Elongated square cupola

In geometry, the elongated square cupola is a polyhedron constructed from an octagonal prism by attaching square cupola onto its base. It is an example of Johnson solid.

Construction
The elongated square cupola is constructed from an octagonal prism by attaching a square cupola onto one of its bases, a process known as the elongation. This cupola covers the octagonal face so that the resulting polyhedron has four equilateral triangles, thirteen squares, and one regular octagon. A convex polyhedron in which all of the faces are regular polygons is the Johnson solid. The elongated square cupola is one of them, enumerated as the nineteenth Johnson solid $$ J_{19} $$.

Properties
The surface area of an elongated square cupola $$ A $$ is the sum of all polygonal faces' area. Its volume $$ V $$ can be ascertained by dissecting it into both square cupola and regular octagon, and then adding their volume. Given the elongated triangular cupola with edge length $$ a $$, its surface area and volume are: $$ \begin{align} A &= \left(15+2\sqrt{2}+\sqrt{3}\right)a^2 \approx 19.561a^2, \\ V &= \left(3+\frac{8\sqrt{2}}{3}\right)a^3 \approx 6.771a^3. \end{align} $$

The dual polyhedron of an elongated square cupola has 20 faces: 8 isosceles triangles, 4 kites, 8 quadrilaterals.

Related polyhedra and honeycombs
The elongated square cupola forms space-filling honeycombs with tetrahedra and cubes; with cubes and cuboctahedra; and with tetrahedra, elongated square pyramids, and elongated square bipyramids. (The latter two units can be decomposed into cubes and square pyramids.)