Rectangular cuboid

A rectangular cuboid, also called rectangular parallelepiped (or orthogonal parallelepiped) is a special case of cuboids and parallelepipeds in which all angles are right angles, and opposite faces are equal. By definition this makes it a right rectangular prism.[lower-alpha 1] Rectangular prisms are often referred to colloquially as "boxes" (after the physical object).

A square box

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

A square rectangular cuboid (also called square cuboid, square box, or right square prism)[lower-alpha 2] is a special case of a rectangular 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.

A cube is a special case of a square rectangular cuboid in which all six faces are squares. It has the 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

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

Rectangular cuboid shapes are often used for boxes, cupboards, rooms, buildings, containers, cabinets, books, sturdy computer chassis, printing devices, electronic calling touchscreen devices, washing and drying machines, etc. They are among those solids that can tessellate three-dimensional space. The shape is fairly versatile in being able to contain multiple smaller rectangular cuboids, e.g. sugar cubes in a box, boxes in a cupboard, cupboards in a room, and rooms in a building.

A rectangular 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.

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 three different lengths.[1]