Triangular cupola

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

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	7	8
Schläfli symbol	{2} || t{2}	{3} || t{3}	{4} || t{4}	{5} || t{5}	{6} || t{6}	{7} || t{7}	{8} || t{8}
Cupola	Digonal cupola	Triangular cupola	Square cupola	Pentagonal cupola	Hexagonal cupola (Flat)	Heptagonal cupola (Non-regular face)	Octagonal cupola (Non-regular face)
Related uniform polyhedra	Rhombohedron	Cuboctahedron	Rhombicuboctahedron	Rhombicosidodecahedron	Rhombitrihexagonal tiling	Rhombitriheptagonal tiling	Rhombitrioctagonal tiling

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