# Elongated pentagonal cupola

Elongated pentagonal cupola
Type Johnson
J19 - J20 - J21
Faces 5 triangles
15 squares
1 pentagon
1 decagon
Edges 45
Vertices 25
Vertex configuration 10(42.10)
10(3.43)
5(3.4.5.4)
Symmetry group C5v
Dual polyhedron -
Properties convex
Net

In geometry, the elongated pentagonal cupola is one of the Johnson solids (J20). As the name suggests, it can be constructed by elongating a pentagonal cupola (J5) by attaching a decagonal prism to its base. The solid can also be seen as an elongated pentagonal orthobicupola (J38) with its "lid" (another pentagonal cupola) removed.

A Johnson solid is one of 92 strictly convex polyhedra that have regular faces but are not uniform (that is, they are not Platonic solids, Archimedean solids, prisms or antiprisms). They were named by Norman Johnson, who first listed these polyhedra in 1966.[1]

## Formulas

The following formulas for the volume and surface area can be used if all faces are regular, with edge length a:[2]

${\displaystyle V=({\frac {1}{6}}(5+4{\sqrt {5}}+15{\sqrt {5+2{\sqrt {5}}}}))a^{3}\approx 10.0183...a^{3}}$

${\displaystyle A=({\frac {1}{4}}(60+{\sqrt {10(80+31{\sqrt {5}}+{\sqrt {2175+930{\sqrt {5}}}})}}))a^{2}\approx 26.5797...a^{2}}$

### Dual polyhedron

The dual of the elongated pentagonal cupola has 25 faces: 10 isosceles triangles, 5 kites, and 10 quadrilaterals.

Dual elongated pentagonal cupola Net of dual

## References

1. ^ Johnson, Norman W. (1966), "Convex polyhedra with regular faces", Canadian Journal of Mathematics, 18: 169–200, doi:10.4153/cjm-1966-021-8, MR 0185507, Zbl 0132.14603.
2. ^ Stephen Wolfram, "Elongated pentagonal cupola" from Wolfram Alpha. Retrieved July 22, 2010.