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.

The elongated square pyramid is a composite, since it can constructed by attaching one equilateral square pyramid onto one of the faces of a cube, a process known as elongation of the pyramid. One square face of each parent body is thus hidden, leaving five squares and four equilateral triangles as faces of the composite.

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.

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

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:

• 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, ${\displaystyle \pi /2=90^{\circ }}$,
• 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 $${\displaystyle \arctan \left({\sqrt {2}}\right)+{\frac {\pi }{2}}\approx 144.74^{\circ }.}$$

• Elongated square bipyramid

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