Gyroelongated pentagonal rotunda

In geometry, the gyroelongated pentagonal rotunda is one of the Johnson solids (J₂₅). As the name suggests, it can be constructed by gyroelongating a pentagonal rotunda (J₆) by attaching a decagonal antiprism to its base. It can also be seen as a gyroelongated pentagonal birotunda (J₄₈) with one pentagonal rotunda removed.

Gyroelongated pentagonal rotunda
Type	Johnson J24 - J25 - J26
Faces	4x5+10 triangles 1+5 pentagons 1 decagon
Edges	65
Vertices	30
Vertex configuration	2.5(3.5.3.5) 2.5(33.10) 10(34.5)
Symmetry group	C5v
Dual polyhedron	see above
Properties	convex
Net

A Johnson solid is one of 92 strictly convex polyhedra that is composed of regular polygon faces but are not uniform polyhedra (that is, they are not Platonic solids, Archimedean solids, prisms, or antiprisms). They were named by Norman Johnson, who first listed these polyhedra in 1966.^[1]

Area and Volume

With edge length a, the surface area is

${\displaystyle A={\frac {1}{2}}\left(15{\sqrt {3}}+\left(5+3{\sqrt {5}}\right){\sqrt {5+2{\sqrt {5}}}}\right)a^{2}\approx 31.007454303...a^{2},}$

and the volume is

${\displaystyle V=\left({\frac {45}{12}}+{\frac {17}{12}}{\sqrt {5}}+{\frac {5}{6}}{\sqrt {2{\sqrt {650+290{\sqrt {5}}}}-2{\sqrt {5}}-2}}\right)a^{3}\approx 13.667050844...a^{3}.}$

Dual polyhedron

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

Dual gyroelongated pentagonal rotunda	Net of dual