Rectangular cuboid

A rectangular cuboid is a special case of a cuboid with rectangular faces in which all of its dihedral angles are right angles. This shape is also called rectangular parallelepiped or orthogonal parallelepiped.^{[lower-alpha 1]}

Rectangular cuboid
Type	Prism Plesiohedron
Faces	6 rectangles
Edges	12
Vertices	8
Properties	convex, zonohedron, isogonal

Properties

A square rectangular prism, a special case of the rectangular prism.

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

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.^[1] The dihedral angles of a rectangular cuboid are all right angles, and its opposite faces are congruent.^[2] 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.^{[lower-alpha 2]} They can be represented as the prism graph ${\displaystyle \Pi _{4}}$.^{[3][lower-alpha 3]} In the case that all six faces are squares, the result is a cube.^[4]

If a rectangular cuboid has length ${\displaystyle a}$, width ${\displaystyle b}$, and height ${\displaystyle c}$, then:^[5]

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

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 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.

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 perfect cuboid is an Euler brick whose space diagonal is also an integer. It is currently unknown whether a perfect cuboid actually exists.^[6]

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.^[7]

See also

• Hyperrectangle — generalization of a rectangle;
• Minimum bounding box — a measurement of a cuboid in which all points exist;
• The spider and the fly problem — a problem asking the shortest path between two points on a cuboid's surface.

References

Notes

1. The terms rectangular prism and oblong prism, however, are ambiguous, since they do not specify all angles.
2. This is also called square cuboid, square box, or right square prism. However, this is sometimes ambiguously called a square prism.
3. The symbol ${\displaystyle \Pi _{n}}$ represents the skeleton of a ${\displaystyle n}$-sided prism.^[3]

Citations

1. Robertson (1984), p. 75.
2. • Dupuis (1893), p. 68
   • Bird (2020), p. 143–144
3. Pisanski & Servatius (2013), p. 21.
4. Mills & Kolf (1999), p. 16.
5. • Bird (2020), p. 144
   • Dupuis (1893), p. 82
6. Webb & Smith (2013), p. 108.
7. Steward, Don (May 24, 2013). "nets of a cuboid". Retrieved December 1, 2018.

Bibliographies

• Bird, John (2020). Science and Mathematics for Engineering (6th ed.). Routledge. ISBN 978-0-429-26170-1.
• Dupuis, Nathan Fellowes (1893). Elements of Synthetic Solid Geometry. Macmillan.
• Mills, Steve; Kolf, Hillary (1999). Maths Dictionary. Heinemann. ISBN 978-0-435-02474-1.
• Pisanski, Tomaž; Servatius, Brigitte (2013). Configuration from a Graphical Viewpoint. Springer. doi:10.1007/978-0-8176-8364-1. ISBN 978-0-8176-8363-4.
• Robertson, Stewart Alexander (1984). Polytopes and Symmetry. Cambridge University Press. ISBN 9780521277396.
• Webb, Charlotte; Smith, Cathy (2013). "Developing subject knowledge". In Lee, Clare; Johnston-Wilder, Sue; Ward-Penny, Robert (eds.). A Practical Guide to Teaching Mathematics in the Secondary School. Routledge.

External links

• Weisstein, Eric W. "Cuboid". MathWorld.
• Rectangular prism and cuboid Paper models and pictures