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Cuboid

RectangularParallelepiped
Cuboid

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 a, b, and c. A rectangular cuboid with all sides equal (a=b=c) is called a cube, and a cuboid with integer edge lengths a>b>c 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

V=abc
(1)

and the total surface area is

S=2(ab+bc+ca).
(2)

The lengths of the face diagonals are

d_(ab)=sqrt(a^2+b^2)
(3)
d_(ac)=sqrt(a^2+c^2)
(4)
d_(bc)=sqrt(b^2+c^2),
(5)

and the length of the space diagonal is

d_(abc)=sqrt(a^2+b^2+c^2).
(6)

See also

Cube, Euler Brick, Parallelepiped, Prism, Spider and Fly Problem

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References

Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 127, 1987.Gellert, W.; Gottwald, S.; Hellwich, M.; Kästner, H.; and Künstner, H. (Eds.). "Cube and Cuboid." §8.2 in VNR Concise Encyclopedia of Mathematics, 2nd ed. New York: Van Nostrand Reinhold, pp. 187-190, 1989.Grünbaum, B. Convex Polytopes, 2nd ed. New York: Springer-Verlag, 2003.Harris, J. W. and Stocker, H. "Cuboid." §4.2.3 in Handbook of Mathematics and Computational Science. New York: Springer-Verlag, p. 97, 1998.Kern, W. F. and Bland, J. R. "Rectangular Parallelepiped." §10 in Solid Mensuration with Proofs, 2nd ed. New York: Wiley, pp. 21-25, 1948.Lines, L. Solid Geometry, with Chapters on Space-Lattices, Sphere-Packs, and Crystals. New York: Dover, 1965.Robertson, S. A. Polytopes and Symmetry. Cambridge, England: Cambridge University Press, p. 75, 1984.

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Cuboid

Cite this as:

Weisstein, Eric W. "Cuboid." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Cuboid.html

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