Elongated pentagonal gyrobicupola

Elongated pentagonal gyrobicupola
Type	Johnson J38 - J39 - J40
Faces	10 triangles 2.10 squares 2 pentagons
Edges	60
Vertices	30
Vertex configuration	20(3.4^3) 10(3.4.5.4)
Symmetry group	D5d
Dual polyhedron	-
Properties	convex
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 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]

$V=\frac{1}{6}\left(10+8\sqrt{5}+15\sqrt{5+2\sqrt{5}}\right)a^3\approx12.3423...a^3$

$A=\left(20+\sqrt{\frac{5}{2}\left(10+\sqrt{5}+\sqrt{75+30\sqrt{5}}\right)}\right)a^2\approx27.7711...a^2$

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.

External links

• Eric W. Weisstein, Elongated pentagonal gyrobicupola (Johnson solid) at MathWorld