# Square pyramid

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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 *C*_{4v} symmetry.

## Johnson solid (J1)

If the sides are all equilateral triangles, the pyramid is one of the Johnson solids (J_{1}). The 92 Johnson solids were named and described by Norman Johnson in 1966.

A Johnson solid is one of 92 strictly convex regular-faced polyhedra, but which is not uniform, i.e., not a Platonic solid, Archimedean solid, prism or antiprism. They are named by Norman Johnson who first enumerated the set in 1966.

The Johnson square pyramid can be characterized by a single edge-length parameter *a*. The height *H* (from the midpoint of the square to the apex), the surface area *A* (including all five faces), and the volume *V* of such a pyramid are:

## Other square pyramids

Other square pyramids have isosceles triangle sides.

For square pyramids in general, with base length *l* and height *h*, the surface area and volume are:

## Related polyhedra and honeycombs

A regular octahedron can be considered a square bipyramid, i.e. two Johnson square pyramids connected base-to-base. | The tetrakis hexahedron can be constructed from a cube with short square pyramids added to each face. | Square frustum is a square pyramid with the apex truncated. |

*Square pyramid* fill the space with Tetrahedron or Truncated cube or Cuboctahedron.^{[1]}

### Dual polyhedron

The square pyramid is topologically a self-dual polyhedron. The dual edge lengths are different due to the polar reciprocation.

Dual Square pyramid | Net of dual |
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## Topology

Like all pyramids, the square pyramid is self-dual, having the same number of vertices as faces.

A square pyramid can be represented by the Wheel graph W_{5}.

## References

## External links

- Template:Mathworld2
- Weisstein, Eric W., "Wheel graph",
*MathWorld*. - Square Pyramid -- Interactive Polyhedron Model
- Virtual Reality Polyhedra www.georgehart.com: The Encyclopedia of Polyhedra (VRML model)