# Gyroelongated triangular cupola

Gyroelongated triangular cupola
Type Johnson
J21 - J22 - J23
Faces 1+3.3+6 triangles
3 squares
1 hexagon
Edges 33
Vertices 15
Vertex configuration 3(3.4.3.4)
2.3(32.6)
6(34.4)
Symmetry group C3v
Dual polyhedron -
Properties convex
Net

In geometry, the gyroelongated triangular cupola is one of the Johnson solids (J22). It can be constructed by attaching a hexagonal antiprism to the base of a triangular cupola (J3). This is called "gyroelongation", which means that an antiprism is joined to the base of a solid, or between the bases of more than one solid.

The gyroelongated triangular cupola can also be seen as a gyroelongated triangular bicupola (J44) with one triangular cupola removed. Like all cupolae, the base polygon has twice as many sides as the top (in this case, the bottom polygon is a hexagon because the top is a triangle).

The 92 Johnson solids were named and described by Norman Johnson in 1966.

## Formulae

The following formulae for volume and surface area can be used if all faces are regular, with edge length a:[1]

$V=(\frac{1}{3}\sqrt{\frac{61}{2}+18\sqrt{3}+30\sqrt{1+\sqrt{3}}})a^3\approx3.51605...a^3$

$A=(3+\frac{11\sqrt{3}}{2})a^2\approx12.5263...a^2$

## Dual polyhedron

The dual of the gyroelongated triangular cupola has 15 faces: 6 kites, 3 rhombi, and 6 quadrilaterals.

Dual gyroelongated triangular cupola Net of dual