geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, the gyroelongated pentagonal rotunda is one of the

Johnson solid In geometry, a Johnson solid, sometimes also known as a Johnson–Zalgaller solid, is a convex polyhedron whose faces are regular polygons. They are sometimes defined to exclude the uniform polyhedrons. There are ninety-two Solid geometry, s ...

s (''J''₂₅). As the name suggests, it can be constructed by gyroelongating a

pentagonal rotunda The pentagonal rotunda is a convex polyhedron with regular polygonal faces. These faces comprise ten equilateral triangles, six regular pentagons, and one regular decagon, making a total of seventeen. The pentagonal rotunda is an example of Johns ...

(''J''₆) by attaching a decagonal antiprism to its base. It can also be seen as a

gyroelongated pentagonal birotunda In geometry, the gyroelongated pentagonal birotunda is one of the Johnson solids (). As the name suggests, it can be constructed by gyroelongating a pentagonal birotunda (either or the icosidodecahedron) by inserting a decagonal antiprism betwee ...

(''J''₄₈) with one pentagonal rotunda removed.

Area and Volume

With edge length a, the surface area is :

A=\frac\left( 15\sqrt+\left(5+3\sqrt\right)\sqrt\right)a^2\approx31.007454303...a^2,

and the volume is :

V=\left(\frac+\frac\sqrt + \frac\sqrt\right) a^3\approx13.667050844...a^3.

Dual polyhedron

The dual of the gyroelongated pentagonal rotunda has 30 faces: 10 pentagons, 10 rhombi, and 10 quadrilaterals.

External links

* {{Johnson solids navigator Johnson solids