# Pentagonal cupola

Pentagonal cupola
Type Johnson
J4 - J5 - J6
Faces 5 triangles
5 squares
1 pentagon
1 decagon
Edges 25
Vertices 15
Vertex configuration 10(3.4.10)
5(3.4.5.4)
Symmetry group C5v, [5], (*55)
Rotation group C5, [5]+, (55)
Dual polyhedron -
Properties convex
Net

In geometry, the pentagonal cupola is one of the Johnson solids (J5). It can be obtained as a slice of the rhombicosidodecahedron.

The 92 Johnson solids were named and described by Norman Johnson in 1966.

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:[1]

$V=\left(\frac{1}{6}\left(5+4\sqrt{5}\right)\right)a^3\approx2.32405...a^3$

$A=\left(\frac{1}{4}\left(20+5\sqrt{3}+\sqrt{5(145+62\sqrt{5})}\right)\right)a^2=\left(\frac{1}{4}\left(20+\sqrt{10\left(80+31\sqrt{5}+\sqrt{15(145+62\sqrt{5})}\right)}\right)\right)a^2\approx16.5797...a^2$

$C=\left(\frac{1}{2}\sqrt{11+4\sqrt{5}}\right)a\approx2.23295...a$

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## Related polyhedra

### Dual polyhedron

The dual of the pentagonal cupola has 20 triangular faces:

Dual pentagonal cupola Net of dual
Family of cupolae
2 3 4 5 6

Digonal cupola

Triangular cupola

Square cupola

Pentagonal cupola

Hexagonal cupola
(Flat)
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## References

1. ^ Stephen Wolfram, "Pentagonal cupola" from Wolfram Alpha. Retrieved July 21, 2010.
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