View source for Rectangular cuboid

{{Short description|Cuboid with all right angles and equal opposite faces}}

{{infobox polyhedron
 | name = Rectangular cuboid
 | image = Cuboid no label.svg
 | type = [[Prism (geometry)|Prism]]<br>[[Plesiohedron]]
 | faces = 6 [[rectangle]]s
 | edges = 12
 | vertices = 8
 | symmetry group = ''D''<sub>''2h''</sub>
 | properties = [[Convex polyhedron|convex]],<br>[[zonohedron]],<br>[[isogonal figure|isogonal]]
}}

A '''rectangular cuboid''' is a special case of a [[cuboid]] with [[rectangular]] faces in which all of its [[dihedral angle]]s are [[right angle]]s. This shape is also called '''rectangular parallelepiped''' or '''orthogonal parallelepiped'''.{{efn|The terms ''rectangular prism'' and ''oblong prism'', however, are ambiguous, since they do not specify all angles.}}

== Properties ==
{{multiple image
 | image1 = Square prism.svg
 | caption1 = A square rectangular prism, a special case of the rectangular prism.
 | image2 = Hexahedron.png
 | caption2 = A cube, a special case of the square rectangular box.
 | align = rigiht
 | total_width = 400
}}
A rectangular cuboid is a [[convex polyhedron]] with six [[rectangle]] faces. These are often called "cuboids", without qualifying them as being rectangular, but a [[cuboid]] can also refer to a more general class of polyhedra, with six [[quadrilateral]] faces.{{sfnp|Robertson|1984|p=[https://archive.org/details/polytopessymmetr0000robe/page/75 75]}} The [[dihedral angle]]s of a rectangular cuboid are all [[right angle]]s, and its opposite faces are [[congruence (geometry)|congruent]].<ref>{{multiref
 |{{harvp|Dupuis|1893|p=[https://archive.org/details/elementssynthet01dupugoog/page/n68 68]}}
 |{{harvp|Bird|2020|p=[https://books.google.com/books?id=GVbGDwAAQBAJ&pg=PA143 143]&ndash;[https://books.google.com/books?id=GVbGDwAAQBAJ&pg=PA144 144]}}
}}
</ref> By definition, this makes it a ''right rectangular [[Prism (geometry)|prism]]''. Rectangular cuboids may be referred to colloquially as "boxes" (after the [[box|physical object]]). If two opposite faces become [[square]]s, the resulting one may obtain another special case of rectangular prism, known as '''square rectangular cuboid'''.{{efn|This is also called ''square cuboid'', ''square box'', or ''right square prism''. However, this is sometimes ambiguously called a ''square prism''.}} They can be represented as the [[prism graph]] <math> \Pi_4 </math>.{{sfnp|Pisanski|Servatius|2013|p=[https://books.google.com/books?id=3vnEcMCx0HkC&pg=PA21 21]}}{{efn|1=The symbol <math> \Pi_n </math> represents the [[Skeleton (topology)|skeleton]] of a {{nowrap|1=<math> n </math>-}}sided prism.{{sfnp|Pisanski|Servatius|2013|p=[https://books.google.com/books?id=3vnEcMCx0HkC&pg=PA21 21]}}}} In the case that all six faces are squares, the result is a [[cube (geometry)|cube]].{{sfnp|Mills|Kolf|1999|p=[https://books.google.com/books?id=dvFfTAR6XwEC&pg=PA16 16]}}

If a rectangular cuboid has length <math> a </math>, width <math> b </math>, and height <math> c </math>, then:<ref>{{multiref
|{{harvp|Bird|2020|p=[https://books.google.com/books?id=GVbGDwAAQBAJ&pg=PA144 144]}}
|{{harvp|Dupuis|1893|p=[https://archive.org/details/elementssynthet01dupugoog/page/n98 82]}}
}}</ref>

* its volume is the product of the rectangular area and its height: <math display="block"> V=abc. </math>
* its surface area is the sum of the area of all faces: <math display="block"> A=2(ab+ac+bc). </math>
* its [[space diagonal]] can be found by constructing a right triangle of height <math> c </math> with its base as the diagonal of the {{Nobreak|<math> a </math>-by-<math> b </math>}} rectangular face, then calculating the hypotenuse's length using the [[Pythagorean theorem]]: <math display="block"> d=\sqrt{a^2 + b^2 + c^2}. </math>

== Appearance ==
Rectangular cuboid shapes are often used for boxes, cupboards, rooms, buildings, containers, cabinets, books, sturdy computer chassis, printing devices, electronic calling touchscreen devices, washing and drying machines, etc. They are among those solids that can [[Honeycomb (geometry)|tessellate three-dimensional space]]. The shape is fairly versatile in being able to contain multiple smaller rectangular cuboids, e.g. [[sugar cube]]s in a box, boxes in a cupboard, cupboards in a room, and rooms in a building.

== Related polyhedra ==
A rectangular cuboid with integer edges, as well as integer face diagonals, is called an [[Euler brick]]; for example with sides 44, 117, and 240.
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.{{sfnp|Webb|Smith|2013|p=[https://books.google.com/books?id=kUT--gR64ekC&pg=PA108 108]}}

The number of different [[Net (polyhedron)|nets]] for a [[Cube#Geometric_relations|simple cube is 11]]. However, this number increases significantly to at least 54 for a rectangular cuboid of three different lengths.<ref>{{cite web |url=https://donsteward.blogspot.com/2013/05/nets-of-cuboid.html |title=nets of a cuboid |first=Don |last=Steward |date=May 24, 2013 |access-date=December 1, 2018}}</ref>

==See also==
* [[Hyperrectangle]] &mdash; generalization of a rectangle;
* [[Minimum bounding box]] &mdash; a measurement of a cuboid in which all points exist;
* [[The spider and the fly problem]] &mdash; a problem asking the shortest path between two points on a cuboid's surface.

== References ==
=== Notes ===
{{notelist}}

=== Citations ===
{{reflist|25em}}

=== Bibliographies ===
* {{cite book
 | last = Bird | first = John
 | year = 2020
 | title = Science and Mathematics for Engineering
 | edition = 6th
 | url = https://books.google.com/books?id=GVbGDwAAQBAJ
 | publisher = Routledge
 | isbn = 978-0-429-26170-1
}}
* {{cite book
 | last = Dupuis | first = Nathan Fellowes
 | year = 1893
 | title = Elements of Synthetic Solid Geometry
 | publisher = Macmillan
}}
* {{cite book
 | last1 = Mills | first1 = Steve
 | last2 = Kolf | first2 = Hillary
 | year = 1999
 | title = Maths Dictionary
 | publisher = Heinemann
 | url = https://books.google.com/books?id=dvFfTAR6XwEC
 | isbn = 978-0-435-02474-1
}}
* {{cite book
 | last1 = Pisanski | first1 = Tomaž
 | last2 = Servatius | first2 = Brigitte
 | title = Configuration from a Graphical Viewpoint
 | year = 2013
 | url = https://books.google.com/books?id=3vnEcMCx0HkC
 | publisher = Springer
 | isbn = 978-0-8176-8363-4
 | doi = 10.1007/978-0-8176-8364-1
}}
* {{cite book
 | last = Robertson | first = Stewart Alexander
 | title = Polytopes and Symmetry
 | year = 1984
 | url = https://archive.org/details/polytopessymmetr0000robe
 | url-access = registration
 | publisher = Cambridge University Press
 | isbn = 9780521277396
}}
* {{cite book
 | last1 = Webb | first1 = Charlotte
 | last2 = Smith | first2 = Cathy
 | contribution = Developing subject knowledge
 | editor-last1 = Lee | editor-first1 = Clare
 | editor-last2 = Johnston-Wilder | editor-first2 = Sue
 | editor-last3 = Ward-Penny | editor-first3 = Robert
 | year = 2013
 | title = A Practical Guide to Teaching Mathematics in the Secondary School
 | url = https://books.google.com/books?id=kUT--gR64ekC
 | publisher = Routledge
 | isbn = 
}}

==External links==
* {{MathWorld | urlname=Cuboid | title=Cuboid}}
* [https://www.korthalsaltes.com/model.php?name_en=rectangular%20prism Rectangular prism and cuboid] Paper models and pictures

{{Commons category|Rectangular cuboids}}

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[[Category:Cuboids| ]]
[[Category:Orthogonality]]