Rectangular cuboid

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| caption2 = A cube, a special case of the square rectangular box.

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A rectangular cuboid is a convex polyhedron with six rectangle faces. The dihedral angles of a rectangular cuboid are all right angles, and its opposite faces are congruent.{{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]–[https://books.google.com/books?id=GVbGDwAAQBAJ&pg=PA144 144]}}

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Because of the faces' orthogonality, the rectangular cuboid is classified as convex orthogonal polyhedron.{{sfnp|Jessen|1967}} By definition, this makes it a right rectangular prism. Rectangular cuboids may be referred to colloquially as "boxes" (after the physical object). If two opposite faces become squares, 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 $\backslash Pi\_4$.{{sfnp|Pisanski|Servatius|2013|p=[https://books.google.com/books?id=3vnEcMCx0HkC&pg=PA21 21]}}{{efn|1=The symbol $\backslash Pi\_n$ represents the skeleton of a {{nowrap|1=$n$-}}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.{{sfnp|Mills|Kolf|1999|p=[https://books.google.com/books?id=dvFfTAR6XwEC&pg=PA16 16]}}

If a rectangular cuboid has length $a$, width $b$, and height $c$, then:{{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]}}

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

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 tessellate three-dimensional space. The shape is fairly versatile in being able to contain multiple smaller rectangular cuboids, e.g. sugar cubes in a box, boxes in a cupboard, cupboards in a room, and rooms in a building.

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 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 nets for a simple cube is 11. However, this number increases significantly to at least 54 for a rectangular cuboid of three different lengths.{{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}}

• Hyperrectangle — generalization of a rectangle;
• Minimum bounding box — a measurement of a cuboid in which all points exist;
• Padovan cuboid spiral — a spiral created by joining the diagonals of faces of successive cuboids added to a unit cube.
• The spider and the fly problem — a problem asking the shortest path between two points on a cuboid's surface.

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{{reflist|25em}}

• {{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 journal

| last = Jessen | first = Børge | author-link = Børge Jessen

| issue = 2

| journal = Nordisk Matematisk Tidskrift

| jstor = 24524998

| mr = 0226494

| pages = 90–96

| title = Orthogonal icosahedra

| volume = 15

| year = 1967

}}

• {{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 = 978-0-415-50820-9

}}

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

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Category:Orthogonality