# Cuboid

In geometry, a **cuboid** is a convex polyhedron bounded by six quadrilateral faces, whose polyhedral graph is the same as that of a cube. While mathematical literature refers to any such polyhedron as a cuboid,^{} other sources use "cuboid" to refer to a shape of this type in which each of the faces is a rectangle (and so each pair of adjacent faces meets in a right angle); this more restrictive type of cuboid is also known as a **rectangular cuboid**, **right cuboid**, **rectangular box**, **rectangular hexahedron**, **right rectangular prism**, or **rectangular parallelepiped**.^{}

## General cuboids

By Euler's formula the numbers of faces, *F*, of vertices *V*, and of edges *E* of any convex polyhedron are related by the formula *F* + *V* = *E* + 2. In the case of a cuboid this gives 6 + 8 = 12 + 2; that is, like a cube, a cuboid has 6 faces, 8 vertices, and 12 edges.

Along with the rectangular cuboids, any parallelepiped is a cuboid of this type, as is a square frustum (the shape formed by truncation of the apex of a square pyramid).

## Rectangular cuboid

Rectangular cuboid | |
---|---|

Type | Prism |

Faces | 6 rectangles |

Edges | 12 |

Vertices | 8 |

Symmetry group | D_{2h}, [2,2], (*222), order 8 |

Schläfli symbol | { } × { } × { } or { }^{3} |

Coxeter diagram | |

Dual polyhedron | Rectangular fusil |

Properties | convex, zonohedron, isogonal |

In a rectangular cuboid, all angles are right angles, and opposite faces of a cuboid are equal. By definition this makes it a **right rectangular prism**, and the terms *rectangular parallelepiped* or *orthogonal parallelepiped* are also used to designate this polyhedron. The terms "rectangular prism" and "oblong prism", however, are ambiguous, since they do not specify all angles.

The **square cuboid**, **square box**, or **right square prism** (also ambiguously called *square prism*) is a special case of the cuboid in which at least two faces are squares. It has Schläfli symbol {4} × { }, and its symmetry is doubled from [2,2] to [4,2], order 16.

The cube is a special case of the square cuboid in which all six faces are squares. It has Schläfli symbol {4,3}, and its symmetry is raised from [2,2], to [4,3], order 48.

If the dimensions of a rectangular cuboid are *a*, *b* and *c*, then its volume is *abc* and its surface area is 2*ab* + 2*ac* + 2*bc*.

The length of the space diagonal is

Cuboid shapes are often used for boxes, cupboards, rooms, buildings, etc. Cuboids are among those solids that can tessellate 3-dimensional space. The shape is fairly versatile in being able to contain multiple smaller cuboids, e.g. sugar cubes in a box, boxes in a cupboard, cupboards in a room, and rooms in a building.

A cuboid with integer edges as well as integer face diagonals is called an Euler brick, for example with sides 44, 117 and 240. A perfect cuboid is an Euler brick whose space diagonal is also an integer. It is currently unknown whether a perfect cuboid actually exists.

### Nets

The number of different nets for a simple cube is 11, however this number increases significantly for 54 for a rectangular cuboid of 3 different lengths.^{}