Pentagonal cupola
Template:Short description {{#invoke:Infobox|infobox}}Template:Template other </math>
| properties = convex, elementary | net = Pentagonal Cupola.PNG
}}
PropertiesEdit
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 produces two or more regular polyhedrons by slicing it with a plane, an elementary polyhedron's example.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 polyhedronEdit
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>, and trigyrate rhombicosidodecahedron <math> J_{75} </math>. Relatedly, a construction from polyhedra by removing one or more pentagonal cupolas is known as diminishment: 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>, and tridiminished rhombicosidodecahedron <math> J_{83} </math>.Template:R