Gyroelongated pentagonal cupolarotunda

In geometry, the gyroelongated pentagonal cupolarotunda is one of the Johnson solids (Template:Math). As the name suggests, it can be constructed by gyroelongating a pentagonal cupolarotunda (Template:Math or Template:Math) by inserting a decagonal antiprism between its two halves.

The gyroelongated pentagonal cupolarotunda is one of five Johnson solids which are chiral, meaning that they have a "left-handed" and a "right-handed" form. In the illustration to the right, each pentagonal face on the bottom half of the figure is connected by a path of two triangular faces to a square face above it and to the left. In the figure of opposite chirality (the mirror image of the illustrated figure), each bottom pentagon would be connected to a square face above it and to the right. The two chiral forms of Template:Math are not considered different Johnson solids.

Area and Volume

With edge length a, the surface area is

$A=\frac{1}{4}(20+35\sqrt{3}+7\sqrt{25+10\sqrt{5}})a^2\approx 32.198786370...a^2,$

and the volume is

$V=(\frac{55}{12}+\frac{25}{12}\sqrt{5}+\frac{5}{6}\sqrt{2\sqrt{650+290\sqrt{5}}-2\sqrt{5}-2})a^3\approx 15.991096162...a^3.$

External links

• Template:Mathworld2