# Elongated square pyramid

In geometry, the elongated square pyramid is one of the Johnson solids (J8). As the name suggests, it can be constructed by elongating a square pyramid (J1) by attaching a cube to its square base. Like any elongated pyramid, it is topologically (but not geometrically) self-dual.

Elongated square pyramid
TypeJohnson
J7 - J8 - J9
Faces4 triangles
1+4 squares
Edges16
Vertices9
Vertex configuration4(43)
1(34)
4(32.42)
Symmetry groupC4v, [4], (*44)
Rotation groupC4, [4]+, (44)
Dual polyhedronself
Propertiesconvex
Net
Johnson solid J8.

A Johnson solid is one of 92 strictly convex polyhedra that is composed of regular polygon faces but are not uniform polyhedra (that is, they are not Platonic solids, Archimedean solids, prisms, or antiprisms). They were named by Norman Johnson, who first listed these polyhedra in 1966.[1]

## Formulae

The following formulae for the height (${\displaystyle H}$ ), surface area (${\displaystyle A}$ ) and volume (${\displaystyle V}$ ) can be used if all faces are regular, with edge length ${\displaystyle L}$ :[2]

${\displaystyle H=L\cdot \left(1+{\frac {\sqrt {2}}{2}}\right)\approx L\cdot 1.707106781}$
${\displaystyle A=L^{2}\cdot \left(5+{\sqrt {3}}\right)\approx L^{2}\cdot 6.732050808}$
${\displaystyle V=L^{3}\left(1+{\frac {\sqrt {2}}{6}}\right)\approx L^{3}\cdot 1.23570226}$

## 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,[3] similar to a modified tetrahedral-octahedral honeycomb.