A rectangular cuboid is a special case of a

cuboid In geometry, a cuboid is a hexahedron with quadrilateral faces, meaning it is a polyhedron with six Face (geometry), faces; it has eight Vertex (geometry), vertices and twelve Edge (geometry), edges. A ''rectangular cuboid'' (sometimes also calle ...

with rectangular faces in which all of its dihedral angles are

right angle In geometry and trigonometry, a right angle is an angle of exactly 90 Degree (angle), degrees or radians corresponding to a quarter turn (geometry), turn. If a Line (mathematics)#Ray, ray is placed so that its endpoint is on a line and the ad ...

s. This shape is also called rectangular parallelepiped or orthogonal parallelepiped. Many writers just call these "cuboids", without qualifying them as being rectangular, but others use

to refer to a more general class of polyhedra with six

quadrilateral In Euclidean geometry, geometry a quadrilateral is a four-sided polygon, having four Edge (geometry), edges (sides) and four Vertex (geometry), corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''l ...

faces.

Properties

A rectangular cuboid is a

convex polyhedron In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal faces, straight edges and sharp corners or vertices. The term "polyhedron" may refer either to a solid figure or to its boundary su ...

with six

rectangle In Euclidean geometry, Euclidean plane geometry, a rectangle is a Rectilinear polygon, rectilinear convex polygon or a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that a ...

faces. The dihedral angles of a rectangular cuboid are all

s, and its opposite faces are congruent. Because of the faces'

orthogonality In mathematics, orthogonality is the generalization of the geometric notion of '' perpendicularity''. Although many authors use the two terms ''perpendicular'' and ''orthogonal'' interchangeably, the term ''perpendicular'' is more specifically ...

, the rectangular cuboid is classified as convex orthogonal polyhedron. By definition, this makes it a ''right rectangular prism''. Rectangular cuboids may be referred to colloquially as "boxes" (after the

physical object In natural language and physical science, a physical object or material object (or simply an object or body) is a contiguous collection of matter, within a defined boundary (or surface), that exists in space and time. Usually contrasted with ...

). If two opposite faces become

square In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...

s, the resulting one may obtain another special case of rectangular prism, known as square rectangular cuboid. They can be represented as the

prism graph In the mathematics, mathematical field of graph theory, a prism graph is a Graph (discrete mathematics), graph that has one of the prism (geometry), prisms as its skeleton. Examples The individual graphs may be named after the associated solid: * ...

\Pi_4

. In the case that all six faces are squares, the result is a

cube A cube or regular hexahedron is a three-dimensional space, three-dimensional solid object in geometry, which is bounded by six congruent square (geometry), square faces, a type of polyhedron. It has twelve congruent edges and eight vertices. It i ...

. If a rectangular cuboid has length

a

, width

b

, and height

c

, then: * 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 rectangular face, then calculating the hypotenuse's length using the

Pythagorean theorem In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite t ...

d=\sqrt.

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 cube Sugar cubes are white sugar granules pressed into small cubes measuring approximately 1 teaspoon each. They are usually used for sweetening drinks such as tea and coffee. They were invented in the early 19th century in response to the difficulti ...

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

perfect cuboid In mathematics, an Euler brick, named after Leonhard Euler, is a rectangular cuboid whose Edge (geometry), edges and face diagonals all have integer lengths. A primitive Euler brick is an Euler brick whose edge lengths are relatively prime. A perfe ...

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.

References

Notes

Citations

Bibliographies

* * * * * * *

External links

*
Rectangular prism and cuboid
Paper models and pictures {{Authority control Orthogonality