Gyroelongated pentagonal cupola

In geometry, the gyroelongated pentagonal cupola is one of the Johnson solids (J₂₄). As the name suggests, it can be constructed by gyroelongating a pentagonal cupola (J₅) by attaching a decagonal antiprism to its base. It can also be seen as a gyroelongated pentagonal bicupola (J₄₆) with one pentagonal cupola removed.

Template:Johnson solid File:J24 gyroelongated pentagonal cupola.stl Template:-

Area and volume

With edge length a, the surface area is

$A=\frac{1}{4}(20+25\sqrt{3}+(10+\sqrt{5})\sqrt{5+2\sqrt{5}})a^2\approx \mathrm{25.240003791...}{a}^2,$

and the volume is

$V=(\frac{5}{6}+\frac{2}{3}\sqrt{5}+\frac{5}{6}\sqrt{2\sqrt{650+290\sqrt{5}}-2\sqrt{5}-2})a^3\approx \mathrm{9.073333194...}{a}^3.$

Dual polyhedron

The dual of the gyroelongated pentagonal cupola has 25 faces: 10 kites, 5 rhombi, and 10 pentagons.

Dual gyroelongated pentagonal cupola	Net of dual

External links

• Template:Mathworld2