There are several definitions for the geometric object known as a cuboid.
By far the most common definition of a cuboid is a closed box composed of three pairs of rectangular faces placed opposite each other and joined at right
angles to each other (e.g., Lines 1965, p. 3; Harris and Stocker 1988, p. 97;
Gellert et al. 1989). The more technical term for such an object is "rectangular
parallelepiped." The cuboid is also a right prism,
a special case of the parallelepiped, and corresponds
to what in everyday parlance is known as a (rectangular) "box" (e.g., Beyer
1987, p. 127). Cuboids are implemented in the Wolfram
Language as Cuboid[
xmin, ymin, zmin
, 
xmax, ymax, zmax
] by giving the coordinates of opposite corners.
The monolith with side lengths 1, 4, and 9 in the book and film version 2001: A Space Odyssey is an example of a cuboid.
Robertson (1984, p. 75) defines a cuboid as a more general object, namely as a hexahedron having six quadrilateral faces .
Grünbaum (2003, p. 59) gives yet a different deifnition of cuboid, namely as a class of convex polytopes obtained by gluing together polytopes that are combinatorially equivalent to hypercubes.
Let the side lengths of a rectangular cuboid be denoted 
, 
,
and 
. A rectangular cuboid with all sides
equal (
) is called a cube,
and a cuboid with integer edge lengths 
and face diagonals
is called an Euler brick. If the space
diagonal is also an integer, the cuboid is called a perfect
cuboid.
The volume of a rectangular cuboid is given by
 |
(1)
|
and the total surface area is
 |
(2)
|
The lengths of the face diagonals are
 |  |  |
(3)
|
 |  |  |
(4)
|
 |  |  |
(5)
|
and the length of the space diagonal is
 |
(6)
|
