Pentagonal cupola



In geometry, the pentagonal cupola is one of the Johnson solids ($J4 – J5 – J6$). 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.

Formulae
The following formulae for volume, surface area and circumradius can be used if all faces are regular, with edge length a:


 * $$V=\left(\frac{1}{6}\left(5+4\sqrt{5}\right)\right)a^3\approx2.32405a^3,$$
 * $$A=\left(\frac{1}{4}\left(20+5\sqrt{3}+\sqrt{5\left(145+62\sqrt{5}\right)}\right)\right)a^2\approx16.57975a^2,$$
 * $$R=\left(\frac{1}{2}\sqrt{11+4\sqrt{5}}\right)a\approx2.23295a.$$

The height of the pentagonal cupola is
 * $$h = \sqrt{\frac{5 - \sqrt{5}}{10}}a \approx 0.52573a$$.

Dual polyhedron
The dual of the pentagonal cupola has 10 triangular faces and 5 kite faces:

Crossed pentagrammic cupola
In geometry, the crossed pentagrammic cupola is one of the nonconvex Johnson solid isomorphs, being topologically identical to the convex pentagonal cupola. It can be obtained as a slice of the nonconvex great rhombicosidodecahedron or quasirhombicosidodecahedron, analogously to how the pentagonal cupola may be obtained as a slice of the rhombicosidodecahedron. As in all cupolae, the base polygon has twice as many edges and vertices as the top; in this case the base polygon is a decagram.

It may be seen as a cupola with a retrograde pentagrammic base, so that the squares and triangles connect across the bases in the opposite way to the pentagrammic cuploid, hence intersecting each other more deeply.