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| In [[geometry]], a '''cuboid''' is a [[convex polyhedron]] bounded by six [[quadrilateral]] faces, whose [[polyhedral graph]] is the same as that of a [[cube]]. While some mathematical literature refers to any such polyhedron as a cuboid,<ref>{{citation|title=Polytopes and Symmetry|first=Stewart Alexander|last=Robertson|publisher=Cambridge University Press|year=1984|isbn=978-0-521-27739-6|page=75}}</ref> other sources use "cuboid" to refer to a shape of this type in which each of the faces is a [[rectangle]] (and so each pair of adjacent faces meets in a [[right angle]]); this more restrictive type of cuboid is also known as a '''rectangular cuboid''', '''right cuboid''', '''rectangular [[box]]''', '''rectangular [[hexahedron]]''', '''right rectangular prism''', or '''rectangular [[parallelepiped]]'''.<ref>{{citation|title=Elements of Synthetic Solid Geometry|first=Nathan Fellowes|last=Dupuis|publisher=Macmillan|year=1893|page=53}}</ref>
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| ==General cuboids==
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| By [[Euler characteristic|Euler's formula]] the number of faces ('F'), vertices (''V''), and edges (''E'') of any convex polyhedron are related by the formula "F + V " = E + 2 . In the case of a cuboid this gives 6 + 8 = 12 + 2; that is, like a cube, a cuboid has 6 [[Face (geometry)|faces]], 8 [[vertex (geometry)|vertices]], and 12 edges.
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| 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]]).
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| ==Rectangular cuboid==
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| {| cellpadding="5" style="background:#fff; float:right; margin-left:10px; width:250px;" class="wikitable"
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| ! style="background:#e7dcc3;" colspan="2"|Rectangular cuboid
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| |colspan=2|[[File:Cuboid simple.svg|240px|Rectangular cuboid]]
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| | style="background:#aeae74;"|Type||[[Prism (geometry)|Prism]]
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| | style="background:#e7dcc3;"|Faces||6 [[rectangle]]s
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| | style="background:#e7dcc3;"|Edges||12
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| |-
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| | style="background:#e7dcc3;"|Vertices||8
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| | style="background:#e7dcc3;"|[[List of spherical symmetry groups|Symmetry group]]||[[Dihedral symmetry|''D''<sub>''2h''</sub>]], [2,2], (*222), order 8
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| | style="background:#e7dcc3;"|[[Schläfli symbol]]||{ }×{ }×{ } or { }<sup>3</sup>
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| | style="background:#e7dcc3;"|[[Coxeter diagram]]||{{CDD|node_1|2|node_1|2|node_1}}
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| | style="background:#e7dcc3;"|[[Dual polyhedron]]||Rectangular fusil
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| | style="background:#e7dcc3;"|Properties||[[Convex polyhedron|convex]], [[zonohedron]], [[isogonal figure|isogonal]]
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| |}In a rectangular cuboid, all angles are [[right angle]]s, and opposite faces of a cuboid are [[congruence (geometry)|equal]]. It is also a '''right rectangular [[Prism (geometry)|prism]]'''. The term "rectangular or oblong prism" is ambiguous. Also the term ''rectangular [[parallelepiped]]'' or orthogonal parallelepiped is used.
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| The '''square cuboid''', '''square box''', or '''right square prism''' (also ambiguously called ''square prism'') is a special case of the cuboid in which at least two faces are squares. The [[cube (geometry)|cube]] is a special case of the square cuboid in which all six faces are squares.
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| If the dimensions of a cuboid are ''a'', ''b'' and ''c'', then its [[volume]] is ''abc'' and its [[surface area]] is 2''ab'' + 2''ac'' + 2''bc''.
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| The length of the [[space diagonal]] is
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| :<math>d = \sqrt{a^2+b^2+c^2}.\ </math>
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| Cuboid shapes are often used for [[box]]es, [[cupboard]]s, [[Room (architecture)|room]]s, buildings, etc. Cuboids are among those solids that can [[Honeycomb (geometry)|tessellate 3-dimensional space]]. The shape is fairly versatile in being able to contain multiple smaller cuboids, e.g. [[sugar]] cubes in a box, boxes in a cupboard, cupboards in a room, and rooms in a building.
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| A cuboid with integer edges as well as integer face diagonals is called an [[Euler brick]], for example with sides 44, 117 and 240.
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| A [[Euler brick#Perfect cuboid|perfect cuboid]] is an Euler brick whose space diagonal is also an integer. It is currently unknown whether a perfect cuboid actually exists.
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| == See also ==
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| * [[Cube]]
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| * [[Hyperrectangle]]
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| * [[Rectangle]]
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| * [[Trapezohedron]]
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| ==References==
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| {{reflist}}
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| == External links ==
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| {{Commons category|Cuboids}}
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| * {{MathWorld | urlname=Cuboid | title=Cuboid}}
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| * [http://www.korthalsaltes.com/model.php?name_en=rectangular%20prism Rectangular prism and cuboid] Paper models and pictures
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| {{Polyhedron navigator}}
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| [[Category:Elementary shapes]]
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| [[Category:Polyhedra]]
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| [[Category:Prismatoid polyhedra]]
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| [[Category:Space-filling polyhedra]]
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| [[Category:Zonohedra]]
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