View source for Cuboid

{{short description|Convex polyhedron with six faces with four edges each}}
{{other uses}}

[[File:Generic quadrilateral hexahedron.svg|thumb|Example of a {{nowrap|quadrilateral-faced}} {{nowrap|non-convex}} hexahedron]]In [[geometry]], a '''cuboid''' is a [[hexahedron]] with [[quadrilateral]] faces, meaning it is a [[polyhedron]] with six [[Face (geometry)|faces]]; it has eight [[Vertex (geometry)|vertices]] and twelve [[Edge (geometry)|edges]]. A ''[[rectangular cuboid]]'' (sometimes also called a "cuboid") has all [[right angle]]s and equal opposite [[rectangular]] faces. Etymologically, "cuboid" means "like a [[cube]]", in the sense of a [[Convex polyhedron|convex]] solid which can be transformed into a cube (by adjusting the lengths of its edges and the [[Dihedral angle|angles]] between its adjacent faces). A cuboid is a convex polyhedron whose [[polyhedral graph]] is the same as that of a cube.{{r|alexander84|grunbaum}}

General cuboids have many different types. 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.{{r|alexander84|dupius}} Along with the rectangular cuboids, ''[[parallelepiped]]'' is a cuboid with six [[parallelogram]] faces. ''[[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 [[Truncation (geometry)|truncating]] the [[Apex (geometry)|apex]] of a [[square pyramid]].
In attempting to classify cuboids by their symmetries, {{harvtxt|Robertson|1983}} found that there were at least 22 different cases, "of which only about half are familiar in the shapes of everyday objects".{{r|robertson}}

There exist quadrilateral-faced hexahedra which are non-[[Convex polyhedron|convex]].

{| class="wikitable center"
|+ style="text-align:center;"|Some notable cuboids<br>([[quadrilateral]]-faced convex [[hexahedra]] • {{math|8}} vertices and {{math|12}} edges each)
|-
!Image||Name||Faces||Symmetry group
|-
|[[File:Hexahedron.png|110px]]||[[Cube]]||{{math|6}} [[Congruence (geometry)|congruent]] squares||{{math|O<sub>h</sub>, [4,3], (*432)}}<br>order {{math|48}}
|-
|[[File:TrigonalTrapezohedron.svg|50px]]||[[Trigonal trapezohedron]]||{{math|6}} congruent [[Rhombus|rhombi]]||{{math|D<sub>3d</sub>, [2<sup>+</sup>,6], (2*3)}}<br>order {{math|12}}
|-
|[[File:Cuboid no label.svg|110px]]||[[Rectangular cuboid]]||{{math|3}} pairs of [[rectangle]]s||rowspan=2|{{math|D<sub>2h</sub>, [2,2], (*222)}}<br>order {{math|8}}
|-
|[[File:Concertina tesseract cell; rhombic prism, upper.png|110px]]||Right rhombic [[Prism (geometry)|prism]]||{{math|1}} pair of rhombi,<br>{{math|4}} congruent [[Square (geometry)|squares]]
|-
|[[File:Usech kvadrat piramid.png|110px]]||Right square [[frustum]]||{{math|2}} non-congruent squares,<br>{{nowrap|{{math|4}} congruent [[isosceles trapezoid]]s}}||{{math|C<sub>4v</sub>, [4], (*44)}}<br>order {{math|8}}
|-
|[[File:Trigonal trapezohedron gyro-side.png|110px]]||Twisted trigonal [[trapezohedron]]||{{math|6}} congruent quadrilaterals||{{math|D<sub>3</sub>, [2,3]<sup>+</sup>, (223)}}<br>order {{math|6}}
|-
|[[File:梯形柱.png|70px]]||Right isosceles-trapezoidal prism||{{math|1}} pair of isosceles trapezoids;<br>{{nowrap|{{math|1}}, {{math|2}} or {{math|3}} (congruent) square(s)}}||{{math|?, ?, ?}}<br>order {{math|4}}
|-
|[[File:Rhombohedron.svg|110px]]||[[Rhombohedron]]||{{math|3}} pairs of rhombi||rowspan=2|{{math|C<sub>i</sub>, [2<sup>+</sup>,2<sup>+</sup>], (×)}}<br>order {{math|2}}
|-
|[[File:Parallelepiped 2013-11-29.svg|110px]]||[[Parallelepiped]]||{{math|3}} pairs of [[parallelogram]]s
|}

== See also ==
* [[Hypercube]]
* [[Lists of shapes]]

== References ==
{{reflist|refs=

<ref name=alexander84>{{cite book
 | title = Polytopes and Symmetry
 | url = https://archive.org/details/polytopessymmetr0000robe
 | url-access = registration
 | last = Robertson | first = Stewart A.
 | publisher = [[Cambridge University Press]]
 | year = 1984
 | isbn = 9780521277396
 | page = [https://archive.org/details/polytopessymmetr0000robe/page/75 75]
}}</ref>

<ref name=dupius>{{cite book
 | last = Dupuis | first = Nathan F.
 | url = https://archive.org/details/elementssynthet01dupugoog/page/n68
 | title = Elements of Synthetic Solid Geometry
 | publisher = Macmillan
 | year = 1893
 | page = 53
 | access-date = December 1, 2018
}}</ref>

<ref name=grunbaum>[[Branko Grünbaum]] has also used the word "cuboid" to describe a more general class of [[convex polytope]]s in three or more dimensions, obtained by gluing together polytopes combinatorially equivalent to [[hypercube]]s. See: {{cite book
 | last = Grünbaum | first = Branko | author-link = Branko Grünbaum
 | doi = 10.1007/978-1-4613-0019-9
 | edition = 2nd
 | isbn = 978-0-387-00424-2
 | location = New York
 | mr = 1976856
 | page = 59
 | publisher = Springer-Verlag
 | series = Graduate Texts in Mathematics
 | title = Convex Polytopes
 | title-link = Convex Polytopes
 | volume = 221
 | year = 2003
}}</ref>

<ref name=robertson>{{cite journal
 | last = Robertson | first = S. A.
 | doi = 10.1007/BF03026511
 | issue = 4
 | journal = [[The Mathematical Intelligencer]]
 | mr = 746897
 | pages = 57–60
 | title = Polyhedra and symmetry
 | volume = 5
 | year = 1983
}}</ref>

}}

{{Commons category|Hexahedra with cube topology}}
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[[Category:Cuboids| ]]
[[Category:Elementary shapes]]
[[Category:Polyhedra]]
[[Category:Prismatoid polyhedra]]
[[Category:Space-filling polyhedra]]
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