In geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, the elongated square pyramid is one of the Johnson solid

Johnson solid

In geometry, a Johnson solid is a strictly convex polyhedron, each face of which is a regular polygon, but which is not uniform, i.e., not a Platonic solid, Archimedean solid, prism or antiprism. There is no requirement that each face must be the same polygon, or that the same polygons join around...

s (J₈). As the name suggests, it can be constructed by elongating a square pyramid

Square pyramid

In geometry, a square pyramid is a pyramid having a square base. If the apex is perpendicularly above the center of the square, it will have C4v symmetry.- Johnson solid :...

 (J₁) by attaching a cube to its square base. Like any elongated pyramid

Pyramid

A pyramid is a structure whose outer surfaces are triangular and converge at a single point. The base of a pyramid can be trilateral, quadrilateral, or any polygon shape, meaning that a pyramid has at least three triangular surfaces...

, it is self-dual

Dual polyhedron

In geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. The dual of the dual is the original polyhedron. The dual of a polyhedron with equivalent vertices is one with equivalent faces, and of one with equivalent edges is another...

.

The 92 Johnson solids were named and described by Norman Johnson in 1966.

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