Gyroelongated triangular cupola

{{Infobox polyhedron

|image=gyroelongated_triangular_cupola.png

|type=Johnson
J₂₁ - J₂₂ - J₂₃

|faces=1+3x3+6 triangles
3 squares
1 hexagon

|edges=33

|vertices=15

|symmetry=C_3v|

|vertex_config=3(3.4.3.4)
2.3(3³.6)
6(3⁴.4)

|dual=-

|properties=convex

|net=Johnson solid 22 net.png

}}

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).

Formulae

The following formulae for volume and surface area can be used if all faces are regular, with edge length a:Stephen Wolfram, "[http://www.wolframalpha.com/input/?i=Gyroelongated+triangular+cupola Gyroelongated triangular cupola]" from Wolfram Alpha. Retrieved July 22, 2010.

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

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

Dual polyhedron

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

class=wikitable width=320

valign=top

!Dual gyroelongated triangular cupola

!Net of dual

valign=top

|160px

References

External links

{{mathworld2 | urlname2 = JohnsonSolid | title2 = Johnson solid| urlname = GyroelongatedTriangularCupola | title = Gyroelongated triangular cupola}}

Category:Johnson solids