Triangular cupola

In geometry, the triangular cupola is the cupola with hexagon as its base and triangle as its top. If the edges are equal in length, the triangular cupola is the Johnson solid. It can be seen as half a cuboctahedron. Many polyhedrons can be constructed involving the attachment of the base of a triangular cupola.

Properties
The triangular cupola has 4 triangles, 3 squares, and 1 hexagon as their faces; the hexagon is the base and one of the four triangles is the top. If all of the edges are equal in length, the triangles and the hexagon becomes regular; the edge length of that hexagon is equal to the edge length of both squares and triangles. The dihedral angle between each triangle and the hexagon is approximately $$ 70.5^\circ $$, that between each square and the hexagon is $$ 54.7^\circ $$, and that between square and triangle is $$ 125.3^\circ $$. A convex polyhedron in which all of the faces are regular is a Johnson solid, and the triangular cupola is among them, enumerated as the third Johnson solid $$ J_{3} $$.

Given that $$ a $$ is the edge length of a triangular cupola. Its surface area $$ A $$ can be calculated by adding the area of four equilateral triangles, three squares, and one hexagon: $$ A = \left(3+\frac{5\sqrt{3}}{2} \right) a^2 \approx 7.33a^2. $$ Its height $$ h $$ and volume $$ V $$ is: $$ \begin{align} h &= \frac{\sqrt{6}}{3} a\approx 0.82a, \\ V &= \left(\frac{5}{3\sqrt{2}}\right)a^3 \approx 1.18a^3. \end{align} $$

It has an axis of symmetry passing through the center of its both top and base, which is symmetrical by rotating around it at one- and two-thirds of a full-turn angle. It is also mirror-symmetric relative to any perpendicular plane passing through a bisector of the hexagonal base. Therefore, it has pyramidal symmetry, the cyclic group $$ C_{3v} $$ of order 6.

Related polyhedra and honeycombs
The dual of the triangular cupola is the polyhedron with 6 triangular and 3 kite faces.

The triangular cupola can be found in the construction of many polyhedrons. An example is the cuboctahedron in which the triangular cupola may be considered as its hemisphere. A construction that involves the attachment of its base to another polyhedron is known as augmentation; attaching it to prisms or antiprisms is known as elongation or gyroelongation. Some of the other Johnson solids constructed in such a way are elongated triangular cupola $$ J_{18} $$, gyroelongated triangular cupola $$ J_{22} $$, triangular orthobicupola $$ J_{27} $$, elongated triangular orthobicupola $$ J_{35} $$, elongated triangular gyrobicupola $$ J_{36} $$, gyroelongated triangular bicupola $$ J_{44} $$, augmented truncated tetrahedron $$ J_{65} $$.

The triangular cupola can be augmented by 3 square pyramids, leaving adjacent coplanar faces. It is not a Johnson solid because the faces are coplanar. 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, the same way octahedra and cuboctahedra can fill space.