Cuboid

For other uses, see Cuboid (disambiguation).

In geometry, a cuboid is a hexahedron with quadrilateral faces, meaning it is a polyhedron with six faces; it has eight vertices and twelve edges. A rectangular cuboid (sometimes also called a "cuboid") has all right angles and equal opposite faces. Etymologically, "cuboid" means "like a cube", in the sense of a convex solid which can be transformed into a cube (by adjusting the lengths of its edges and the angles between its adjacent faces). A cuboid is a convex polyhedron whose polyhedral graph is the same as that of a cube.[1][2]

Cuboids have different types. A special case of a cuboid is a rectangular cuboid, with six rectangle faces and adjacent faces meeting at right angles. When all of the rectangular cuboid's edges are equal in length, it results in a cube, with six square faces and adjacent faces meeting at right angles.[1][3] Along with the rectangular cuboids, parallelepiped is a cuboid with six parallelogram. Rhombohedron is a cuboid with six rhombus faces. A square frustum is a frustum with a square base, but the rest of its faces are quadrilaterals. The square frustum is formed by truncating the apex of a square pyramid.

In attempting to classify cuboids by their symmetries, Robertson (1983) found that there were at least 22 different cases, "of which only about half are familiar in the shapes of everyday objects".[4]

Some notable cuboids
(quadrilateral-faced convex hexahedra • 8 vertices and 12 edges each)
Image Name Faces Symmetry group
Cube 6 congruent squares Oh, [4,3], (*432)
order 48
Trigonal trapezohedron 6 congruent rhombi D3d, [2+,6], (2*3)
order 12
Rectangular cuboid 3 pairs of rectangles D2h, [2,2], (*222)
order 8
Right rhombic prism 1 pair of rhombi,
4 congruent squares
Right square frustum 2 non-congruent squares,
4 congruent isosceles trapezoids		 C4v, [4], (*44)
order 8
Twisted trigonal trapezohedron 6 congruent quadrilaterals D3, [2,3]+, (223)
order 6
Right isosceles-trapezoidal prism 1 pair of isosceles trapezoids;
1, 2 or 3 (congruent) square(s)		 ?, ?, ?
order 4
Rhombohedron 3 pairs of rhombi Ci, [2+,2+], (×)
order 2
Parallelepiped 3 pairs of parallelograms
Example of a quadrilateral-faced non-convex hexahedron

There exist quadrilateral-faced hexahedra which are non-convex.

  1. ^ a b Cite error: The named reference alexander84 was invoked but never defined (see the help page).
  2. ^ Cite error: The named reference grunbaum was invoked but never defined (see the help page).
  3. ^ Cite error: The named reference dupius was invoked but never defined (see the help page).
  4. ^ Cite error: The named reference robertson was invoked but never defined (see the help page).