Rectangular cuboid

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

A rectangular cuboid is a special case of a cuboid with rectangular faces in which all of its dihedral angle are right angle. This shape is also called rectangular parallelepiped or orthogonal parallelepiped.

Properties

A cuboid is a six-faced polyhedron that is quadrilateral. If all of the faces become rectangles, the resulting polyhedron is a special case of cuboid, known as rectangular cuboid. As a result, the dihedral angle of a rectangular cuboid are all right angles, and its opposite faces are equal. By definition, this makes it a right rectangular prism. Rectangular cuboids are often called simply "cuboids", or referred to colloquially as "boxes" (after the physical object). If the two opposite faces become squares, the resulting one may obtain another special case of rectangular prism, known as square rectangular cuboid. Considering that the faces of a rectangular prism are all squares, this resulting a cube. The dihedral angles of these special cases of such polyhedrons are all right angles.

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

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

The volume of a rectangular cuboid is calculated by the product of the rectangular area and its height. The area of a rectangle is the product of length $a$ and width $b$, and let $c$ be the height of a rectangular cuboid. The surface area of a rectangular cuboid can be calculated by adding the area of all faces. The space diagonal of a rectangular cuboid can be calculated by finding the hypothenuse of a right triangle with its base as the diagonal of a rectangular. Therefore, its volume, surface area, and space diagonal can be formulated in the following: $$V=abc,\qquad A=2(ab+ac+bc),\qquad d={\sqrt {a^{2}+b^{2}+c^{2}}}.$$

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 polyhedron

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.

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.

See also

• Hyperrectangle
• Minimum bounding box
• Rectangle

References

Bibliographies

• Robertson, Stewart Alexander (1984). Polytopes and Symmetry. Cambridge University Press. ISBN 9780521277396.
• Dupuis, Nathan Fellowes (1893). Elements of Synthetic Solid Geometry. Macmillan.
• Bird, John (2020). Science and Mathematics for Engineering (6th ed.). Routledge. ISBN 978-0-429-26170-1.

External links

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