geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...

, a cuboid is a

hexahedron A hexahedron (plural: hexahedra or hexahedrons) or sexahedron (plural: sexahedra or sexahedrons) is any polyhedron with six faces. A cube, for example, is a regular hexahedron with all its faces square, and three squares around each vertex. Ther ...

, a six-faced solid. Its faces are

quadrilateral In geometry a quadrilateral is a four-sided polygon, having four edges (sides) and four corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''latus'', meaning "side". It is also called a tetragon, ...

s. Cuboid means "like a cube", in the sense that by adjusting the length of the edges or the angles between edges and faces a cuboid can be transformed into a cube. In mathematical language a cuboid is a

convex polyhedron A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n. Most texts. use the term "polytope" for a bounded convex polytope, and the wo ...

, whose

polyhedral graph In geometric graph theory, a branch of mathematics, a polyhedral graph is the undirected graph formed from the vertices and edges of a convex polyhedron. Alternatively, in purely graph-theoretic terms, the polyhedral graphs are the 3-vertex-co ...

is the same as that of a cube. Special cases are a cube, with 6 squares as faces, a rectangular prism, rectangular cuboid or rectangular box, with 6

rectangle In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a parallelogram containin ...

s as faces, for both, cube and rectangular prism, adjacent faces meet in a

right angle In geometry and trigonometry, a right angle is an angle of exactly 90 degrees or radians corresponding to a quarter turn. If a ray is placed so that its endpoint is on a line and the adjacent angles are equal, then they are right angles. T ...

General cuboids

Euler's formula Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that for a ...

the numbers of faces ''F'', of vertices ''V'', and of edges ''E'' of any convex polyhedron are related by the formula ''F'' + ''V'' = ''E'' + 2. In the case of a cuboid this gives 6 + 8 = 12 + 2; that is, like a cube, a cuboid has 6

faces The face is the front of an animal's head that features the eyes, nose and mouth, and through which animals express many of their emotions. The face is crucial for human identity, and damage such as scarring or developmental deformities may af ...

, 8 vertices, and 12 edges. Along with the rectangular cuboids, any

parallelepiped In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms (the term ''rhomboid'' is also sometimes used with this meaning). By analogy, it relates to a parallelogram just as a cube relates to a square. In Euclidean ...

is a cuboid of this type, as is a square

frustum In geometry, a (from the Latin for "morsel"; plural: ''frusta'' or ''frustums'') is the portion of a solid (normally a pyramid or a cone) that lies between two parallel planes cutting this solid. In the case of a pyramid, the base faces are ...

(the shape formed by truncation of the apex of a

square pyramid In geometry, a square pyramid is a pyramid having a square base. If the apex is perpendicularly above the center of the square, it is a right square pyramid, and has symmetry. If all edge lengths are equal, it is an equilateral square pyrami ...

Rectangular cuboid

In a rectangular cuboid, all angles are

s, and opposite faces of a cuboid are

equal Equal(s) may refer to: Mathematics * Equality (mathematics). * Equals sign (=), a mathematical symbol used to indicate equality. Arts and entertainment * ''Equals'' (film), a 2015 American science fiction film * ''Equals'' (game), a board game ...

. By definition this makes it a right rectangular

prism Prism usually refers to: * Prism (optics), a transparent optical component with flat surfaces that refract light * Prism (geometry), a kind of polyhedron Prism may also refer to: Science and mathematics * Prism (geology), a type of sedimentar ...

, and the terms ''rectangular

'' or ''orthogonal parallelepiped'' are also used to designate this polyhedron. The terms "rectangular prism" and "oblong prism", however, are ambiguous, since they do not specify all angles. The square cuboid, square box, or right square prism (also ambiguously called ''square prism'') is a special case of the cuboid in which at least two faces are squares. It has

Schläfli symbol In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to more ...

× , and its symmetry is doubled from ,2to ,2 order 16. The cube is a special case of the square cuboid in which all six faces are squares. It has Schläfli symbol , and its symmetry is raised from ,2 to ,3 order 48. If the dimensions of a rectangular cuboid are ''a'', ''b'' and ''c'', then its

volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). The def ...

is ''abc'' and its

surface area The surface area of a solid object is a measure of the total area that the surface of the object occupies. The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the definition of ar ...

is 2(''ab'' + ''ac'' + ''bc''). The length of the

space diagonal In geometry, a space diagonal (also interior diagonal or body diagonal) of a polyhedron is a line connecting two vertices that are not on the same face. Space diagonals contrast with ''face diagonals'', which connect vertices on the same face (but ...

is :

d = \sqrt.

Cuboid shapes are often used for

box A box ( plural: boxes) is a container used for the storage or transportation of its contents. Most boxes have flat, parallel, rectangular sides. Boxes can be very small (like a matchbox) or very large (like a shipping box for furniture), and ...

es,

cupboard A cupboard is a piece of furniture for enclosing dishware or grocery items that are stored in a home. The term gradually evolved from its original meaning: an open-shelved side table for displaying dishware, more specifically plates, cups and sa ...

room In a building or large vehicle, like a ship, a room is any enclosed space within a number of walls to which entry is possible only via a door or other dividing structure that connects it to either a passageway, another room, or the outdoors, tha ...

s, buildings, containers, cabinets, books, a sturdy computer chassis, printing devices, electronic calling touchscreen devices, washing and drying machines, etc. Cuboids are among those solids that can tessellate 3-dimensional space. The shape is fairly versatile in being able to contain multiple smaller cuboids, e.g.

sugar Sugar is the generic name for sweet-tasting, soluble carbohydrates, many of which are used in food. Simple sugars, also called monosaccharides, include glucose, fructose, and galactose. Compound sugars, also called disaccharides or double s ...

cubes in a box, boxes in a cupboard, cupboards in a room, and rooms in a building. A cuboid with integer edges as well as integer face diagonals is called an

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

, 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.

Nets

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 3 different lengths.

References

External links

*
Rectangular prism and cuboid
Paper models and pictures {{Authority control Elementary shapes Polyhedra Prismatoid polyhedra Space-filling polyhedra Zonohedra

General cuboids

Rectangular cuboid

Nets

See also

References

External links