$
• trigyrate rhombicosidodecahedron $J_{75}$

Relatedly, a construction from polyhedra by removing one or more pentagonal cupolas is known as diminishmentTemplate:R:

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

External links

• Template:Mathworld2