Triangular cupola

Triangular cupola
Type Johnson
J2 - J3 - J4
Faces 1+3 triangles
3 squares
1 hexagon
Edges 15
Vertices 9
Vertex configuration 6(3.4.6)
3(3.4.3.4)
Symmetry group C3v
Dual polyhedron -
Properties convex
Net

In geometry, the triangular cupola is one of the Johnson solids (J3). It can be seen as half a cuboctahedron.

A Johnson solid is one of 92 strictly convex regular-faced polyhedra, but which is not uniform, i.e., not a Platonic solid, Archimedean solid, prism or antiprism. They are named by Norman Johnson who first enumerated the set in 1966.

Formulae

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

$V=(\frac{5}{3\sqrt{2}})a^3\approx1.17851...a^3$

$A=(3+\frac{5\sqrt{3}}{2})a^2\approx7.33013...a^2$

Dual polyhedron

The dual of the triangular cupola has 6 triangular and 3 kite faces:

Dual triangular cupola Net of dual

Related polyhedra

The triangular cupola can be augmented by 3 square pyramids, leaving adjacent coplanar faces. This isn't a Johnson solid because of its coplanar faces. Merging those coplanar triangles into larger ones, topologically this is another triangular cupola with isosceles trapezoidal side faces. If all the triangle are retained and the base hexagon is replaced by 6 triangles, it generates a coplanar deltahedron with 22 faces.

The family of cupolae with regular polygons exists up to 5-sides, and higher for isosceles triangle version.

Family of convex cupolae
2 3 4 5 6

Digonal cupola

Triangular cupola

Square cupola

Pentagonal cupola

Hexagonal cupola
(Flat)

References

1. ^ Stephen Wolfram, "Triangular cupola" from Wolfram Alpha. Retrieved July 20, 2010.