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.

Construction

The elongated square bipyramid is a composite, since it can 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 edge length of a cube's side, and the height of an equilateral square pyramid is. Therefore, the height of an elongated square bipyramid is: Its surface area can be calculated by adding all the area of four equilateral triangles and four squares: Its volume is obtained by slicing it into an equilateral square pyramid and a cube, and then adding them: 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: