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Concept
Pentagonal cupola
Summary
In geometry, the pentagonal cupola is one of the Johnson solids (J_5). 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.
The following formulae for volume, surface area and circumradius can be used if all faces are regular, with edge length a:
The height of the pentagonal cupola is
The dual of the pentagonal cupola has 10 triangular faces and 5 kite faces:
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.
Official source
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