Pentagonal cupola

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

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]