Cuboid

This article is about polyhedra with six quadrilateral faces. For polyhedra with six rectangle faces, see rectangular cuboid. For other uses, see Cuboid (disambiguation).

In geometry, a cuboid is a hexahedron, a solid with six quadrilateral faces. "Cuboid" means "like a cube", in the sense that by adjusting the lengths of edges or the angles between adjacent faces, a cuboid can be transformed into a cube. In general 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 rectangle faces and adjacent faces meeting at right angles. A special case of a rectangular cuboid is a cube, with six square faces and adjacent faces meeting at right angles.

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 attempting to classify cuboids by their symmetries, S. A. Robinson found that there were at least 22 different cases, "of which only about half are familiar in the shapes of everyday objects".

The numbers F of faces, V of vertices, and E of edges of any convex polyhedron are related by Euler's formula: F + V − E = 2.

In the case of a cuboid, this formula gives: 6 + 8 – 12 = 2;
that is, like a cube, a cuboid has six faces, eight vertices, and twelve edges.

Gallery

• 
  Cube
  (congruent squares)
• 
  Rectangular cuboid
  (three pairs of
  rectangles)
• 
  Trigonal trapezohedron
  (congruent rhombi)
• 
  Twisted trigonal trapezohedron
  (congruent quadrilaterals)
• 
  Square frustum
  (apex-truncated
  square pyramid)
• 
  Parallelepiped
  (three pairs of
  parallelograms)
• 
  Rhombohedron
  (three pairs of
  rhombi)
• 
  Trapezoidal prism
• 
  Non-convex cuboid

See also

• Hypercube
• Lists of shapes