In Euclidean plane geometry, a rectangle is a quadrilateral with four

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. 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 In Euclidean geometry, a parallelogram is a simple (non- self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equa ...

containing a right angle. A rectangle with four sides of equal length is a '' square''. The term " oblong" is occasionally used to refer to a non- square rectangle. A rectangle with vertices ''ABCD'' would be denoted as . The word rectangle comes from the Latin ''rectangulus'', which is a combination of ''rectus'' (as an adjective, right, proper) and ''angulus'' ( angle). A

crossed 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 ...

is a crossed (self-intersecting) quadrilateral which consists of two opposite sides of a rectangle along with the two diagonals (therefore only two sides are parallel). It is a special case of an antiparallelogram, and its angles are not right angles and not all equal, though opposite angles are equal. Other geometries, such as spherical, elliptic, and hyperbolic, have so-called rectangles with opposite sides equal in length and equal angles that are not right angles. Rectangles are involved in many tiling problems, such as tiling the plane by rectangles or tiling a rectangle by polygons.

Characterizations

A convex quadrilateral is a rectangle if and only if it is any one of the following: * a

with at least one

* a parallelogram with diagonals of equal length * a parallelogram ''ABCD'' where triangles ''ABD'' and ''DCA'' are congruent * an equiangular quadrilateral * a quadrilateral with four right angles * a quadrilateral where the two diagonals are equal in length and

bisect Bisect, or similar, may refer to: Mathematics * Bisection, in geometry, dividing something into two equal parts * Bisection method, a root-finding algorithm * Equidistant set Other uses * Bisect (philately), the use of postage stamp halves * Bis ...

each other * a convex quadrilateral with successive sides ''a'', ''b'', ''c'', ''d'' whose area is

\tfrac(a+c)(b+d)

. * a convex quadrilateral with successive sides ''a'', ''b'', ''c'', ''d'' whose area is

\tfrac \sqrt.

Classification

Traditional hierarchy

A rectangle is a special case of a

in which each pair of adjacent sides is perpendicular. A parallelogram is a special case of a trapezium (known as a trapezoid in North America) in which ''both'' pairs of opposite sides are

parallel Parallel is a geometric term of location which may refer to: Computing * Parallel algorithm * Parallel computing * Parallel metaheuristic * Parallel (software), a UNIX utility for running programs in parallel * Parallel Sysplex, a cluster of IBM ...

and

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 ...

length Length is a measure of distance. In the International System of Quantities, length is a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the Interna ...

. A trapezium is a convex quadrilateral which has at least one pair of

opposite sides. A convex quadrilateral is * Simple: The boundary does not cross itself. *

Star-shaped In geometry, a set S in the Euclidean space \R^n is called a star domain (or star-convex set, star-shaped set or radially convex set) if there exists an s_0 \in S such that for all s \in S, the line segment from s_0 to s lies in S. This defini ...

: The whole interior is visible from a single point, without crossing any edge.

Alternative hierarchy

De Villiers defines a rectangle more generally as any quadrilateral with axes of symmetry through each pair of opposite sides. This definition includes both right-angled rectangles and crossed rectangles. Each has an axis of symmetry parallel to and equidistant from a pair of opposite sides, and another which is the perpendicular bisector of those sides, but, in the case of the crossed rectangle, the first axis is not an axis of

symmetry Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...

for either side that it bisects. Quadrilaterals with two axes of symmetry, each through a pair of opposite sides, belong to the larger class of quadrilaterals with at least one axis of symmetry through a pair of opposite sides. These quadrilaterals comprise

isosceles trapezia In Euclidean geometry, an isosceles trapezoid (isosceles trapezium in British English) is a convex quadrilateral with a line of symmetry bisecting one pair of opposite sides. It is a special case of a trapezoid. Alternatively, it can be define ...

and crossed isosceles trapezia (crossed quadrilaterals with the same vertex arrangement as isosceles trapezia).

Properties

Symmetry

A rectangle is cyclic: all corners lie on a single circle. It is equiangular: all its corner angles are equal (each of 90 degrees). It is isogonal or

vertex-transitive In geometry, a polytope (e.g. a polygon or polyhedron) or a tiling is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries of the figure. This implies that each vertex is surrounded by the same kinds of face in ...

: all corners lie within the same

symmetry orbit In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism ...

. It has two

line Line most often refers to: * Line (geometry), object with zero thickness and curvature that stretches to infinity * Telephone line, a single-user circuit on a telephone communication system Line, lines, The Line, or LINE may also refer to: Arts ...

s of

reflectional symmetry In mathematics, reflection symmetry, line symmetry, mirror symmetry, or mirror-image symmetry is symmetry with respect to a reflection. That is, a figure which does not change upon undergoing a reflection has reflectional symmetry. In 2D ther ...

and rotational symmetry of order 2 (through 180°).

Rectangle-rhombus duality

The

dual polygon In geometry, polygons are associated into pairs called duals, where the vertices of one correspond to the edges of the other. Properties Regular polygons are self-dual. The dual of an isogonal (vertex-transitive) polygon is an isotoxal (edge ...

of a rectangle is a rhombus, as shown in the table below. * The figure formed by joining, in order, the midpoints of the sides of a rectangle is a rhombus and vice versa.

Miscellaneous

A rectangle is a rectilinear polygon: its sides meet at right angles. A rectangle in the plane can be defined by five independent

degrees of freedom Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a thermodynamic system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or ...

consisting, for example, of three for position (comprising two of translation and one of

rotation Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...

), one for shape ( aspect ratio), and one for overall size (area). Two rectangles, neither of which will fit inside the other, are said to be incomparable.

Formulae

Illustration for the area of a rectangle

If a rectangle has length

\ell

and width

w

* it has area

A = \ell w\,

, * it has perimeter

P = 2\ell + 2w = 2(\ell + w)\,

, * each diagonal has length

d=\sqrt

, * and when

\ell = w\,

, the rectangle is a square.

Theorems

The

isoperimetric theorem In mathematics, the isoperimetric inequality is a geometric inequality involving the perimeter of a set and its volume. In n-dimensional space \R^n the inequality lower bounds the surface area or perimeter \operatorname(S) of a set S\subset\R^n by ...

for rectangles states that among all rectangles of a given perimeter, the square has the largest area. The midpoints of the sides of any quadrilateral with perpendicular

diagonals In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ''diagonal'' derives from the ancient Greek δ� ...

form a rectangle. A

with equal

is a rectangle. The

Japanese theorem for cyclic quadrilaterals In geometry, the Japanese theorem states that the centers of the incircles of certain triangles inside a cyclic quadrilateral are vertices of a rectangle. Triangulating an arbitrary cyclic quadrilateral by its diagonals yields four overlapping tr ...

states that the incentres of the four triangles determined by the vertices of a cyclic quadrilateral taken three at a time form a rectangle. The British flag theorem states that with vertices denoted ''A'', ''B'', ''C'', and ''D'', for any point ''P'' on the same plane of a rectangle: :

\displaystyle (AP)^2 + (CP)^2 = (BP)^2 + (DP)^2.

For every convex body ''C'' in the plane, we can inscribe a rectangle ''r'' in ''C'' such that a homothetic copy ''R'' of ''r'' is circumscribed about ''C'' and the positive homothety ratio is at most 2 and

0.5 \text(R) \leq \text(C) \leq 2 \text(r)

Crossed rectangles

A ''crossed'' ''quadrilateral'' (self-intersecting) consists of two opposite sides of a non-self-intersecting quadrilateral along with the two diagonals. Similarly, a crossed rectangle is a ''crossed quadrilateral'' which consists of two opposite sides of a rectangle along with the two diagonals. It has the same vertex arrangement as the rectangle. It appears as two identical triangles with a common vertex, but the geometric intersection is not considered a vertex. A ''crossed quadrilateral'' is sometimes likened to a bow tie or butterfly, sometimes called an "angular eight". A three-dimensional rectangular wire frame that is twisted can take the shape of a bow tie. The interior of a ''crossed rectangle'' can have a

polygon density In geometry, the density of a star polyhedron is a generalization of the concept of winding number from two dimensions to higher dimensions, representing the number of windings of the polyhedron around the center of symmetry of the polyhedron. It ...

of ±1 in each triangle, dependent upon the winding orientation as clockwise or counterclockwise. A ''crossed rectangle'' may be considered equiangular if right and left turns are allowed. As with any ''crossed quadrilateral'', the sum of its interior angles is 720°, allowing for internal angles to appear on the outside and exceed 180°. A rectangle and a crossed rectangle are quadrilaterals with the following properties in common: * Opposite sides are equal in length. * The two diagonals are equal in length. * It has two lines of reflectional symmetry and rotational symmetry of order 2 (through 180°). Crossed rectangles

Other rectangles

spherical geometry 300px, A sphere with a spherical triangle on it. Spherical geometry is the geometry of the two-dimensional surface of a sphere. In this context the word "sphere" refers only to the 2-dimensional surface and other terms like "ball" or "solid sp ...

, a spherical rectangle is a figure whose four edges are

great circle In mathematics, a great circle or orthodrome is the circular intersection of a sphere and a plane passing through the sphere's center point. Any arc of a great circle is a geodesic of the sphere, so that great circles in spherical geomet ...

arcs which meet at equal angles greater than 90°. Opposite arcs are equal in length. The surface of a sphere in Euclidean solid geometry is a non-Euclidean surface in the sense of elliptic geometry. Spherical geometry is the simplest form of elliptic geometry. In elliptic geometry, an elliptic rectangle is a figure in the elliptic plane whose four edges are elliptic arcs which meet at equal angles greater than 90°. Opposite arcs are equal in length. In hyperbolic geometry, a hyperbolic rectangle is a figure in the hyperbolic plane whose four edges are hyperbolic arcs which meet at equal angles less than 90°. Opposite arcs are equal in length.

Tessellations

The rectangle is used in many periodic tessellation patterns, in brickwork, for example, these tilings:

Squared, perfect, and other tiled rectangles

A rectangle tiled by squares, rectangles, or triangles is said to be a "squared", "rectangled", or "triangulated" (or "triangled") rectangle respectively. The tiled rectangle is ''perfect'' if the tiles are similar and finite in number and no two tiles are the same size. If two such tiles are the same size, the tiling is ''imperfect''. In a perfect (or imperfect) triangled rectangle the triangles must be right triangles. A database of all known perfect rectangles, perfect squares and related shapes can be found a
squaring.net
The lowest number of squares need for a perfect tiling of a rectangle is 9 and the lowest number needed for a perfect tilling a square is 21, found in 1978 by computer search. A rectangle has commensurable sides if and only if it is tileable by a finite number of unequal squares. The same is true if the tiles are unequal isosceles right triangles. The tilings of rectangles by other tiles which have attracted the most attention are those by congruent non-rectangular polyominoes, allowing all rotations and reflections. There are also tilings by congruent

polyabolo In recreational mathematics, a polyabolo (also known as a polytan) is a shape formed by gluing isosceles right triangles edge-to-edge, making a polyform with the isosceles right triangle as the base form. Polyaboloes were introduced by Martin Gar ...

es.

Unicode

U+25AC ▬ BLACK RECTANGLE U+25AD ▭ WHITE RECTANGLE U+25AE ▮ BLACK VERTICAL RECTANGLE U+25AF ▯ WHITE VERTICAL RECTANGLE

References

External links

*
Definition and properties of a rectangle
with interactive animation.

with interactive animation. {{Authority control Types of quadrilaterals Elementary shapes