# Elongated pentagonal cupola

Elongated pentagonal cupola TypeJohnson
J19 - J20 - J21
Faces5 triangles
15 squares
1 pentagon
1 decagon
Edges45
Vertices25
Vertex configuration10(42.10)
10(3.43)
5(3.4.5.4)
Symmetry groupC5v
Dual polyhedron-
Propertiesconvex
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.

## Formulas

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

$V=({\frac {1}{6}}(5+4{\sqrt {5}}+15{\sqrt {5+2{\sqrt {5}}}}))a^{3}\approx 10.0183...a^{3}$ $A=({\frac {1}{4}}(60+{\sqrt {10(80+31{\sqrt {5}}+{\sqrt {2175+930{\sqrt {5}}}})}}))a^{2}\approx 26.5797...a^{2}$ ### Dual polyhedron

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