Elongated pentagonal rotunda

Elongated pentagonal rotunda
Type Johnson
J20 - J21 - J22
Faces 2.5 triangles
2.5 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

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.

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]

Formulae

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

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

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

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

References

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