# Cuboid

(Redirected from Rectangular cuboid)

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

Special cases are a cube, with 6 squares as faces, a rectangular prism, rectangular cuboid or rectangular box, with 6 rectangles as faces, for both, cube and rectangular prism, adjacent faces meet in a right angle.[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
(apex-truncated
square pyramid)
Parallelepiped
(three pairs of
parallelograms)
Rhombohedron
(three pairs of
rhombi)
Oh, [4,3], (*432)
order 48
D2h, [2,2], (*222)
order 8
D3d, [2+,6], (2*3)
order 12
D3, [2,3]+, (223)
order 6
C4v, [4], (*44)
order 8
Ci, [2+,2+], (×)
order 2

## Rectangular cuboid

Rectangular cuboid

Type Prism
Plesiohedron
Faces 6 rectangles
Edges 12
Vertices 8
Symmetry group D2h, [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

${\displaystyle d={\sqrt {a^{2}+b^{2}+c^{2}}}.}$

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 54 for a rectangular cuboid of 3 different lengths.[3]