Elongated square pyramid

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.^[1]. 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 ${\displaystyle J_{15}}$ , the fifteenth Johnson solid.^[2]

Given that ${\displaystyle a}$ 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 ${\displaystyle a}$ , and the height of an equilateral square pyramid is ${\displaystyle (1/{\sqrt {2}})a}$ . Therefore, the height of an elongated square bipyramid is:^[3]

$${\displaystyle a+{\frac {1}{\sqrt {2}}}a=\left(1+{\frac {\sqrt {2}}{2}}\right)a\approx 1.707a.}$$

$${\displaystyle \left(5+{\sqrt {3}}\right)a^{2}\approx 6.732a^{2}.}$$

$${\displaystyle \left(1+{\frac {\sqrt {2}}{6}}\right)a^{3}\approx 1.236a^{3}.}$$

The elongated square pyramid has the same three-dimensional symmetry group as the equilateral square pyramid, the cyclic group ${\displaystyle C_{4v}}$ of order eight. Its dihedral angle can be obtained by adding the angle of an equilateral square pyramid and a cube:^[5]

• The dihedral angle of an elongated square bipyramid between two adjacent triangles is the dihedral angle of an equilateral triangle between its lateral faces, ${\displaystyle \arccos(-1/3)\approx 109.47^{\circ }}$
• The dihedral angle of an elongated square bipyramid between two adjacent squares is the dihedral angle of a cube between those, ${\displaystyle \pi /2}$
• The dihedral angle of an equilateral square pyramid between square and triangle is ${\displaystyle \arctan \left({\sqrt {2}}\right)\approx 54.74^{\circ }}$ . 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
  $${\displaystyle \arctan \left({\sqrt {2}}\right)+{\frac {\pi }{2}}\approx 144.74^{\circ }.}$$

  

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

Dual elongated square pyramid	Net of dual

The elongated square pyramid can form a tessellation of space with tetrahedra,^[6] similar to a modified tetrahedral-octahedral honeycomb.

• Elongated square bipyramid

• Weisstein, Eric W., "Johnson solid" ("Elongated square pyramid") at MathWorld.