TheInfoList

In
geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mat ...

, a cuboid is a
convex polyhedron A convex polytope is a special case of a polytope In elementary geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest bran ...
bounded by six
quadrilateral A quadrilateral is a polygon in Euclidean geometry, Euclidean plane geometry with four Edge (geometry), edges (sides) and four Vertex (geometry), vertices (corners). Other names for quadrilateral include quadrangle (in analogy to triangle) and ...

faces, whose
polyhedral graph In geometric graph theory, a branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change ( ...
is the same as that of a
cube In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...

. While mathematical literature refers to any such polyhedron as a cuboid, other sources use "cuboid" to refer to a shape of this type in which each of the faces is a
rectangle In Euclidean geometry, 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 para ...

(and so each pair of adjacent faces meets in a
right angle In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...

); this more restrictive type of cuboid is also known as a rectangular cuboid, right cuboid, rectangular box, rectangular
hexahedronA hexahedron (plural: hexahedra) is any polyhedron with six Face (geometry), faces. A cube, for example, is a Regular polyhedron, regular hexahedron with all its faces Square (geometry), square, and three squares around each Vertex (geometry), vertex ...

, right rectangular prism, or rectangular
parallelepiped In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of f ...

.

# General cuboids

By
Euler's formula Euler's formula, named after Leonhard Euler, is a mathematics, mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex number, complex exponential function. Euler's ...
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 three of the head's Sense, sense organs, the eyes, nose, and mouth, and through which animals express many of their Emotion, emotions. The face is crucial for human Personal identity, ident ...
, 8 vertices, and 12 edges. Along with the rectangular cuboids, any
parallelepiped In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of f ...

is a cuboid of this type, as is a square
frustum In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space th ...

(the shape formed by truncation of the apex of a
square pyramid In geometry, a square pyramid is a pyramid (geometry), pyramid having a square base. If the apex (geometry), apex is perpendicularly above the center of the square, it is a right square pyramid, and has ''C''4v symmetry. If all edges are equal, it ...

).

# Rectangular cuboid

In a rectangular cuboid, all angles are
right angle In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...

s, and opposite faces of a cuboid are
equal Equal or equals may refer to: Arts and entertainment * Equals (film), ''Equals'' (film), a 2015 American science fiction film * Equals (game), ''Equals'' (game), a board game * The Equals, a British pop group formed in 1965 * "Equal", a 2016 song b ...
. By definition this makes it a right rectangular
prism A prism An optical prism is a transparent optics, optical element with flat, polished surfaces that refraction, refract light. At least one surface must be angled—elements with two parallel surfaces are not prisms. The traditional geometrical ...
, and the terms ''rectangular
parallelepiped In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of f ...

'' 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 Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of ...
× , and its symmetry is doubled from ,2to ,2 order 16. The
cube In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...
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 scalar quantity expressing the amount Quantity or amount is a property that can exist as a multitude Multitude is a term for a group of people who cannot be classed under any other distinct category, except for their shared fact ...

is ''abc'' and its
surface area The surface area of a solid Solid is one of the four fundamental states of matter 4 (four) is a number A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of ...

is 2(''ab'' + ''ac'' + ''bc''). The length of the
space diagonalImage:Cube diagonals.svg, 250px, AC' (shown in blue) is a space diagonal while AC (shown in red) is a face diagonal In geometry, a space diagonal (also interior diagonal or body diagonal) of a polyhedron is a line connecting two vertex (geometry), v ...
is :$d = \sqrt.\$ Cuboid shapes are often used for
box File:Box with cover MET DP241878.jpg, alt=A small, elaborate box, featuring a hinged lid, two swing doors at the front and a small pull-out drawer; the interior is entirely red and features small items that seem to be part of a toilette set, An el ...

es,
cupboard The term cupboard was originally used to describe an open-shelved side table for displaying dishware, more specifically plates, cups and saucers. These open cupboards typically had between one and three display tiers, and at the time, a drawer o ...

s,
room In a building, a room is any space Space is the boundless extent in which and events have relative and . In , physical space is often conceived in three s, although modern s usually consider it, with , to be part of a boundless known ...
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 carbohydrate is a disaccharide A disaccharide (also called a double sugar or ''biose'') is the sugar formed when two monosaccharides are joined by glycosidic linkage. Like monosacc ...

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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

, 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 54 for a rectangular cuboid of 3 different lengths.

*
Hyperrectangle In geometry, an orthotopeCoxeter, 1973 (also called a hyperrectangle or a box) is the generalization of a rectangle to higher dimensions. It is formally defined as the Cartesian product of orthogonal interval (mathematics), intervals. Types A thr ...
*
Trapezohedron The ''n''-gonal trapezohedron, antidipyramid, antibipyramid, or deltohedron is the dual polyhedron of an ''n''-gonal antiprism. The 2''n'' faces of the ''n''-trapezohedron are congruent and symmetrically staggered; they are called ''twisted ki ...
* Lists of shapes