Rectangle

<h2 id="characterizations">Characterizations</h2>
A <a href="/facts/Convex_polygon/7x35w610">convex</a> <a href="/facts/Quadrilateral/QCGqrkp2">quadrilateral</a> is a rectangle <a href="/facts/If_and_only_if/bYSxGJ66">if and only if</a> it is any one of the following:<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a><a class="footnote-ref" id="fnref:6" href="#fn:6">6</a>

<ul><li>a <a href="/facts/Parallelogram/DRuktq60">parallelogram</a> with at least one <a href="/facts/Right_angle/psaslRq7">right angle</a></li>
<li>a parallelogram with <a href="/facts/Diagonal/39z9sIRb">diagonals</a> of equal length</li>
<li>a parallelogram ABCD where <a href="/facts/Triangle/cRXUwsMn">triangles</a> ABD and DCA are <a href="/facts/Congruence_(geometry)/2tKJ6AhB">congruent</a></li>
<li>an equiangular quadrilateral</li>
<li>a quadrilateral with four right angles</li>
<li>a quadrilateral where the two diagonals are equal in length and <a href="/facts/Bisection/JTsya7Ca">bisect</a> each other<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a></li>
<li>a convex quadrilateral with successive sides a, b, c, d whose area is 
 
 
 
 
 
 
 1
 4
 
 
 
 (
 a
 +
 c
 )
 (
 b
 +
 d
 )
 
 
 {\displaystyle {\tfrac {1}{4}}(a+c)(b+d)}
 
.<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a>: fn.1 </li>
<li>a convex quadrilateral with successive sides a, b, c, d whose area is 
 
 
 
 
 
 
 1
 2
 
 
 
 
 
 (
 
 a
 
 2
 
 
 +
 
 c
 
 2
 
 
 )
 (
 
 b
 
 2
 
 
 +
 
 d
 
 2
 
 
 )
 
 
 .
 
 
 {\displaystyle {\tfrac {1}{2}}{\sqrt {(a^{2}+c^{2})(b^{2}+d^{2})}}.}
 
<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a></li></ul>
<h2 id="classification">Classification</h2>

<h3>Traditional hierarchy</h3>
A rectangle is a special case of a <a href="/facts/Parallelogram/DRuktq60">parallelogram</a> in which each pair of adjacent <a href="/facts/Edge_(geometry)/3lFN6sEz">sides</a> is <a href="/facts/Perpendicular/u0k8xqx4">perpendicular</a>.
A parallelogram is a special case of a trapezium (known as a <a href="/facts/Trapezoid/nCBKsauz">trapezoid</a> in North America) in which both pairs of opposite sides are <a href="/facts/Parallel_(geometry)/16r5yq7D">parallel</a> and <a href="/facts/Equality_(mathematics)/AQCuumLg">equal</a> in <a href="/facts/Length/EwFyAqGH">length</a>.
A trapezium is a <a href="/facts/Convex_polygon/7x35w610">convex</a> <a href="/facts/Quadrilateral/QCGqrkp2">quadrilateral</a> which has at least one pair of <a href="/facts/Parallel_(geometry)/16r5yq7D">parallel</a> opposite sides.
A convex quadrilateral is

<ul><li><a href="/facts/Simple_polygon/C5CfrPWf">Simple</a>: The boundary does not cross itself.</li>
<li><a href="/facts/Star-shaped_polygon/yW0cSRzz">Star-shaped</a>: The whole interior is visible from a single point, without crossing any edge.</li></ul>
<h3>Alternative hierarchy</h3>
De Villiers defines a rectangle more generally as any quadrilateral with <a href="/facts/Reflection_symmetry/eSeR8nbD">axes of symmetry</a> through each pair of opposite sides.<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a> 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 <a href="/facts/Perpendicular/u0k8xqx4">perpendicular</a> bisector of those sides, but, in the case of the crossed rectangle, the first <a href="/facts/Axis_of_symmetry/bpvJqcnS">axis</a> is not an axis of <a href="/facts/Symmetry/hpu7DK8p">symmetry</a> 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 <a href="/facts/Isosceles_trapezia/Izn5AA4p">isosceles trapezia</a> and crossed isosceles trapezia (crossed quadrilaterals with the same <a href="/facts/Vertex_arrangement/DFgCsP0n">vertex arrangement</a> as isosceles trapezia).

<h2 id="properties">Properties</h2>
<h3>Symmetry</h3>
A rectangle is <a href="/facts/Cyclic_polygon/RafYT4am">cyclic</a>: all <a href="/facts/Corner_angle/gFUTEQYq">corners</a> lie on a single <a href="/facts/Circle/9fy5ggKn">circle</a>.
It is <a href="/facts/Equiangular_polygon/Cuo92HxJ">equiangular</a>: all its corner <a href="/facts/Angle/gFUTEQYq">angles</a> are equal (each of 90 <a href="/facts/Degree_(angle)/f48Ns8SX">degrees</a>).
It is isogonal or <a href="/facts/Vertex-transitive/5nC8PdBm">vertex-transitive</a>: all corners lie within the same <a href="/facts/Symmetry_orbit/Vh9VtUUw">symmetry orbit</a>.
It has two <a href="/facts/Line_(geometry)/gJqsr4Gj">lines</a> of <a href="/facts/Reflectional_symmetry/eSeR8nbD">reflectional symmetry</a> and <a href="/facts/Rotational_symmetry/bpvJqcnS">rotational symmetry</a> of order 2 (through 180°).

<h3>Rectangle-rhombus duality</h3>
The <a href="/facts/Dual_polygon/mbR8QK0n">dual polygon</a> of a rectangle is a <a href="/facts/Rhombus/T4pU50GF">rhombus</a>, as shown in the table below.<a class="footnote-ref" id="fnref:11" href="#fn:11">11</a>

<table><tbody><tr><th>Rectangle</th><th>Rhombus</th></tr><tr><td>All angles are equal.</td><td>All sides are equal.</td></tr><tr><td>Alternate sides are equal.</td><td>Alternate angles are equal.</td></tr><tr><td>Its centre is equidistant from its <a href="/facts/Vertex_(geometry)/ID3cea9z">vertices</a>, hence it has a <a href="/facts/Circumcircle/RafYT4am">circumcircle</a>.</td><td>Its centre is equidistant from its sides, hence it has an incircle.</td></tr><tr><td>Two axes of symmetry bisect opposite sides.</td><td>Two axes of symmetry bisect opposite angles.</td></tr><tr><td>Diagonals are equal in length.</td><td>Diagonals intersect at equal angles.</td></tr></tbody></table>
<ul><li>The figure formed by joining, in order, the midpoints of the sides of a rectangle is a <a href="/facts/Rhombus/T4pU50GF">rhombus</a> and vice versa.</li></ul>
<h3>Miscellaneous</h3>
A rectangle is a <a href="/facts/Rectilinear_polygon/gV7lyUlA">rectilinear polygon</a>: its sides meet at right angles.
A rectangle in the plane can be defined by five independent <a href="/facts/Degrees_of_freedom_(mechanics)/2BqcdnLA">degrees of freedom</a> consisting, for example, of three for position (comprising two of <a href="/facts/Translation_(geometry)/KlpWmnfX">translation</a> and one of <a href="/facts/Rotation/yChkz49x">rotation</a>), one for shape (<a href="/facts/Aspect_ratio/1LVBgtfT">aspect ratio</a>), and one for overall size (area).
Two rectangles, neither of which will fit inside the other, are said to be <a href="/facts/Comparability/PvLeJXol">incomparable</a>.

<h2 id="formulae">Formulae</h2>

If a rectangle has length 
 
 
 
 ℓ
 
 
 {\displaystyle \ell }
 
 and width 
 
 
 
 w
 
 
 {\displaystyle w}
 
, then:<a class="footnote-ref" id="fnref:12" href="#fn:12">12</a>

<ul><li>it has <a href="/facts/Area/KsNG1G9t">area</a> 
 
 
 
 A
 =
 ℓ
 w
 
 
 
 {\displaystyle A=\ell w\,}
 
;</li>
<li>it has <a href="/facts/Perimeter/yQv1AajA">perimeter</a> 
 
 
 
 P
 =
 2
 ℓ
 +
 2
 w
 =
 2
 (
 ℓ
 +
 w
 )
 
 
 
 {\displaystyle P=2\ell +2w=2(\ell +w)\,}
 
;</li>
<li>each diagonal has length 
 
 
 
 d
 =
 
 
 
 ℓ
 
 2
 
 
 +
 
 w
 
 2
 
 
 
 
 
 
 {\displaystyle d={\sqrt {\ell ^{2}+w^{2}}}}
 
; and</li>
<li>when 
 
 
 
 ℓ
 =
 w
 
 
 
 {\displaystyle \ell =w\,}
 
, the rectangle is a <a href="/facts/Square_(geometry)/JHJeMvmL">square</a>.<a class="footnote-ref" id="fnref:13" href="#fn:13">13</a></li></ul>
<h2 id="theorems">Theorems</h2>
The <a href="/facts/Isoperimetric_theorem/u02PG11C">isoperimetric theorem</a> for rectangles states that among all rectangles of a given <a href="/facts/Perimeter/yQv1AajA">perimeter</a>, the square has the largest <a href="/facts/Area/KsNG1G9t">area</a>.
The midpoints of the sides of any <a href="/facts/Quadrilateral/QCGqrkp2">quadrilateral</a> with <a href="/facts/Perpendicular/u0k8xqx4">perpendicular</a> <a href="/facts/Diagonals/39z9sIRb">diagonals</a> form a rectangle.
A <a href="/facts/Parallelogram/DRuktq60">parallelogram</a> with equal <a href="/facts/Diagonals/39z9sIRb">diagonals</a> is a rectangle.
The <a href="/facts/Japanese_theorem_for_cyclic_quadrilaterals/45JBozUM">Japanese theorem for cyclic quadrilaterals</a><a class="footnote-ref" id="fnref:14" href="#fn:14">14</a> 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 <a href="/facts/British_flag_theorem/xhVeCuIz">British flag theorem</a> states that with vertices denoted A, B, C, and D, for any point P on the same plane of a rectangle:<a class="footnote-ref" id="fnref:15" href="#fn:15">15</a>

(
          A
          P
          
            )
            
              2
            
          
          +
          (
          C
          P
          
            )
            
              2
            
          
          =
          (
          B
          P
          
            )
            
              2
            
          
          +
          (
          D
          P
          
            )
            
              2
            
          
          .
        
      
    
    {\displaystyle \displaystyle (AP)^{2}+(CP)^{2}=(BP)^{2}+(DP)^{2}.}

For every convex body C in the plane, we can <a href="/facts/Inscribed_figure/ltbko1Eh">inscribe</a> a rectangle r in C such that a <a href="/facts/Homothetic_transformation/WUmuRuzC">homothetic</a> copy R of r is circumscribed about C and the positive homothety ratio is at most 2 and 
 
 
 
 0.5
 
  × Area
 
 (
 R
 )
 ≤
 
 Area
 
 (
 C
 )
 ≤
 2
 
  × Area
 
 (
 r
 )
 
 
 {\displaystyle 0.5{\text{ × Area}}(R)\leq {\text{Area}}(C)\leq 2{\text{ × Area}}(r)}
 
.<a class="footnote-ref" id="fnref:16" href="#fn:16">16</a>
There exists a unique rectangle with sides 
 
 
 
 a
 
 
 {\displaystyle a}
 
 and 
 
 
 
 b
 
 
 {\displaystyle b}
 
, where 
 
 
 
 a
 
 
 {\displaystyle a}
 
 is less than 
 
 
 
 b
 
 
 {\displaystyle b}
 
, 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 – a triangle and a pentagon. The unique ratio of side lengths is 
 
 
 
 
 
 
 a
 b
 
 
 =
 0.815023701...
 
 
 
 {\displaystyle \displaystyle {\frac {a}{b}}=0.815023701...}
 
.<a class="footnote-ref" id="fnref:17" href="#fn:17">17</a>

<h2 id="crossed-rectangles">Crossed rectangles</h2>
A <a href="/facts/List_of_self-intersecting_polygons/zySCnDLM">crossed</a> 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 <a href="/facts/Vertex_arrangement/DFgCsP0n">vertex arrangement</a> 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 <a href="/facts/Bow_tie/YHUKslLD">bow tie</a> or <a href="/facts/Butterfly/qlSgwlRx">butterfly</a>, sometimes called an "angular eight". A <a href="/facts/Three-dimensional/KAqKWUbS">three-dimensional</a> rectangular <a href="/facts/Wire/yyy05H1S">wire</a> <a href="/facts/Space_frame/aKwYtCj6">frame</a> that is twisted can take the shape of a bow tie.
The interior of a crossed rectangle can have a <a href="/facts/Polygon_density/LLQmbxty">polygon density</a> of ±1 in each triangle, dependent upon the winding orientation as clockwise or counterclockwise.
A crossed rectangle may be considered <a href="/facts/Equiangular_polygon/Cuo92HxJ">equiangular</a> if right and left turns are allowed. As with any crossed quadrilateral, the sum of its <a href="/facts/Interior_angle/jd78Qkbt">interior angles</a> is 720°, allowing for internal angles to appear on the outside and exceed 180°.<a class="footnote-ref" id="fnref:18" href="#fn:18">18</a>
A rectangle and a crossed rectangle are quadrilaterals with the following properties in common:

<ul><li>Opposite sides are equal in length.</li>
<li>The two diagonals are equal in length.</li>
<li>It has two lines of reflectional symmetry and rotational symmetry of order 2 (through 180°).</li></ul>

<h2 id="other-rectangles">Other rectangles</h2>

In <a href="/facts/Spherical_geometry/WGLQb1Ys">spherical geometry</a>, a spherical rectangle is a figure whose four edges are <a href="/facts/Great_circle/x4fq6eNm">great circle</a> 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 <a href="/facts/Elliptic_geometry/oMnFAoVb">elliptic geometry</a>, 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 <a href="/facts/Hyperbolic_geometry/DjTMKdgy">hyperbolic geometry</a>, 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.

<h2 id="tessellations">Tessellations</h2>
The rectangle is used in many periodic <a href="/facts/Tessellation/7BWDywXC">tessellation</a> patterns, in <a href="/facts/Brickwork/Gt2lU5SJ">brickwork</a>, for example, these tilings:

<table><tbody><tr><td>Stacked bond</td><td>Running bond</td><td>Basket weave</td><td>Basket weave</td><td><a href="/facts/Herringbone_pattern/enPBfEI4">Herringbone pattern</a></td></tr></tbody></table>
<h2 id="squared-perfect-and-other-tiled-rectangles">Squared, perfect, and other tiled rectangles</h2>

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<a class="footnote-ref" id="fnref:19" href="#fn:19">19</a><a class="footnote-ref" id="fnref:20" href="#fn:20">20</a> if the tiles are <a href="/facts/Similarity_(geometry)/Sbd6hEAN">similar</a> 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 <a href="/facts/Right_triangle/dD6M9GAn">right triangles</a>. A database of all known perfect rectangles, perfect squares and related shapes can be found at <a href="http://www.squaring.net/">squaring.net</a>. The lowest number of squares need for a perfect tiling of a rectangle is 9<a class="footnote-ref" id="fnref:21" href="#fn:21">21</a> and the lowest number needed for a <a href="/facts/Squaring_the_square/8LPIGM4t">perfect tilling a square</a> is 21, found in 1978 by computer search.<a class="footnote-ref" id="fnref:22" href="#fn:22">22</a>
A rectangle has <a href="/facts/Commensurability_(mathematics)/yBFCLN1Q">commensurable</a> sides if and only if it is tileable by a finite number of unequal squares.<a class="footnote-ref" id="fnref:23" href="#fn:23">23</a><a class="footnote-ref" id="fnref:24" href="#fn:24">24</a> 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 <a href="/facts/Polyomino/XUlHFIRY">polyominoes</a>, allowing all rotations and reflections. There are also tilings by congruent <a href="/facts/Polyabolo/jvwFcjks">polyaboloes</a>.

<h2 id="unicode">Unicode</h2>
The following <a href="/facts/Unicode/DDu3MfPV">Unicode</a> code points depict rectangles:

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

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Cuboid/eLLKX4rh">Cuboid</a></li>
<li><a href="/facts/Golden_rectangle/Y7YQsbYh">Golden rectangle</a></li>
<li><a href="/facts/Hyperrectangle/5cbD4khC">Hyperrectangle</a></li>
<li><a href="/facts/Superellipse/W30jhjo5">Superellipse</a> (includes a rectangle with rounded corners)</li></ul>

<h2 id="external-links">External links</h2>

Wikimedia Commons has media related to Rectangles.

<ul><li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a> <a href="https://mathworld.wolfram.com/Rectangle.html">"Rectangle"</a>. <a href="/facts/MathWorld/yc1jodbq">MathWorld</a>.</li>
<li><a href="http://www.mathopenref.com/rectangle.html">Definition and properties of a rectangle</a> with interactive animation.</li>
<li><a href="http://www.mathopenref.com/rectanglearea.html">Area of a rectangle</a> with interactive animation.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Tapson, Frank (July 1999). "A Miscellany of Extracts from a Dictionary of Mathematics" (PDF). Oxford University Press. Archived from the original (PDF) on 2014-05-14. Retrieved 2013-06-20. <a href="https://web.archive.org/web/20140514200449/http://www.cimt.plymouth.ac.uk/resources/topics/art002.pdf" target="_blank">https://web.archive.org/web/20140514200449/http://www.cimt.plymouth.ac.uk/resources/topics/art002.pdf</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">"Definition of Oblong". Math Is Fun. Retrieved 2011-11-13. <a href="http://www.mathsisfun.com/definitions/oblong.html" target="_blank">http://www.mathsisfun.com/definitions/oblong.html</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Oblong – Geometry – Math Dictionary. Icoachmath.com. Retrieved 2011-11-13. <a href="http://www.icoachmath.com/SiteMap/Oblong.html" target="_blank">http://www.icoachmath.com/SiteMap/Oblong.html</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Coxeter, Harold Scott MacDonald; Longuet-Higgins, M.S.; Miller, J.C.P. (1954). "Uniform polyhedra". Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences. 246 (916). The Royal Society: 401–450. Bibcode:1954RSPTA.246..401C. doi:10.1098/rsta.1954.0003. ISSN 0080-4614. JSTOR 91532. MR 0062446. S2CID 202575183. <a href="/wiki/Harold_Scott_MacDonald_Coxeter" target="_blank">/wiki/Harold_Scott_MacDonald_Coxeter</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">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. <a href="/wiki/ISBN_(identifier)" target="_blank">/wiki/ISBN_(identifier)</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Owen Byer; Felix Lazebnik; Deirdre L. Smeltzer (19 August 2010). Methods for Euclidean Geometry. MAA. pp. 53–. ISBN 978-0-88385-763-2. Retrieved 2011-11-13. <a href="978-0-88385-763-2" target="_blank">978-0-88385-763-2</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Gerard Venema, "Exploring Advanced Euclidean Geometry with GeoGebra", MAA, 2013, p. 56. <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Josefsson Martin (2013). "Five Proofs of an Area Characterization of Rectangles" (PDF). Forum Geometricorum. 13: 17–21. Archived from the original (PDF) on 2016-03-04. Retrieved 2013-02-08. <a href="https://web.archive.org/web/20160304001152/http://forumgeom.fau.edu/FG2013volume13/FG201304.pdf" target="_blank">https://web.archive.org/web/20160304001152/http://forumgeom.fau.edu/FG2013volume13/FG201304.pdf</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Josefsson Martin (2013). "Five Proofs of an Area Characterization of Rectangles" (PDF). Forum Geometricorum. 13: 17–21. Archived from the original (PDF) on 2016-03-04. Retrieved 2013-02-08. <a href="https://web.archive.org/web/20160304001152/http://forumgeom.fau.edu/FG2013volume13/FG201304.pdf" target="_blank">https://web.archive.org/web/20160304001152/http://forumgeom.fau.edu/FG2013volume13/FG201304.pdf</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">An Extended Classification of Quadrilaterals Archived 2019-12-30 at the Wayback Machine (An excerpt from De Villiers, M. 1996. Some Adventures in Euclidean Geometry. University of Durban-Westville.) <a href="http://mysite.mweb.co.za/residents/profmd/quadclassify.pdf" target="_blank">http://mysite.mweb.co.za/residents/profmd/quadclassify.pdf</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">de Villiers, Michael, "Generalizing Van Aubel Using Duality", Mathematics Magazine 73 (4), Oct. 2000, pp. 303–307. <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">"Rectangle". Math Is Fun. Retrieved 2024-03-22. <a href="https://www.mathsisfun.com/geometry/rectangle.html" target="_blank">https://www.mathsisfun.com/geometry/rectangle.html</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
<li id="fn:13">Tapson, Frank (July 1999). "A Miscellany of Extracts from a Dictionary of Mathematics" (PDF). Oxford University Press. Archived from the original (PDF) on 2014-05-14. Retrieved 2013-06-20. <a href="https://web.archive.org/web/20140514200449/http://www.cimt.plymouth.ac.uk/resources/topics/art002.pdf" target="_blank">https://web.archive.org/web/20140514200449/http://www.cimt.plymouth.ac.uk/resources/topics/art002.pdf</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></li>
<li id="fn:14">Cyclic Quadrilateral Incentre-Rectangle Archived 2011-09-28 at the Wayback Machine 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. <a href="http://math.kennesaw.edu/~mdevilli/cyclic-incentre-rectangle.html" target="_blank">http://math.kennesaw.edu/~mdevilli/cyclic-incentre-rectangle.html</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></li>
<li id="fn:15">Hall, Leon M. & Robert P. Roe (1998). "An Unexpected Maximum in a Family of Rectangles" (PDF). Mathematics Magazine. 71 (4): 285–291. doi:10.1080/0025570X.1998.11996653. JSTOR 2690700. Archived from the original (PDF) on 2010-07-23. Retrieved 2011-11-13. <a href="https://web.archive.org/web/20100723134734/http://web.mst.edu/~lmhall/Personal/HallRoe/Hall_Roe.pdf" target="_blank">https://web.archive.org/web/20100723134734/http://web.mst.edu/~lmhall/Personal/HallRoe/Hall_Roe.pdf</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></li>
<li id="fn:16">Lassak, M. (1993). "Approximation of convex bodies by rectangles". Geometriae Dedicata. 47: 111–117. doi:10.1007/BF01263495. S2CID 119508642. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></li>
<li id="fn:17">Sloane, N. J. A. (ed.). "Sequence A366185 (Decimal expansion of the real root of the quintic equation 
 
 
 
  
 
 x
 
 5
 
 
 +
 3
 
 x
 
 4
 
 
 +
 4
 
 x
 
 3
 
 
 +
 x
 −
 1
 =
 0
 
 
 {\displaystyle \ x^{5}+3x^{4}+4x^{3}+x-1=0}
 
)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. <a href="/wiki/Neil_Sloane" target="_blank">/wiki/Neil_Sloane</a> <a href="#fnref:17" class="footnote-back-ref">↩</a></li>
<li id="fn:18">Stars: A Second Look. (PDF). Retrieved 2011-11-13. <a href="https://web.archive.org/web/20150723004135/http://mysite.mweb.co.za/residents/profmd/stars.pdf" target="_blank">https://web.archive.org/web/20150723004135/http://mysite.mweb.co.za/residents/profmd/stars.pdf</a> <a href="#fnref:18" class="footnote-back-ref">↩</a></li>
<li id="fn:19">R.L. Brooks; C.A.B. Smith; A.H. Stone & W.T. Tutte (1940). "The dissection of rectangles into squares". Duke Math. J. 7 (1): 312–340. doi:10.1215/S0012-7094-40-00718-9. <a href="http://projecteuclid.org/euclid.dmj/1077492259" target="_blank">http://projecteuclid.org/euclid.dmj/1077492259</a> <a href="#fnref:19" class="footnote-back-ref">↩</a></li>
<li id="fn:20">J.D. Skinner II; C.A.B. Smith & W.T. Tutte (November 2000). "On the Dissection of Rectangles into Right-Angled Isosceles Triangles". Journal of Combinatorial Theory, Series B. 80 (2): 277–319. doi:10.1006/jctb.2000.1987. <a href="https://doi.org/10.1006%2Fjctb.2000.1987" target="_blank">https://doi.org/10.1006%2Fjctb.2000.1987</a> <a href="#fnref:20" class="footnote-back-ref">↩</a></li>
<li id="fn:21">Sloane, N. J. A. (ed.). "Sequence A219766 (Number of nonsquare simple perfect squared rectangles of order n up to symmetry)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. <a href="/wiki/Neil_Sloane" target="_blank">/wiki/Neil_Sloane</a> <a href="#fnref:21" class="footnote-back-ref">↩</a></li>
<li id="fn:22">"Squared Squares; Perfect Simples, Perfect Compounds and Imperfect Simples". www.squaring.net. Retrieved 2021-09-26. <a href="http://www.squaring.net/sq/ss/spss/o21/spsso21.html" target="_blank">http://www.squaring.net/sq/ss/spss/o21/spsso21.html</a> <a href="#fnref:22" class="footnote-back-ref">↩</a></li>
<li id="fn:23">R.L. Brooks; C.A.B. Smith; A.H. Stone & W.T. Tutte (1940). "The dissection of rectangles into squares". Duke Math. J. 7 (1): 312–340. doi:10.1215/S0012-7094-40-00718-9. <a href="http://projecteuclid.org/euclid.dmj/1077492259" target="_blank">http://projecteuclid.org/euclid.dmj/1077492259</a> <a href="#fnref:23" class="footnote-back-ref">↩</a></li>
<li id="fn:24">R. Sprague (1940). "Ũber die Zerlegung von Rechtecken in lauter verschiedene Quadrate". Journal für die reine und angewandte Mathematik (in German). 1940 (182): 60–64. doi:10.1515/crll.1940.182.60. S2CID 118088887. <a href="/wiki/Crelle%27s_Journal" target="_blank">/wiki/Crelle%27s_Journal</a> <a href="#fnref:24" class="footnote-back-ref">↩</a></li>
</ol>

Rectangle open-in-new

Rectangle