Triangular cupola

{{Infobox polytope The triangular cupola, or tricu, is one of the 92 Johnson solids (J3). It consists of 1+3 triangles, 3 squares, and 1 hexagon. It is a cupola based on the equilateral triangle, and is one of three Johnson solid cupolas, the other two being the square cupola and the pentagonal cupola.
 * type=CRF
 * img=Triangular cupola 2.png
 * 3d=J3 triangular cupola.stl
 * off=Triangular cupola.off
 * dim = 3
 * obsa = Tricu
 * faces = 1+3 triangles, 3 squares, 1 hexagon
 * edges = 3+3+3+6
 * vertices = 3+6
 * verf = 3 rectangles, edge lengths 1 and $\sqrt{2}$; 6 scalene triangles, edge lengths 1, $\sqrt{2}$, $\sqrt{3}$
 * coxeter = ox3xx&#x
 * army=Tricu
 * reg=Tricu
 * symmetry = A2×I, order 6
 * circum=1
 * height = $$\frac{\sqrt6}{3] ≈ 0.81650$$
 * volume = $$\frac{5\sqrt2}{6} ≈ 1.17851$$
 * dih = 3–4: $$\arccos\left(-\frac{\sqrt3}{3}\right) ≈ 125.26439°$$
 * dih2 = 3–6: $$\arccos\left(\frac13\right) ≈ 70.52878°$$
 * dih3 = 4–6: $$\arccos\left(-\frac{\sqrt3}{3}\right) ≈ 54.73561°$$
 * dual = Semibisected trigonal trapezohedron
 * conjugate = Triangular cupola
 * conv=Yes
 * orientable=Yes
 * nat=Tame}}

It can also be constructed as one of the halves formed by cutting a cuboctahedron in half along a hexagonal section.

Vertex coordinates
A triangular cupola of edge length 1 has vertices given by the following coordinates:


 * $$\left(±\frac12,\,-\frac{\sqrt3}{6},\,\frac{\sqrt6}{3}\right),$$
 * $$\left(0,\,\frac{\sqrt3}{3},\,\frac{\sqrt6}{3}\right),$$
 * $$\left(±\frac12,\,±\frac{\sqrt3}{2},\,0\right),$$
 * $$\left(0,\,±1,\,0\right).$$

These coordinates can be formed by putting a triangle and hexagon in parallel planes and setting edge lengths to be equal.

Alternatively, it can be formed by removing the vertices of a triangular face from the cuboctahedron, producing the following coordinates:


 * $$\left(0,\,±\frac{\sqrt2}{2},\,\frac{\sqrt2}{2}\right),$$
 * $$\left(0,\,\frac{\sqrt2}{2},\,-\frac{\sqrt2}{2}\right),$$
 * $$\left(±\frac{\sqrt2}{2},\,0\,\frac{\sqrt2}{2}\right),$$
 * $$\left(\frac{\sqrt2}{2},\,0,\,-\frac{\sqrt2}{2}\right),$$
 * $$\left(±\frac{\sqrt2}{2},\,\frac{\sqrt2}{2},\,0\right),$$
 * $$\left(\frac{\sqrt2}{2},\,\frac{\sqrt2}{2],\,0\right).$$

Representations
A triangular cupola can be represented by the following Coxeter diagrams:


 * ox3xx&#xt
 * so6ox&#xt

Related polyhedra
Two triangular cupolas can be attached at their hexagonal bases in the same orientation to form a triangular orthobicupola. If the second cupola is rotated by 60º the result is the triangular gyrobicupola, better known as the cuboctahedron.

A hexagonal prism can be attached to the triangular cupola's hexagonal base to form the elongated triangular cupola. If a hexagonal antiprism is attached instead, the result is the gyroelongated triangular cupola.