Gyroelongated pentagonal rotunda

Gyroelongated pentagonal rotunda
Type	Johnson J24 – J25 – J26
Faces	4×5+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

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.

A Johnson solid is one of 92 strictly convex polyhedra that are 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]

3D model of a gyroelongated pentagonal rotunda

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

References

1. Johnson, Norman W. (1966), "Convex polyhedra with regular faces", Canadian Journal of Mathematics , 18: 169–200, doi: 10.4153/cjm-1966-021-8 , MR   0185507, Zbl   0132.14603 .

External links

• Weisstein, Eric W., "Gyroelongated pentagonal rotunda" ("Johnson solid") at MathWorld .