# Triangular cupola

Triangular cupola
TypeJohnson
J2 - J3 - J4
Faces1+3 triangles
3 squares
1 hexagon
Edges15
Vertices9
Vertex configuration6(3.4.6)
3(3.4.3.4)
Symmetry groupC3v
Dual polyhedronhttps://levskaya.github.io/polyhedronisme/?recipe=C1000dJ3
Propertiesconvex
Net
3D model of a triangular cupola

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 is composed of regular polygon faces but are not uniform polyhedra (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 (${\displaystyle V}$), the surface area (${\displaystyle A}$) and the height (${\displaystyle H}$) can be used if all faces are regular, with edge length a:[2][3]

${\displaystyle V=\left({\frac {5}{3{\sqrt {2}}}}\right)a^{3}\approx 1.17851...a^{3}}$
${\displaystyle A=\left(3+{\frac {5{\sqrt {3}}}{2}}\right)a^{2}\approx 7.33013...a^{2}}$
${\displaystyle H={\frac {\sqrt {6}}{3}}a\approx 0.816496...a}$

### 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,[4] 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. ^ Sapiña, R. "Area and volume of the Johnson solid J₃". Problemas y ecuaciones (in Spanish). ISSN 2659-9899. Retrieved 2020-09-08.
4. ^ http://woodenpolyhedra.web.fc2.com/J3.html