Elongated pentagonal rotunda

In geometry, the elongated pentagonal rotunda is one of the Johnson solids (J₂₁). As the name suggests, it can be constructed by elongating a pentagonal rotunda (J₆) by attaching a decagonal prism to its base. It can also be seen as an elongated pentagonal orthobirotunda (J₄₂) with one pentagonal rotunda removed.

Elongated pentagonal rotunda
Type	Johnson J20 - J21 - J22
Faces	2x5 triangles 2x5 squares 1+5 pentagons 1 decagon
Edges	55
Vertices	30
Vertex configuration	10(42.10) 10(3.42.5) 2.5(3.5.3.5)
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]

Formulae

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

${\displaystyle V={\frac {1}{12}}\left(45+17{\sqrt {5}}+30{\sqrt {5+2{\sqrt {5}}}}\right)a^{3}\approx 14.612...a^{3}}$

${\displaystyle A={\frac {1}{2}}\left(20+{\sqrt {5\left(145+58{\sqrt {5}}+2{\sqrt {30\left(65+29{\sqrt {5}}\right)}}\right)}}\right)a^{2}\approx 32.3472...a^{2}}$

Dual polyhedron

The dual of the elongated pentagonal rotunda has 30 faces: 10 isosceles triangles, 10 rhombi, and 10 quadrilaterals.

Dual elongated pentagonal rotunda	Net of dual