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 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]

Formulae

The following formulae for the volume and surface area can be used if all faces are regular, with edge length a:^[2]

${\displaystyle V=\left({\frac {5}{3{\sqrt {2}}}}\right)a^{3}\approx 1.17851...a^{3}}$

${\displaystyle A=\left(3+{\frac {5{\sqrt {3}}}{2}}\right)a^{2}\approx 7.33013...a^{2}}$

Dual polyhedron

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

Dual triangular cupola	Net of dual

Related polyhedra and honeycombs

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.

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	Digonal cupola	Triangular cupola	Square cupola	Pentagonal cupola	Hexagonal cupola (Flat)
Related uniform polyhedra	Triangular prism	Cubocta- hedron	Rhombi- cubocta- hedron	Rhomb- icosidodeca- hedron	Rhombi- trihexagonal tiling

References

1. Johnson, Norman W. (1966), "Convex polyhedra with regular faces", Canadian Journal of Mathematics, 18: 169–200, doi:10.4153/cjm-1966-021-8, MR 0185507, Zbl 0132.14603.
2. Stephen Wolfram, "Triangular cupola" from Wolfram Alpha. Retrieved July 20, 2010.
3. http://woodenpolyhedra.web.fc2.com/J3.html

External links

• Eric W. Weisstein, Triangular cupola (Johnson solid) at MathWorld.