In geometry, the gyroelongated square cupola is one of the

Johnson solid In geometry, a Johnson solid is a strictly convex polyhedron each face of which is a regular polygon. There is no requirement that isohedral, each face must be the same polygon, or that the same polygons join around each Vertex (geometry), ver ...

s (''J''₂₃). As the name suggests, it can be constructed by gyroelongating a square cupola (''J''₄) by attaching an

octagonal antiprism In geometry, the octagonal antiprism is the 6th in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps. Antiprisms are similar to prisms except the bases are twisted relative to each other ...

to its base. It can also be seen as a

gyroelongated square bicupola In geometry, the gyroelongated square bicupola is one of the Johnson solids (). As the name suggests, it can be constructed by gyroelongating a square bicupola ( or ) by inserting an octagon In geometry, an octagon (from the Greek ὀκτάγ� ...

(''J''₄₅) with one square bicupola removed.

Area and Volume

The surface area is, :

A=\left(7+2\sqrt+5\sqrt\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\sqrt + \frac\sqrt\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.

External links

* {{Johnson solids navigator Johnson solids