# Cuboid

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Convex polyhedron with six faces with four edges each

For other uses, see [Cuboid (disambiguation)](/source/Cuboid_(disambiguation)).

Example of a quadrilateral-faced non-convex hexahedron

In [geometry](/source/Geometry), a **cuboid** is a [hexahedron](/source/Hexahedron) with [quadrilateral](/source/Quadrilateral) faces, meaning it is a [polyhedron](/source/Polyhedron) with six [faces](/source/Face_(geometry)); it has eight [vertices](/source/Vertex_(geometry)) and twelve [edges](/source/Edge_(geometry)). A *[rectangular cuboid](/source/Rectangular_cuboid)* (sometimes also called a "cuboid") has all [right angles](/source/Right_angle) and equal opposite [rectangular](/source/Rectangular) faces. Etymologically, "cuboid" means "like a [cube](/source/Cube)", in the sense of a [convex](/source/Convex_polyhedron) solid which can be transformed into a cube (by adjusting the lengths of its edges and the [angles](/source/Dihedral_angle) between its adjacent faces). A cuboid is a convex polyhedron whose [polyhedral graph](/source/Polyhedral_graph) is the same as that of a cube.[1][2]

General cuboids have many different types. When all of the rectangular cuboid's edges are equal in length, it results in a [cube](/source/Cube), with six [square](/source/Square) faces and adjacent faces meeting at right angles.[1][3] Along with the rectangular cuboids, a *[parallelepiped](/source/Parallelepiped)* is a cuboid with six [parallelogram](/source/Parallelogram) faces. A *[rhombohedron](/source/Rhombohedron)* is a cuboid with six [rhombus](/source/Rhombus) faces. A *[square frustum](/source/Square_frustum)* is a frustum with a square base, but the rest of its faces are quadrilaterals; the square frustum is formed by [truncating](/source/Truncation_(geometry)) the [apex](/source/Apex_(geometry)) of a [square pyramid](/source/Square_pyramid). In attempting to classify cuboids by their symmetries, [Robertson (1983)](#CITEREFRobertson1983) found that there were at least 22 different cases, "of which only about half are familiar in the shapes of everyday objects".[4]

There exist quadrilateral-faced hexahedra which are non-[convex](/source/Convex_polyhedron).

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

## See also

- [Hypercube](/source/Hypercube)

- [Lists of shapes](/source/Lists_of_shapes)

## References

1. ^ [***a***](#cite_ref-alexander84_1-0) [***b***](#cite_ref-alexander84_1-1) Robertson, Stewart A. (1984). [*Polytopes and Symmetry*](https://archive.org/details/polytopessymmetr0000robe). [Cambridge University Press](/source/Cambridge_University_Press). p. [75](https://archive.org/details/polytopessymmetr0000robe/page/75). [ISBN](/source/ISBN_(identifier)) [9780521277396](https://en.wikipedia.org/wiki/Special:BookSources/9780521277396).

1. **[^](#cite_ref-grunbaum_2-0)** [Branko Grünbaum](/source/Branko_Gr%C3%BCnbaum) has also used the word "cuboid" to describe a more general class of [convex polytopes](/source/Convex_polytope) in three or more dimensions, obtained by gluing together polytopes combinatorially equivalent to [hypercubes](/source/Hypercube). See: [Grünbaum, Branko](/source/Branko_Gr%C3%BCnbaum) (2003). [*Convex Polytopes*](/source/Convex_Polytopes). Graduate Texts in Mathematics. Vol. 221 (2nd ed.). New York: Springer-Verlag. p. 59. [doi](/source/Doi_(identifier)):[10.1007/978-1-4613-0019-9](https://doi.org/10.1007%2F978-1-4613-0019-9). [ISBN](/source/ISBN_(identifier)) [978-0-387-00424-2](https://en.wikipedia.org/wiki/Special:BookSources/978-0-387-00424-2). [MR](/source/MR_(identifier)) [1976856](https://mathscinet.ams.org/mathscinet-getitem?mr=1976856).

1. **[^](#cite_ref-dupius_3-0)** Dupuis, Nathan F. (1893). [*Elements of Synthetic Solid Geometry*](https://archive.org/details/elementssynthet01dupugoog/page/n68). Macmillan. p. 53. Retrieved December 1, 2018.

1. **[^](#cite_ref-robertson_4-0)** Robertson, S. A. (1983). "Polyhedra and symmetry". *[The Mathematical Intelligencer](/source/The_Mathematical_Intelligencer)*. **5** (4): 57–60. [doi](/source/Doi_(identifier)):[10.1007/BF03026511](https://doi.org/10.1007%2FBF03026511). [MR](/source/MR_(identifier)) [0746897](https://mathscinet.ams.org/mathscinet-getitem?mr=0746897).

Wikimedia Commons has media related to [Hexahedra with cube topology](https://commons.wikimedia.org/wiki/Category:Hexahedra_with_cube_topology).

v t e Convex polyhedra Platonic solids (regular) tetrahedron cube octahedron dodecahedron icosahedron Archimedean solids (semiregular or uniform) truncated tetrahedron cuboctahedron truncated cube truncated octahedron rhombicuboctahedron truncated cuboctahedron snub cube icosidodecahedron truncated dodecahedron truncated icosahedron rhombicosidodecahedron truncated icosidodecahedron snub dodecahedron Catalan solids (duals of Archimedean) triakis tetrahedron rhombic dodecahedron triakis octahedron tetrakis hexahedron deltoidal icositetrahedron disdyakis dodecahedron pentagonal icositetrahedron rhombic triacontahedron triakis icosahedron pentakis dodecahedron deltoidal hexecontahedron disdyakis triacontahedron pentagonal hexecontahedron Dihedral regular dihedron hosohedron Dihedral uniform prisms antiprisms duals: bipyramids trapezohedra Dihedral others pyramids truncated trapezohedra gyroelongated bipyramid cupola bicupola frustum bifrustum rotunda birotunda prismatoid scutoid Johnson solids v t e Johnson solids Pyramids, cupolae and rotundae square pyramid pentagonal pyramid triangular cupola square cupola pentagonal cupola pentagonal rotunda Modified pyramids elongated triangular pyramid elongated square pyramid elongated pentagonal pyramid gyroelongated square pyramid gyroelongated pentagonal pyramid triangular bipyramid pentagonal bipyramid elongated triangular bipyramid elongated square bipyramid elongated pentagonal bipyramid gyroelongated square bipyramid Modified cupolae and rotundae elongated triangular cupola elongated square cupola elongated pentagonal cupola elongated pentagonal rotunda gyroelongated triangular cupola gyroelongated square cupola gyroelongated pentagonal cupola gyroelongated pentagonal rotunda gyrobifastigium triangular orthobicupola square orthobicupola square gyrobicupola pentagonal orthobicupola pentagonal gyrobicupola pentagonal orthocupolarotunda pentagonal gyrocupolarotunda pentagonal orthobirotunda elongated triangular orthobicupola elongated triangular gyrobicupola elongated square gyrobicupola elongated pentagonal orthobicupola elongated pentagonal gyrobicupola elongated pentagonal orthocupolarotunda elongated pentagonal gyrocupolarotunda elongated pentagonal orthobirotunda elongated pentagonal gyrobirotunda gyroelongated triangular bicupola gyroelongated square bicupola gyroelongated pentagonal bicupola gyroelongated pentagonal cupolarotunda gyroelongated pentagonal birotunda Augmented prisms augmented triangular prism biaugmented triangular prism triaugmented triangular prism augmented pentagonal prism biaugmented pentagonal prism augmented hexagonal prism parabiaugmented hexagonal prism metabiaugmented hexagonal prism triaugmented hexagonal prism Modified Platonic solids augmented dodecahedron parabiaugmented dodecahedron metabiaugmented dodecahedron triaugmented dodecahedron metabidiminished icosahedron tridiminished icosahedron augmented tridiminished icosahedron Modified Archimedean solids augmented truncated tetrahedron augmented truncated cube biaugmented truncated cube augmented truncated dodecahedron parabiaugmented truncated dodecahedron metabiaugmented truncated dodecahedron triaugmented truncated dodecahedron gyrate rhombicosidodecahedron parabigyrate rhombicosidodecahedron metabigyrate rhombicosidodecahedron trigyrate rhombicosidodecahedron diminished rhombicosidodecahedron paragyrate diminished rhombicosidodecahedron metagyrate diminished rhombicosidodecahedron bigyrate diminished rhombicosidodecahedron parabidiminished rhombicosidodecahedron metabidiminished rhombicosidodecahedron gyrate bidiminished rhombicosidodecahedron tridiminished rhombicosidodecahedron Other elementary solids snub disphenoid snub square antiprism sphenocorona augmented sphenocorona sphenomegacorona hebesphenomegacorona disphenocingulum bilunabirotunda triangular hebesphenorotunda (See also List of Johnson solids, a sortable table) Degenerate polyhedra are in italics.

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Adapted from the Wikipedia article [Cuboid](https://en.wikipedia.org/wiki/Cuboid) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Cuboid?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.