In
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 c ...
, a cuboid is a
hexahedron, a six-faced solid. Its faces are
quadrilaterals. 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, whose
polyhedral graph 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 contain ...
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. Th ...
.
General cuboids
By
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 ...
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, 8
vertices, and 12 edges.
Along with the rectangular cuboids, any
parallelepiped is a cuboid of this type, as is a square
frustum (the shape formed by truncation of the apex of a
square pyramid).
Rectangular cuboid
In a rectangular cuboid, all angles are
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. Th ...
s, and opposite faces of a cuboid are
equal. By definition this makes it a right rectangular
prism, and the terms ''rectangular
parallelepiped'' 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 × , 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). ...
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 ...
is 2(''ab'' + ''ac'' + ''bc'').
The length of the
space diagonal is
:
Cuboid shapes are often used for
boxes,
cupboards,
rooms, 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 do ...
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 bri ...
, 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.
See also
*
Hyperrectangle
In geometry, an orthotopeCoxeter, 1973 (also called a hyperrectangle or a box) is the generalization of a rectangle to higher dimensions.
A necessary and sufficient condition is that it is congruent to the Cartesian product of intervals. If al ...
*
Trapezohedron
In geometry, an trapezohedron, -trapezohedron, -antidipyramid, -antibipyramid, or -deltohedron is the dual polyhedron of an antiprism. The faces of an are congruent and symmetrically staggered; they are called ''twisted kites''. With a hi ...
*
Lists of shapes
References
External links
*
Rectangular prism and cuboidPaper models and pictures
{{Authority control
Elementary shapes
Polyhedra
Prismatoid polyhedra
Space-filling polyhedra
Zonohedra