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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).
FormulaeEdit
The following formulae for volume and surface area can be used if all faces are regular, with edge length a:<ref>Stephen Wolfram, "Gyroelongated triangular cupola" from Wolfram Alpha. Retrieved July 22, 2010.</ref>
- <math>V=\left(\frac{1}{3}\sqrt{\frac{61}{2}+18\sqrt{3}+30\sqrt{1+\sqrt{3}}}\right)a^3\approx3.51605...a^3</math>
- <math>A=\left(3+\frac{11\sqrt{3}}{2}\right)a^2\approx12.5263...a^2</math>
Dual polyhedronEdit
The dual of the gyroelongated triangular cupola has 15 faces: 6 kites, 3 rhombi, and 6 pentagons.
| Dual gyroelongated triangular cupola | Net of dual |
|---|---|
| File:Dual gyroelongated triangular cupola.png | File:Dual gyroelongated triangular cupola net.png |
