| Type: | Johnson |
| Faces: | 5 triangles 5 squares 1 pentagon 1 decagon |
| Edges: | 25 |
| Vertices: | 15 |
| Dual: | - |
| Properties: | convex |
| Net: | Pentagonal Cupola.PNG |
In geometry, the pentagonal cupola is one of the Johnson solids . 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:[1]
| V=\left( | 1 |
| 6 |
\left(5+4\sqrt{5}\right)\right)a3 ≈ 2.32405a3,
| A=\left( | 1 |
| 4 |
\left(20+5\sqrt{3}+\sqrt{5\left(145+62\sqrt{5}\right)}\right)\right)a2 ≈ 16.57975a2,
| R=\left( | 1 |
| 2 |
\sqrt{11+4\sqrt{5}}\right)a ≈ 2.23295a.
The height of the pentagonal cupola is [2]
h=\sqrt{
| 5-\sqrt{5 | |
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.