Elongated square pyramid

In geometry, the elongated square pyramid is a convex polyhedron constructed from a cube by attaching an equilateral square pyramid onto one of its faces. It is an example of Johnson solid. It is topologically (but not geometrically) self-dual.

Construction
The elongated square bipyramid is constructed by attaching two equilateral square pyramids onto the faces of a cube that are opposite each other, a process known as elongation. This construction involves the removal of those two squares and replacing them with those pyramids, resulting in eight equilateral triangles and four squares as their faces.. A convex polyhedron in which all of its faces are regular is a Johnson solid, and the elongated square bipyramid is one of them, denoted as $$ J_{15} $$, the fifteenth Johnson solid.

Properties
Given that $$ a $$ is the edge length of an elongated square pyramid. The height of an elongated square pyramid can be calculated by adding the height of an equilateral square pyramid and a cube. The height of a cube is the same as the given edge length $$ a $$, and the height of an equilateral square pyramid is $$ (1/\sqrt{2})a $$. Therefore, the height of an elongated square bipyramid is: $$ a + \frac{1}{\sqrt{2}}a = \left(1 + \frac{\sqrt{2}}{2}\right)a \approx 1.707a. $$ Its surface area can be calculated by adding all the area of four equilateral triangles and four squares: $$ \left(5 + \sqrt{3}\right)a^2 \approx 6.732a^2. $$ Its volume is obtained by slicing it into an equilateral square pyramid and a cube, and then adding them: $$ \left(1 + \frac{\sqrt{2}}{6}\right)a^3 \approx 1.236a^3. $$

The elongated square pyramid has the same three-dimensional symmetry group as the equilateral square pyramid, the cyclic group $$ C_{4v} $$ of order eight. Its dihedral angle can be obtained by adding the angle of an equilateral square pyramid and a cube:
 * The dihedral angle of an elongated square bipyramid between two adjacent triangles is the dihedral angle of an equilateral triangle between its lateral faces, $$ \arccos(-1/3) \approx 109.47^\circ $$
 * The dihedral angle of an elongated square bipyramid between two adjacent squares is the dihedral angle of a cube between those, $$ \pi/2 $$
 * The dihedral angle of an equilateral square pyramid between square and triangle is $$ \arctan \left(\sqrt{2}\right) \approx 54.74^\circ $$. Therefore, the dihedral angle of an elongated square bipyramid between triangle-to-square, on the edge where the equilateral square pyramids attach the cube, is $$ \arctan\left(\sqrt{2}\right) + \frac{\pi}{2} \approx 144.74^\circ. $$

Dual polyhedron
The dual of the elongated square pyramid has 9 faces: 4 triangular, 1 square, and 4 trapezoidal.

Related polyhedra and honeycombs
The elongated square pyramid can form a tessellation of space with tetrahedra, similar to a modified tetrahedral-octahedral honeycomb.