# 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 is a **right square pyramid**, and has *C*_{4v} symmetry. If all edges are equal, it is an **equilateral square pyramid**,[1] the Johnson solid J_{1}.

Square pyramid | |
---|---|

Type | Johnson J _{92} – J – J_{1}_{2} |

Faces | 4 triangles 1 square |

Edges | 8 |

Vertices | 5 |

Vertex configuration | 4(3^{2}.4)(3 ^{4}) |

Schläfli symbol | ( ) ∨ {4} |

Symmetry group | C_{4v}, [4], (*44) |

Rotation group | C_{4}, [4]^{+}, (44) |

Dual polyhedron | self |

Properties | convex |

Net | |

## General square pyramid

A possibly oblique square pyramid with base length *l* and perpendicular height *h* has volume:

- .

## Right square pyramid

In a right square pyramid, all the lateral edges have the same length, and the sides other than the base are congruent isosceles triangles.

A right square pyramid with base length *l* and height *h* has surface area and volume:

- .

The lateral edge length is:

- ,

and the slant height is:

- .

The dihedral angles are:

- between the base and a side: ;
- between two sides: .

## Equilateral square pyramid, Johnson solid J_{1}

_{1}

If all the edges have the same length, then the sides are equilateral triangles, and the pyramid is an equilateral square pyramid, Johnson solid J_{1}.

The Johnson square pyramid can be characterized by a single edge-length parameter *l*. 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:

## Graph

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

## Related polyhedra and honeycombs

Regular pyramids | ||||||||
---|---|---|---|---|---|---|---|---|

Digonal | Triangular | Square | Pentagonal | Hexagonal | Heptagonal | Octagonal | Enneagonal | Decagonal... |

Improper | Regular | Equilateral | Isosceles | |||||

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 pyramids* fill space with tetrahedra, truncated cubes or cuboctahedra.[2]

### 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 |
---|---|

## References

- Franz Hocevar,
*Solid Geometry*, 1903, p. 44 - http://woodenpolyhedra.web.fc2.com/JohnsonHoneycomb.pdf

## External links

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