Pentagonal orthobicupola

In geometry, the pentagonal orthobicupola is one of the Johnson solids (J₃₀). As the name suggests, it can be constructed by joining two pentagonal cupolae (J₅) along their decagonal bases, matching like faces. A 36-degree rotation of one cupola before the joining yields a pentagonal gyrobicupola (J₃₁).

Pentagonal orthobicupola
Type	Bicupola, Johnson J29 – J30 – J31
Faces	10 triangles 10 squares 2 pentagons
Edges	40
Vertices	20
Vertex configuration	10(32.42) 10(3.4.5.4)
Symmetry group	D5h
Dual polyhedron	-
Properties	convex
Net

The pentagonal orthobicupola is the third in an infinite set of orthobicupolae.

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]

${\displaystyle V={\frac {1}{3}}\left(5+4{\sqrt {5}}\right)a^{3}\approx 4.64809...a^{3}}$

${\displaystyle A=\left(10+{\sqrt {{\frac {5}{2}}\left(10+{\sqrt {5}}+{\sqrt {75+30{\sqrt {5}}}}\right)}}\right)a^{2}\approx 17.7711...a^{2}}$