Elongated pentagonal cupola

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.

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

A Johnson solid is one of 92 strictly convex polyhedra that is composed of regular polygon faces but are not uniform polyhedra (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]

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

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

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