Gyroelongated pentagonal cupolarotunda

In geometry, the gyroelongated pentagonal cupolarotunda is one of the Johnson solids ($J46 – J47 – J48$). As the name suggests, it can be constructed by gyroelongating a pentagonal cupolarotunda ($C5$ or $5(3.4.5.4) 2.5(3.5.3.5) 2.5(34.4) 2.5(34.5)$) 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 $J47$ are not considered different Johnson solids.

Area and Volume
With edge length a, the surface area is


 * $$A=\frac{1}{4}\left(20+35\sqrt{3}+7\sqrt{25+10\sqrt{5}}\right)a^2\approx32.198786370...a^2,$$

and the volume is


 * $$V=\left(\frac{55}{12}+\frac{25}{12}\sqrt{5}+ \frac{5}{6}\sqrt{2\sqrt{650+290\sqrt{5}}-2\sqrt{5}-2}\right) a^3\approx15.991096162...a^3.$$