Pentagonal cupola
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In geometry, the pentagonal cupola is one of the Johnson solids (Template:Math). 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 <math> R </math>, and height <math> h </math>, surface area <math> A </math>, and volume <math> V </math> may be applied if all faces are regular with edge length <math> a </math>:Template:R <math display="block"> \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} </math>
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 <math> C_{5\mathrm{v}} </math> 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 <math> J_{20} </math>
- gyroelongated pentagonal cupola <math> J_{24} </math>
- pentagonal orthobicupola <math> J_{30} </math>
- pentagonal gyrobicupola <math> J_{31} </math>
- pentagonal orthocupolarotunda <math> J_{32} </math>
- pentagonal gyrocupolarotunda <math> J_{33} </math>
- elongated pentagonal orthobicupola <math> J_{38} </math>
- elongated pentagonal gyrobicupola <math> J_{39} </math>
- elongated pentagonal orthocupolarotunda <math> J_{40} </math>
- gyroelongated pentagonal bicupola <math> J_{46} </math>
- gyroelongated pentagonal cupolarotunda <math> J_{47} </math>
- augmented truncated dodecahedron <math> J_{68} </math>
- parabiaugmented truncated dodecahedron <math> J_{69} </math>
- metabiaugmented truncated dodecahedron <math> J_{70} </math>
- triaugmented truncated dodecahedron <math> J_{71} </math>
- gyrate rhombicosidodecahedron <math> J_{72} </math>
- parabigyrate rhombicosidodecahedron <math> J_{73} </math>
- metabigyrate rhombicosidodecahedron <math> J_{74} </math>
- trigyrate rhombicosidodecahedron <math> J_{75} </math>
Relatedly, a construction from polyhedra by removing one or more pentagonal cupolas is known as diminishmentTemplate:R:
- diminished rhombicosidodecahedron <math> J_{76} </math>
- paragyrate diminished rhombicosidodecahedron <math> J_{77} </math>
- metagyrate diminished rhombicosidodecahedron <math> J_{78} </math>
- bigyrate diminished rhombicosidodecahedron <math> J_{79} </math>
- parabidiminished rhombicosidodecahedron <math> J_{80} </math>
- metabidiminished rhombicosidodecahedron <math> J_{81} </math>
- gyrate bidiminished rhombicosidodecahedron <math> J_{82} </math>
- tridiminished rhombicosidodecahedron <math> J_{83} </math>