Cuboid

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

thumb|Example of a {{nowrap|quadrilateral-faced}} {{nowrap|non-convex}} hexahedronIn 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 rectangular 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.{{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, a ''parallelepiped'' is a cuboid with six parallelogram faces. A ''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, {{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.

{| 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 |- |110px||Cube||{{math|6}} congruent squares||{{math|O_h, [4,3], (*432)}}<br>order {{math|48}} |- |50px||Trigonal trapezohedron||{{math|6}} congruent rhombi||{{math|D_3d, [2⁺,6], (2*3)}}<br>order {{math|12}} |- |110px||Rectangular cuboid||{{math|3}} pairs of rectangles||rowspan=2|{{math|D_2h, [2,2], (*222)}}<br>order {{math|8}} |- |110px||Right rhombic prism||{{math|1}} pair of rhombi,<br>{{math|4}} congruent squares |- |110px||Right square frustum||{{math|2}} non-congruent squares,<br>{{nowrap|{{math|4}} congruent isosceles trapezoids}}||{{math|C_4v, [4], (*44)}}<br>order {{math|8}} |- |110px||Twisted trigonal trapezohedron||{{math|6}} congruent quadrilaterals||{{math|D₃, [2,3]⁺, (223)}}<br>order {{math|6}} |- |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}} |- |110px||Rhombohedron||{{math|3}} pairs of rhombi||rowspan=2|{{math|C_i, [2⁺,2⁺], (×)}}<br>order {{math|2}} |- |110px||Parallelepiped||{{math|3}} pairs of parallelograms |}

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

== References == <references>

<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 polytopes in three or more dimensions, obtained by gluing together polytopes combinatorially equivalent to hypercubes. 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>

</references>

{{Commons category|Hexahedra with cube topology}} {{Convex polyhedron navigator}} {{Authority control}}

Category:Cuboids Category:Elementary shapes Category:Polyhedra Category:Prismatoid polyhedra Category:Space-filling polyhedra Category:Zonohedra