Gyroelongated square cupola



In geometry, the gyroelongated square cupola is one of the Johnson solids (J23). As the name suggests, it can be constructed by gyroelongating a square cupola (J4) by attaching an octagonal antiprism to its base. It can also be seen as a gyroelongated square bicupola (J45) with one square bicupola removed.

Area and Volume
The surface area is,


 * $$A=\left(7+2\sqrt{2}+5\sqrt{3}\right)a^2\approx 18.4886811...a^2.$$

The volume is the sum of the volume of a square cupola and the volume of an octagonal prism,


 * $$V=\left(1+\frac{2}{3}\sqrt{2} + \frac{2}{3}\sqrt{4+2\sqrt{2}+2\sqrt{146+103\sqrt{2}}}\right)a^3\approx6.2107658...a^3.$$

Dual polyhedron
The dual of the gyroelongated square cupola has 20 faces: 8 kites, 4 rhombi, and 8 pentagons.