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.

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Elongated square pyramid
Type	Johnson J7 – J8 – J9
Faces	4 triangles 1+4 squares
Edges	16
Vertices	9
Vertex configuration	${\displaystyle {\begin{aligned}4\times (4^{3})&+\\1\times (3^{4})&+\\4\times (3^{2}\times 4^{2})\end{aligned}}}$
Symmetry group	${\displaystyle C_{4v}}$
Dual polyhedron	self-dual
Properties	convex
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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]

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

Its surface area can be calculated by adding all the area of four equilateral triangles and four squares:^[4]

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

Its volume is obtained by slicing it into an equilateral square pyramid and a cube, and then adding them:^[4]

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

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. In an equilateral square pyramid, the dihedral angle between square and triangle is ${\displaystyle \arctan \left({\sqrt {2}}\right)\approx 54.74^{\circ }}$, and that between two adjacent triangles is ${\displaystyle \arccos(-1/3)\approx 109.47^{\circ }}$. The dihedral angle between two adjacent squares in a cube is ${\displaystyle \pi /2}$. Therefore, for the elongated square pyramid, the dihedral angle of the triangle and square, on the edge where the equilateral square pyramid attaches the cube, is:^[5]

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

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

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

• Elongated square bipyramid