Triangular cupola

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Triangular cupola
Triangular cupola.png
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
Triangular cupola net.PNG

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[edit]

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[edit]

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

Dual triangular cupola Net of dual
Dual triangular cupola.png Dual triangular cupola net.png

Related polyhedra[edit]

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.

Augmented triangular cupola.png

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
Triangular prism wedge.png
Digonal cupola
Triangular cupola.png
Triangular cupola
Square cupola.png
Square cupola
Pentagonal cupola.png
Pentagonal cupola
Hexagonal cupola flat.png
Hexagonal cupola
(Flat)

References[edit]

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

External links[edit]