# Elongated square pyramid

Elongated square pyramid
TypeJohnson
J7J8J9
Faces4 triangles
1+4 squares
Edges16
Vertices9
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 polyhedronself-dual
Propertiesconvex
Net

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.[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]

## Properties

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

## Dual polyhedron

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

Dual elongated square pyramid Net of dual

## Related polyhedra and honeycombs

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

3. ^ Sapiña, R. "Area and volume of the Johnson solid ${\displaystyle J_{8}}$". Problemas y Ecuaciones (in Spanish). ISSN 2659-9899. Retrieved 2020-09-09.