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)
Symmetry group C3v
Dual polyhedron -
Properties convex
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 polyhedra that have regular faces but are not uniform (that is, they are not Platonic solids, Archimedean solids, prisms or antiprisms). They were named by Norman Johnson, who first listed these polyhedra in 1966.[1]


The following formulae for the volume and surface area can be used if all faces are regular, with edge length 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 and honeycombs[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 triangles are retained and the base hexagon is replaced by 6 triangles, it generates a coplanar deltahedron with 22 faces.

Augmented triangular cupola.png

The triangular cupola can form a tessellation of space with square pyramids and/or octahedra,[3] the same way octahedra and cuboctahedra can fill space.

The family of cupolae with regular polygons exists up to n=5 (pentagons), and higher if isosceles triangles are used in the cupolae.

Family of convex cupolae
n 2 3 4 5 6
Name {2} || t{2} {3} || t{3} {4} || t{4} {5} || t{5} {6} || t{6}
Cupola 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
Triangular prism
CDel node 1.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node 1.png

CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png

CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png

CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.png

CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.png


External links[edit]