View source for Rectangle

{{Short description|Quadrilateral with four right angles}}
{{For|the record label|Rectangle (label)}}
{{pp|small=yes}}
{{Infobox Polygon
| name       = Rectangle
| image      = Rectangle_Geometry_Vector.svg
| caption    = Rectangle
| type       = [[quadrilateral]], [[trapezoid|trapezium]], [[parallelogram]], [[orthotope]]
| edges      = 4
| symmetry   = [[Dihedral symmetry|Dihedral]] (D<sub>2</sub>), [2], (*22), order 4
| schläfli   = {&nbsp;} × {&nbsp;}
| wythoff    =
| coxeter    = {{CDD|node_1|2|node_1}}
| area       =
| dual       = [[rhombus]]
| properties = [[convex polygon|convex]], [[isogonal figure|isogonal]], [[Cyclic polygon|cyclic]] Opposite angles and sides are congruent
}}

In [[Euclidean geometry|Euclidean plane geometry]], a '''rectangle''' is a [[Rectilinear polygon|rectilinear]] [[convex polygon]] or a [[quadrilateral]] with four [[right angle]]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]] containing a right angle. A rectangle with four sides of equal length is a ''[[square]]''. The term "[[wikt:oblong|oblong]]" is used to refer to a non-[[square]] rectangle.<ref name=":0">{{cite web |last=Tapson |first=Frank |date=July 1999 |title=A Miscellany of Extracts from a Dictionary of Mathematics |url=http://www.cimt.plymouth.ac.uk/resources/topics/art002.pdf |url-status=dead |archive-url=https://web.archive.org/web/20140514200449/http://www.cimt.plymouth.ac.uk/resources/topics/art002.pdf |archive-date=2014-05-14 |access-date=2013-06-20 |publisher=Oxford University Press}}</ref><ref>[http://www.mathsisfun.com/definitions/oblong.html "Definition of Oblong"]. ''Math Is Fun''. Retrieved 2011-11-13.</ref><ref>[http://www.icoachmath.com/SiteMap/Oblong.html Oblong – Geometry – Math Dictionary]. Icoachmath.com. Retrieved 2011-11-13.</ref> A rectangle with [[Vertex (geometry)|vertices]] ''ABCD'' would be denoted as {{rectanglenotation|ABCD}}.

The word rectangle comes from the [[Latin]] ''rectangulus'', which is a combination of ''rectus'' (as an adjective, right, proper) and ''angulus'' ([[angle]]).

A '''[[#Crossed rectangles|crossed rectangle]]''' is a crossed (self-intersecting) quadrilateral which consists of two opposite sides of a rectangle along with the two diagonals<ref>{{Cite journal |doi=10.1098/rsta.1954.0003 |last1=Coxeter |first1=Harold Scott MacDonald |author1-link=Harold Scott MacDonald Coxeter |last2=Longuet-Higgins |first2=M.S. |last3=Miller |first3=J.C.P. |title=Uniform polyhedra |jstor=91532 |mr=0062446 |year=1954 |journal=Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences |issn=0080-4614 |volume=246 |pages=401–450 |issue=916 |publisher=The Royal Society|bibcode=1954RSPTA.246..401C |s2cid=202575183 }}</ref> (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 geometry|spherical]], [[Elliptic geometry|elliptic]], and [[Hyperbolic geometry|hyperbolic]], have so-called rectangles with opposite sides equal in length and equal angles that are not right angles.

Rectangles are involved in many [[#Tessellations|tiling]] problems, such as tiling the plane by rectangles or tiling a rectangle by [[polygon]]s.

==Characterizations==
A [[Convex polygon|convex]] [[quadrilateral]] is a rectangle [[if and only if]] it is any one of the following:<ref>Zalman Usiskin and Jennifer Griffin, "The Classification of Quadrilaterals. A Study of Definition", Information Age Publishing, 2008, pp. 34–36 {{isbn|1-59311-695-0}}.
</ref><ref>{{cite book |author1=Owen Byer |author2=Felix Lazebnik |author3=Deirdre L. Smeltzer|author3-link=Deirdre Smeltzer |title=Methods for Euclidean Geometry |url=https://books.google.com/books?id=W4acIu4qZvoC&pg=PA53 |access-date=2011-11-13 |date=19 August 2010 |publisher=MAA |isbn=978-0-88385-763-2 |pages=53–}}</ref>
* a [[parallelogram]] with at least one [[right angle]]
* a parallelogram with [[diagonal]]s of equal length
* a parallelogram ''ABCD'' where [[triangle]]s ''ABD'' and ''DCA'' are [[Congruence (geometry)|congruent]]
* an equiangular quadrilateral
* a quadrilateral with four right angles
* a quadrilateral where the two diagonals are equal in length and [[Bisection|bisect]] each other<ref>Gerard Venema, "Exploring Advanced Euclidean Geometry with GeoGebra", MAA, 2013, p. 56.</ref>
* a convex quadrilateral with successive sides ''a'', ''b'', ''c'', ''d'' whose area is <math>\tfrac{1}{4}(a+c)(b+d)</math>.<ref name=Josefsson/>{{rp|fn.1}}
* a convex quadrilateral with successive sides ''a'', ''b'', ''c'', ''d'' whose area is <math>\tfrac{1}{2} \sqrt{(a^2+c^2)(b^2+d^2)}.</math><ref name=Josefsson>{{cite journal | author = Josefsson Martin | year = 2013 | title = Five Proofs of an Area Characterization of Rectangles | url = http://forumgeom.fau.edu/FG2013volume13/FG201304.pdf | journal = Forum Geometricorum | volume = 13 | pages = 17–21 | access-date = 2013-02-08 | archive-date = 2016-03-04 | archive-url = https://web.archive.org/web/20160304001152/http://forumgeom.fau.edu/FG2013volume13/FG201304.pdf | url-status = dead }}</ref>

==Classification==
[[File:Symmetries_of_square.svg|280px|thumb|A rectangle is a special case of both [[parallelogram]] and [[trapezoid]]. A [[square]] is a special case of a rectangle.]]

===Traditional hierarchy===
A rectangle is a special case of a [[parallelogram]] in which each pair of adjacent [[Edge (geometry)|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 (geometry)|parallel]] and [[equality (mathematics)|equal]] in [[length]].

A trapezium is a [[Convex polygon|convex]] [[quadrilateral]] which has at least one pair of [[parallel (geometry)|parallel]] opposite sides.

A convex quadrilateral is
* '''[[Simple polygon|Simple]]''': The boundary does not cross itself.
* '''[[Star-shaped polygon|Star-shaped]]''': 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 [[Reflection symmetry|axes of symmetry]] through each pair of opposite sides.<ref>[http://mysite.mweb.co.za/residents/profmd/quadclassify.pdf An Extended Classification of Quadrilaterals] {{Webarchive|url=https://web.archive.org/web/20191230004754/http://mysite.mweb.co.za/residents/profmd/quadclassify.pdf |date=2019-12-30 }} (An excerpt from De Villiers, M. 1996. ''Some Adventures in Euclidean Geometry.'' University of Durban-Westville.)</ref>  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 of symmetry|axis]] is not an axis of [[symmetry]] 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]] and crossed isosceles trapezia (crossed quadrilaterals with the same [[vertex arrangement]] as isosceles trapezia).

==Properties==

===Symmetry===
A rectangle is [[Cyclic polygon|cyclic]]: all [[Corner angle|corner]]s lie on a single [[circle]].

It is [[equiangular polygon|equiangular]]: all its corner [[angle]]s are equal (each of 90 [[Degree (angle)|degrees]]).

It is isogonal or [[vertex-transitive]]: all corners lie within the same [[symmetry orbit]].

It has two [[line (geometry)|line]]s of [[reflectional symmetry]] and [[rotational symmetry]] of order 2 (through 180°).

===Rectangle-rhombus duality===
The [[dual polygon]] of a rectangle is a [[rhombus]], as shown in the table below.<ref>de Villiers, Michael, "Generalizing Van Aubel Using Duality", ''Mathematics Magazine'' 73 (4), Oct. 2000, pp. 303–307.</ref>

{|class="wikitable" style="text-align:center"
|-
!Rectangle !! Rhombus
|-
|All ''angles'' are equal.
||All ''sides'' are equal.
|-
|Alternate ''sides'' are equal.
||Alternate ''angles'' are equal.
|-
|Its centre is equidistant from its ''[[Vertex (geometry)|vertices]]'', hence it has a ''[[circumcircle]]''.
||Its centre is equidistant from its ''sides'', hence it has an ''incircle''.
|-
|Two axes of symmetry bisect opposite ''sides''.
||Two axes of symmetry bisect opposite ''angles''.
|-
|Diagonals are equal in ''length''.
||Diagonals intersect at equal ''angles''.
|}
* 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 (mechanics)|degrees of freedom]] consisting, for example, of three for position (comprising two of [[Translation (geometry)|translation]] and one of [[rotation]]), one for shape ([[Aspect ratio#Rectangles|aspect ratio]]), and one for overall size (area).

Two rectangles, neither of which will fit inside the other, are said to be [[Comparability|incomparable]].

==Formulae==
[[File:PerimeterRectangle.svg|thumb|150px|The formula for the perimeter of a rectangle]]
[[File:Illustration for the area of a rectangle.svg|thumb|150px|The area of a rectangle is the product of the length and width.]]

If a rectangle has length <math>\ell</math> and width <math>w</math>, then:<ref>{{Cite web |title=Rectangle |url=https://www.mathsisfun.com/geometry/rectangle.html |access-date=2024-03-22 |website=Math Is Fun}}</ref>
* it has [[area]] <math>A = \ell w\,</math>;
* it has [[perimeter]] <math>P = 2\ell + 2w = 2(\ell + w)\,</math>;
* each diagonal has length <math>d=\sqrt{\ell^2 + w^2}</math>; and
* when <math>\ell = w\,</math>, the rectangle is a [[Square (geometry)|square]].<ref name=":0" />

==Theorems==
The [[isoperimetric theorem]] 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]] form a rectangle.

A [[parallelogram]] with equal [[diagonals]] is a rectangle.

The [[Japanese theorem for cyclic quadrilaterals]]<ref>[http://math.kennesaw.edu/~mdevilli/cyclic-incentre-rectangle.html Cyclic Quadrilateral Incentre-Rectangle] {{Webarchive|url=https://web.archive.org/web/20110928154652/http://math.kennesaw.edu/~mdevilli/cyclic-incentre-rectangle.html |date=2011-09-28 }} with interactive animation illustrating a rectangle that becomes a 'crossed rectangle', making a good case for regarding a 'crossed rectangle' as a type of rectangle.</ref> 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:<ref>{{cite journal |author1=Hall, Leon M. |author2=Robert P. Roe |name-list-style=amp |title=An Unexpected Maximum in a Family of Rectangles |journal=Mathematics Magazine |volume=71 |issue=4 |year=1998 |pages=285–291 |doi=10.1080/0025570X.1998.11996653 |url=http://web.mst.edu/~lmhall/Personal/HallRoe/Hall_Roe.pdf |jstor=2690700 |access-date=2011-11-13 |archive-date=2010-07-23 |archive-url=https://web.archive.org/web/20100723134734/http://web.mst.edu/~lmhall/Personal/HallRoe/Hall_Roe.pdf |url-status=dead }}</ref>
:<math>\displaystyle (AP)^2 + (CP)^2 = (BP)^2 + (DP)^2.</math>

For every convex body ''C'' in the plane, we can [[Inscribed figure|inscribe]] a rectangle ''r'' in ''C'' such that a [[homothetic transformation|homothetic]] copy ''R'' of ''r'' is circumscribed about ''C'' and the positive homothety ratio is at most 2 and <math>0.5 \text{ × Area}(R) \leq \text{Area}(C) \leq 2 \text{ × Area}(r)</math>.<ref>{{Cite journal | doi = 10.1007/BF01263495| title = Approximation of convex bodies by rectangles| journal = Geometriae Dedicata| volume = 47| pages = 111–117| year = 1993| last1 = Lassak | first1 = M. | s2cid = 119508642}}</ref>

There exists a unique rectangle with sides <math>a</math> and <math>b</math>, where <math>a</math> is less than <math>b</math>, with two ways of being folded along a line through its center such that the area of overlap is minimized and each area yields a different shape{{Snd}}a triangle and a pentagon. The unique ratio of side lengths is <math>\displaystyle \frac {a} {b}=0.815023701...</math>.<ref>{{cite OEIS|A366185|	Decimal expansion of the real root of the quintic equation <math>\ x^5 + 3x^4 + 4x^3 + x -1 = 0</math> }}</ref>

==Crossed rectangles==
A [[list of self-intersecting polygons|''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]] [[Space frame|frame]] that is twisted can take the shape of a bow tie.

The interior of a ''crossed rectangle'' can have a [[polygon density]] of ±1 in each triangle, dependent upon the winding orientation as clockwise or counterclockwise.

A ''crossed rectangle'' may be considered [[equiangular polygon|equiangular]] if right and left turns are allowed. As with any ''crossed quadrilateral'', the sum of its [[interior angle]]s is 720°, allowing for internal angles to appear on the outside and exceed 180°.<ref>[https://web.archive.org/web/20150723004135/http://mysite.mweb.co.za/residents/profmd/stars.pdf Stars: A Second Look]. (PDF). Retrieved 2011-11-13.</ref>

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°).

[[File:Crossed rectangles.png|320px]]

==Other rectangles==
[[File:Saddle rectangle example.png|thumb|A '''saddle rectangle''' has 4 nonplanar vertices, [[Alternation (geometry)|alternated]] from vertices of a [[rectangular cuboid]], with a unique [[minimal surface]] interior defined as a linear combination of the four vertices, creating a saddle surface. This example shows 4 blue edges of the rectangle, and two [[green]] diagonals, all being diagonal of the cuboid rectangular faces.]]

In [[spherical geometry]], a '''spherical rectangle''' is a figure whose four edges are [[great circle]] 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:
{|class=wikitable
|- align=center
|[[File:Stacked bond.png|182px]]<br />Stacked bond
|[[File:Wallpaper group-cmm-1.jpg|150px]]<br />Running bond
|[[File:Wallpaper group-p4g-1.jpg|150px]]<br />Basket weave
|[[File:Basketweave bond.svg|150px]]<br />Basket weave
|[[File:Herringbone bond.svg|150px]]<br />[[Herringbone pattern]]
|}

==Squared, perfect, and other tiled rectangles==
[[File:Perfektes_Rechteck.svg|thumb|right|A perfect rectangle of order 9]]
[[File:smallest_perfect_squared_squares.svg|thumb|Lowest-order perfect squared square (1) and the three smallest perfect squared squares (2&ndash;4) &ndash; {{nowrap|all are simple squared squares}}]]
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''<ref name="BSST"/><ref>{{cite journal |author=J.D. Skinner II |author2=C.A.B. Smith |author3=W.T. Tutte |name-list-style=amp |date=November 2000 |title=On the Dissection of Rectangles into Right-Angled Isosceles Triangles |journal=[[Journal of Combinatorial Theory, Series B]] |volume=80 |issue=2 |pages=277–319 |doi=10.1006/jctb.2000.1987|doi-access=free }}</ref> if the tiles are [[Similarity (geometry)|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 triangle]]s. A database of all known perfect rectangles, perfect squares and related shapes can be found at [http://www.squaring.net/ squaring.net]. The lowest number of squares need for a perfect tiling of a rectangle is 9<ref>{{cite OEIS|A219766|Number of nonsquare simple perfect squared rectangles of order n up to symmetry}}</ref> and the lowest number needed for a [[squaring the square|perfect tilling a square]] is 21, found in 1978 by computer search.<ref>{{cite web|access-date=2021-09-26|title=Squared Squares; Perfect Simples, Perfect Compounds and Imperfect Simples|url=http://www.squaring.net/sq/ss/spss/o21/spsso21.html|website=www.squaring.net}}</ref>

A rectangle has [[Commensurability (mathematics)|commensurable]] sides if and only if it is tileable by a finite number of unequal squares.<ref name="BSST">{{cite journal |author=R.L. Brooks |author2=C.A.B. Smith |author3=A.H. Stone |author4=W.T. Tutte |name-list-style=amp |year=1940 |title=The dissection of rectangles into squares |journal=[[Duke Mathematical Journal|Duke Math. J.]] |volume=7 |issue=1 |pages=312–340 |doi=10.1215/S0012-7094-40-00718-9 |url=http://projecteuclid.org/euclid.dmj/1077492259}}</ref><ref>{{cite journal |author=R. Sprague |year=1940 |title=Ũber die Zerlegung von Rechtecken in lauter verschiedene Quadrate |journal=[[Crelle's Journal|Journal für die reine und angewandte Mathematik]] |language=de |volume=1940 |issue=182 |pages=60–64 |doi=10.1515/crll.1940.182.60 |s2cid=118088887}}</ref> The same is true if the tiles are unequal isosceles [[wikt:right triangle|right triangles]].

The tilings of rectangles by other tiles which have attracted the most attention are those by congruent non-rectangular [[polyomino]]es, allowing all rotations and reflections. There are also tilings by congruent [[polyabolo]]es.

==Unicode==
The following [[Unicode]] code points depict rectangles:
    U+25AC ▬ BLACK RECTANGLE
    U+25AD ▭ WHITE RECTANGLE
    U+25AE ▮ BLACK VERTICAL RECTANGLE
    U+25AF ▯ WHITE VERTICAL RECTANGLE

==See also==
* [[Cuboid]]
* [[Golden rectangle]]
* [[Hyperrectangle]]
* [[Superellipse]] (includes a rectangle with rounded corners)

==References==
{{Reflist|35em}}

==External links==
{{Commons category|Rectangles}}
* {{MathWorld |urlname=Rectangle |title=Rectangle}}
* [http://www.mathopenref.com/rectangle.html Definition and properties of a rectangle] with interactive animation.
* [http://www.mathopenref.com/rectanglearea.html Area of a rectangle] with interactive animation.

{{Polygons}}
{{Authority control}}

[[Category:Types of quadrilaterals]]
[[Category:Elementary shapes]]