# Elongated pentagonal gyrobicupola

Elongated pentagonal gyrobicupola
TypeJohnson
J38 - J39 - J40
Faces10 triangles
2x10 squares
2 pentagons
Edges60
Vertices30
Vertex configuration20(3.43)
10(3.4.5.4)
Symmetry groupD5d
Dual polyhedron-
Propertiesconvex
Net

In geometry, the elongated pentagonal gyrobicupola is one of the Johnson solids (J39). As the name suggests, it can be constructed by elongating a pentagonal gyrobicupola (J31) by inserting a decagonal prism between its congruent halves. Rotating one of the pentagonal cupolae (J5) through 36 degrees before inserting the prism yields an elongated pentagonal orthobicupola (J38).

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]

${\displaystyle V={\frac {1}{6}}\left(10+8{\sqrt {5}}+15{\sqrt {5+2{\sqrt {5}}}}\right)a^{3}\approx 12.3423...a^{3}}$

${\displaystyle A=\left(20+{\sqrt {{\frac {5}{2}}\left(10+{\sqrt {5}}+{\sqrt {75+30{\sqrt {5}}}}\right)}}\right)a^{2}\approx 27.7711...a^{2}}$

## References

1. ^ Johnson, Norman W. (1966), "Convex polyhedra with regular faces", Canadian Journal of Mathematics, 18: 169–200, doi:10.4153/cjm-1966-021-8, MR 0185507, Zbl 0132.14603.
2. ^ Stephen Wolfram, "Elongated pentagonal gyrobicupola" from Wolfram Alpha. Retrieved July 25, 2010.