geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, the gyroelongated pentagonal 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''₂₄). As the name suggests, it can be constructed by gyroelongating a

pentagonal cupola In geometry, the pentagonal cupola is one of the Johnson solids (). It can be obtained as a slice of the rhombicosidodecahedron. The pentagonal cupola consists of 5 equilateral triangles, 5 squares, 1 pentagon, and 1 decagon. Formulae The fol ...

(''J''₅) by attaching a decagonal antiprism to its base. It can also be seen as a

gyroelongated pentagonal bicupola In geometry, the gyroelongated pentagonal bicupola 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 each face must be the same ...

(''J''₄₆) with one pentagonal cupola removed.

Area and Volume

With edge length a, the surface area is :

A=\frac\left( 20+25\sqrt+\left(10+\sqrt\right)\sqrt\right)a^2\approx25.240003791...a^2,

and the volume is :

V=\left(\frac+\frac\sqrt + \frac\sqrt\right) a^3\approx 9.073333194...a^3.

Dual polyhedron

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

External links

* {{Johnson solids navigator Johnson solids