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Concept
Cuboid
Résumé
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 six 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.
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 six faces, eight vertices, and twelve 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).
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.
A square cuboid, square box, or right square prism (also ambiguously called a square prism) is a special case of a 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 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).
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