Elongated triangular cupola

In geometry, the elongated triangular cupola is a polyhedron constructed from a hexagonal prism by attaching a triangular cupola. It is an example of a Johnson solid.

Construction
The elongated triangular cupola is constructed from a hexagonal prism by attaching a triangular cupola onto one of its bases, a process known as the elongation. This cupola covers the hexagonal face so that the resulting polyhedron has four equilateral triangles, nine squares, and one regular hexagon. A convex polyhedron in which all of the faces are regular polygons is the Johnson solid. The elongated triangular cupola is one of them, enumerated as the eighteenth Johnson solid $$ J_{18} $$.

Properties
The surface area of an elongated triangular cupola $$ A $$ is the sum of all polygonal face's area. The volume of an elongated triangular cupola can be ascertained by dissecting it into a cupola and a hexagonal prism, after which summing their volume. Given the edge length $$ a $$, its surface and volume can be formulated as: $$ \begin{align} A &= \frac{18 + 5\sqrt{3}}{2}a^2 &\approx 13.330a^2, \\ V &= \frac{5\sqrt{2} + 9\sqrt{3}}{6}a^3 &\approx 3.777a^3. \end{align} $$

It has the three-dimensional same symmetry as the triangular cupola, the cyclic group $$ C_{3\mathrm{v}} $$ of order 6. Its dihedral angle can be calculated by adding the angle of a triangular cupola and a hexagonal prism:
 * the dihedral angle of an elongated triangular cupola between square-to-triangle is that of a triangular cupola between those: 125.3&deg;;
 * the dihedral angle of an elongated triangular cupola between two adjacent squares is that of a hexagonal prism, the internal angle of its base 120&deg;;
 * the dihedral angle of a hexagonal prism between square-to-hexagon is 90&deg;, that of a triangular cupola between square-to-hexagon is 54.7&deg;, and that of a triangular cupola between triangle-to-hexagonal is an 70.5&deg;. Therefore, the elongated triangular cupola between square-to-square and triangle-to-square, on the edge where a triangular cupola is attached to a hexagonal prism, is 90&deg; + 54.7&deg; = 144.7&deg; and 90&deg; + 70.5&deg; = 166.5&deg; respectively.

Dual polyhedron
The dual of the elongated triangular cupola has 15 faces: 6 isosceles triangles, 3 rhombi, and 6 quadrilaterals.

Related polyhedra and honeycombs
The elongated triangular cupola can form a tessellation of space with tetrahedra and square pyramids.