# Cuboid

In geometry, a **cuboid** is a hexahedron, a six-faced solid. Its faces are quadrilaterals. Cuboid means "like a cube". A cuboid is like a cube in the sense that by adjusting the lengths of the edges or the angles between faces a cuboid can be transformed into a cube. In mathematical language a cuboid is a convex polyhedron whose polyhedral graph is the same as that of a cube.

A special case of a cuboid is a rectangular cuboid, with 6 rectangles as faces. Its adjacent faces meet at right angles. A special case of a rectangular cuboid is a cube, with six square faces meeting at right angles.[1][2]

## 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).

Quadrilaterally-faced hexahedron (cuboid) 6 faces, 12 edges, 8 vertices | ||||||
---|---|---|---|---|---|---|

Cube (square) |
Rectangular cuboid (three pairs of rectangles) |
Trigonal trapezohedron (congruent rhombi) |
Trigonal trapezohedron (congruent quadrilaterals) |
Quadrilateral frustum (apex-truncated square pyramid) |
Parallelepiped (three pairs of parallelograms) |
Rhombohedron (three pairs of rhombi) |

O_{h}, [4,3], (*432)order 48 |
D_{2h}, [2,2], (*222)order 8 |
D_{3d}, [2^{+},6], (2*3)order 12 |
D_{3}, [2,3]^{+}, (223)order 6 |
C_{4v}, [4], (*44)order 8 |
C_{i}, [2^{+},2^{+}], (×)order 2 |

## Rectangular cuboid

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

Type | Prism Plesiohedron |

Faces | 6 rectangles |

Edges | 12 |

Vertices | 8 |

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

Schläfli symbol | { } × { } × { } |

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* + *ac* + *bc*).

The length of the space diagonal is

Cuboid shapes are often used for boxes, cupboards, rooms, buildings, containers, cabinets, books, a sturdy computer chassis, printing devices, electronic calling touchscreen devices, washing and drying machines, 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 to (at least) 54 for a rectangular cuboid of 3 different lengths.[3]

## See also

- Hyperrectangle
- Trapezohedron
- Lists of shapes

## References

- Robertson, Stewart Alexander (1984).
*Polytopes and Symmetry*. Cambridge University Press. p. 75. ISBN 9780521277396. - Dupuis, Nathan Fellowes (1893).
*Elements of Synthetic Solid Geometry*. Macmillan. p. 53. Retrieved December 1, 2018. - Steward, Don (May 24, 2013). "nets of a cuboid". Retrieved December 1, 2018.

## External links

- Weisstein, Eric W. "Cuboid".
*MathWorld*. - Rectangular prism and cuboid Paper models and pictures