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{{Short description|Quadrilateral symmetric across a diagonal}}
{{Good article}}
{{Use dmy dates|cs1-dates=ly|date=October 2022}}
{{Use list-defined references|date=October 2022}}
{{Infobox polygon
| name       = Kite
| image      = GeometricKite.svg
| caption    = A kite, showing its pairs of equal-length sides and its inscribed circle.
| type       = [[Quadrilateral]]
| euler      =
| edges      = 4
| schläfli   = 
| wythoff    = 
| coxeter    = 
| symmetry   = [[Reflection symmetry|D<sub>1</sub>]] (*)
| area       = 
| angle      = 
| dual       = [[Isosceles trapezoid]]
| properties = }}
In [[Euclidean geometry]], a '''kite''' is a [[quadrilateral]] with [[reflection symmetry]] across a [[diagonal]]. Because of this symmetry, a kite has two equal angles and two pairs of adjacent equal-length sides. Kites are also known as '''deltoids''',{{r|halsted}} but the word ''deltoid'' may also refer to a [[deltoid curve]], an unrelated geometric object sometimes studied in connection with quadrilaterals.{{r|goormaghtigh}}<ref>See [[H. S. M. Coxeter]]'s review of {{harvtxt|Grünbaum|1960}} in {{MR|0125489}}: "It is unfortunate that the author uses, instead of 'kite', the name 'deltoid', which belongs more properly to a curve, the three-cusped hypocycloid."</ref> A kite may also be called a '''dart''',{{r|charter-rogers}} particularly if it is not convex.{{r|gardner|thurston}}

Every kite is an [[orthodiagonal quadrilateral]] (its diagonals are at right angles) and, when convex, a [[tangential quadrilateral]] (its sides are tangent to an inscribed circle). The convex kites are exactly the quadrilaterals that are both orthodiagonal and tangential. They include as special cases the [[right kite]]s, with two opposite right angles; the [[rhombus|rhombi]], with two diagonal axes of symmetry; and the [[square]]s, which are also special cases of both right kites and rhombi.

The quadrilateral with the greatest ratio of [[perimeter]] to [[diameter]] is a kite, with 60°, 75°, and 150° angles. Kites of two shapes (one convex and one non-convex) form the [[prototile]]s of one of the forms of the [[Penrose tiling]]. Kites also form the faces of several [[Isohedral figure|face-symmetric]] polyhedra and [[tessellation]]s, and have been studied in connection with [[outer billiards]], a problem in the advanced mathematics of [[dynamical system]]s.

== Definition and classification ==
[[File:Deltoid.svg|thumb|Convex and concave kites]]
A kite is a [[quadrilateral]] with [[reflection symmetry]] across one of its diagonals. Equivalently, it is a quadrilateral whose four sides can be grouped into two pairs of adjacent equal-length sides.{{r|halsted|devilliers-adventures}} A kite can be constructed from the centers and crossing points of any two intersecting [[circle]]s.{{r|idiot}} Kites as described here may be either [[Convex polygon|convex]] or [[Concave polygon|concave]], although some sources restrict ''kite'' to mean only convex kites. A quadrilateral is a kite [[if and only if]] any one of the following conditions is true:
* The four sides can be split into two pairs of adjacent equal-length sides.{{r|devilliers-adventures}}
* One diagonal crosses the midpoint of the other diagonal at a right angle, forming its [[perpendicular bisector]].{{r|usiskin-griffin}} (In the concave case, the line through one of the diagonals bisects the other.)
* One diagonal is a line of symmetry. It divides the quadrilateral into two congruent triangles that are mirror images of each other.{{r|devilliers-adventures}}
* One diagonal [[Angle bisector|bisects]] both of the angles at its two ends.{{r|devilliers-adventures}}
Kite quadrilaterals are named for the wind-blown, flying [[kite]]s, which often have this shape{{r|beamer|alexander-koeberlein}} and which are in turn named for [[kite (bird)|a hovering bird]] and the sound it makes.{{r|nuncius|liberman}} According to [[Olaus Henrici]], the name "kite" was given to these shapes by [[James Joseph Sylvester]].{{r|henrici}}

Quadrilaterals can be classified ''hierarchically'', meaning that some classes of quadrilaterals include other classes, or ''partitionally'', meaning that each quadrilateral is in only one class. Classified hierarchically, kites include the [[rhombus|rhombi]] (quadrilaterals with four equal sides), [[square (geometry)|squares]],{{r|devilliers-role}} and [[Apollonius quadrilateral]]s (in which the products of opposite sides are equal).{{r|harras}} All [[equilateral polygon|equilateral]] kites are rhombi, and all [[equiangular polygon|equiangular]] kites are squares. When classified partitionally, rhombi and squares would not be kites, because they belong to a different class of quadrilaterals; similarly, the [[right kite]]s discussed below would not be kites. The remainder of this article follows a hierarchical classification; rhombi, squares, and right kites are all considered kites. By avoiding the need to consider special cases, this classification can simplify some facts about kites.{{r|devilliers-role}}

Like kites, a [[parallelogram]] also has two pairs of equal-length sides, but they are opposite to each other rather than adjacent. Any [[simple polygon|non-self-crossing]] quadrilateral that has an axis of symmetry must be either a kite, with a diagonal axis of symmetry; or an [[isosceles trapezoid]], with an axis of symmetry through the midpoints of two sides. These include as special cases the [[rhombus]] and the [[rectangle]] respectively, and the square, which is a special case of both.{{r|halsted}} The self-crossing quadrilaterals include another class of symmetric quadrilaterals, the [[antiparallelogram]]s.{{r|alsina-nelson}}

== Special cases ==
{{multiple image
|image1=Bicentric kite 001.svg|caption1=Right kite
|image2=Reuleaux kite.svg|caption2=Equidiagonal kite in a [[Reuleaux triangle]]
|image3=Lute of Pythagoras.svg|caption3=[[Lute of Pythagoras]]
|total_width=500}}

The [[right kite]]s have two opposite [[right angle]]s.{{r|devilliers-role|alsina-nelson}} The right kites are exactly the kites that are [[cyclic quadrilateral]]s, meaning that there is a circle that passes through all their vertices.{{r|gant}} The cyclic quadrilaterals may equivalently defined as the quadrilaterals in which two opposite angles are [[supplementary angles|supplementary]] (they add to 180°); if one pair is supplementary the other is as well.{{r|usiskin-griffin}} Therefore, the right kites are the kites with two opposite supplementary angles, for either of the two opposite pairs of angles. Because right kites circumscribe one circle and are inscribed in another circle, they are [[bicentric quadrilateral]]s (actually tricentric, as they also have a third circle externally tangent to the [[Extended side|extensions of their sides]]).{{r|alsina-nelson}} If the sizes of an inscribed and a circumscribed circle are fixed, the right kite has the largest area of any quadrilateral trapped between them.{{r|josefsson-area}}

Among all quadrilaterals, the shape that has the greatest ratio of its [[perimeter]] to its [[diameter]] (maximum distance between any two points) is an [[equidiagonal quadrilateral|equidiagonal]] kite with angles 60°, 75°, 150°, 75°. Its four vertices lie at the three corners and one of the side midpoints of the [[Reuleaux triangle]].{{r|ball|griffiths-culpin}} An equidiagonal kite is a special case of a [[midsquare quadrilateral]]. When an equidiagonal kite has side lengths less than or equal to its diagonals, like this one or the square, it is one of the quadrilaterals with the [[biggest little polygon|greatest ratio of area to diameter]].{{r|audet-hansen-svrtan}}

A kite with three 108° angles and one 36° angle forms the [[convex hull]] of the [[lute of Pythagoras]], a [[fractal]] made of nested [[pentagram]]s.{{r|darling}} The four sides of this kite lie on four of the sides of a [[regular pentagon]], with a [[Golden triangle (mathematics)|golden triangle]] glued onto the fifth side.{{r|alsina-nelson}}

[[File:Aperiodic monotile smith 2023.svg|thumb|Part of an aperiodic tiling with prototiles made from eight kites]]
There are only eight polygons that can tile the plane such that reflecting any tile across any one of its edges produces another tile; this arrangement is called an [[edge tessellation]]. One of them is a tiling by a right kite, with 60°, 90°, and 120° angles. It produces the [[deltoidal trihexagonal tiling]] (see {{slink||Tilings and polyhedra}}).{{r|kirby-umble}} A [[prototile]] made by eight of these kites tiles the plane only [[aperiodic tiling|aperiodically]], key to a claimed solution of the [[einstein problem]].{{r|smkg}}

In [[non-Euclidean geometry]], a kite can have three right angles and one non-right angle, forming a special case of a [[Lambert quadrilateral]]. The fourth angle is acute in [[hyperbolic geometry]] and obtuse in [[spherical geometry]].{{r|eves}}

== Properties ==
=== Diagonals, angles, and area ===
Every kite is an [[orthodiagonal quadrilateral]], meaning that its two diagonals are [[perpendicular|at right angles]] to each other. Moreover, one of the two diagonals (the symmetry axis) is the [[perpendicular bisector]] of the other, and is also the [[angle bisector]] of the two angles it meets.{{r|halsted}} Because of its symmetry, the other two angles of the kite must be equal.{{r|beamer|alexander-koeberlein}} The diagonal symmetry axis of a convex kite divides it into two [[congruent triangles]]; the other diagonal divides it into two [[isosceles triangle]]s.{{r|halsted}}

As is true more generally for any orthodiagonal quadrilateral, the area {{tmath|A}} of a kite may be calculated as half the product of the lengths of the diagonals {{tmath|p}} and {{tmath|q}}:{{r|beamer}}
<math display=block>A =\frac{p \cdot q}{2}.</math>
Alternatively, the area can be calculated by dividing the kite into two congruent triangles and applying the [[Area of a triangle#Using trigonometry|SAS formula]] for their area. If <math>a</math> and <math>b</math> are the lengths of two sides of the kite, and <math>\theta</math> is the [[angle]] between, then the area is{{r|crux}}
<math display=block>\displaystyle A = ab \cdot \sin\theta.</math>

=== Inscribed circle ===
{{multiple_image|direction=vertical
|image1=Kite inexcircles.svg
|image2=Dart inexcircles.svg
|image3=Antipar inexcircles.svg
|width=300px
|footer=Two circles tangent to the sides and extended sides of a convex kite (top), non-convex kite (middle), and [[antiparallelogram]] (bottom). The four lines through the sides of each quadrilateral are [[bitangent]]s of the circles.}}
Every ''convex'' kite is also a [[tangential quadrilateral]], a quadrilateral that has an [[inscribed circle]]. That is, there exists a circle that is [[tangent]] to all four sides. Additionally, if a convex kite is not a rhombus, there is a circle outside the kite that is tangent to the extensions of the four sides; therefore, every convex kite that is not a rhombus is an [[ex-tangential quadrilateral]]. The convex kites that are not rhombi are exactly the quadrilaterals that are both tangential and ex-tangential.{{r|alsina-nelson}} For every ''concave'' kite there exist two circles tangent to two of the sides and the extensions of the other two: one is interior to the kite and touches the two sides opposite from the concave angle, while the other circle is exterior to the kite and touches the kite on the two edges incident to the concave angle.{{r|wheeler}}

For a convex kite with diagonal lengths {{tmath|p}} and {{tmath|q}} and side lengths {{tmath|a}} and {{tmath|b}}, the radius {{tmath|r}} of the inscribed circle is <math display=block>r=\frac{pq}{2(a+b)},</math>
and the radius {{tmath|\rho}} of the ex-tangential circle is{{r|alsina-nelson}}
<math display=block>\rho=\frac{pq}{2|a-b|}.</math>

A tangential quadrilateral is also a kite [[if and only if]] any one of the following conditions is true:{{r|josefsson-when}}
* The area is one half the product of the [[diagonal]]s.
* The diagonals are [[perpendicular]]. (Thus the kites are exactly the quadrilaterals that are both tangential and [[orthodiagonal quadrilateral|orthodiagonal]].)
* The two line segments connecting opposite points of tangency have equal length.
* The [[tangential quadrilateral#Special line segments|tangent lengths]], distances from a point of tangency to an adjacent vertex of the quadrilateral, are equal at two opposite vertices of the quadrilateral. (At each vertex, there are two adjacent points of tangency, but they are the same distance as each other from the vertex, so each vertex has a single tangent length.)
* The two [[Quadrilateral#Special line segments|bimedians]], line segments connecting midpoints of opposite edges, have equal length.
* The products of opposite side lengths are equal.
* The center of the incircle lies on a line of symmetry that is also a diagonal.

If the diagonals in a tangential quadrilateral {{tmath|ABCD}} intersect at {{tmath|P}}, and the [[Incircle and excircles of a triangle|incircle]]s of triangles {{tmath|ABP}}}, {{tmath|BCP}}, {{tmath|CDP}}, {{tmath|DAP}} have radii {{tmath|r_1}}, {{tmath|r_2}}, {{tmath|r_3}}, and {{tmath|r_4}} respectively, then the quadrilateral is a kite if and only if{{r|josefsson-when}}
<math display=block>r_1+r_3=r_2+r_4.</math>
If the [[Incircle and excircles of a triangle|excircle]]s to the same four triangles opposite the vertex {{tmath|P}} have radii {{tmath|R_1}}, {{tmath|R_2}}, {{tmath|R_3}}, and {{tmath|R_4}} respectively, then the quadrilateral is a kite if and only if{{r|josefsson-when}}
<math display=block>R_1+R_3=R_2+R_4.</math>

=== Duality ===
[[File:Kite isotrap duality.svg|thumb|upright|A kite and its dual isosceles trapezoid]]
Kites and [[isosceles trapezoid]]s are dual to each other, meaning that there is a correspondence between them that reverses the dimension of their parts, taking vertices to sides and sides to vertices. From any kite, the inscribed circle is tangent to its four sides at the four vertices of an isosceles trapezoid. For any isosceles trapezoid, tangent lines to the circumscribing circle at its four vertices form the four sides of a kite. This correspondence can also be seen as an example of [[polar reciprocation]], a general method for corresponding points with lines and vice versa given a fixed circle. Although they do not touch the circle, the four vertices of the kite are reciprocal in this sense to the four sides of the isosceles trapezoid.{{r|robertson}} The features of kites and isosceles trapezoids that correspond to each other under this duality are compared in the table below.{{r|devilliers-adventures}}

{| class=wikitable style="text-align: center;"
|-
! scope="col" | Isosceles trapezoid
! scope="col" | Kite
|-
| Two pairs of equal adjacent angles
| Two pairs of equal adjacent sides
|-
| Two equal opposite sides
| Two equal opposite angles
|-
| Two opposite sides with a shared perpendicular bisector
| Two opposite angles with a shared angle bisector
|-
| An axis of symmetry through two opposite sides
| An axis of symmetry through two opposite angles
|-
| Circumscribed circle through all vertices
| Inscribed circle tangent to all sides
|}

=== Dissection ===
The [[equidissection]] problem concerns the subdivision of polygons into triangles that all have equal areas. In this context, the ''spectrum'' of a polygon is the set of numbers <math>n</math> such that the polygon has an equidissection into <math>n</math> equal-area triangles. Because of its symmetry, the spectrum of a kite contains all even integers. Certain special kites also contain some odd numbers in their spectra.{{r|kasimitis-stein|jepsen-sedberry-hoyer}}

Every triangle can be subdivided into three right kites meeting at the center of its inscribed circle. More generally, a method based on [[circle packing]] can be used to subdivide any polygon with <math>n</math> sides into <math>O(n)</math> kites, meeting edge-to-edge.{{r|bern-eppstein}}

== Tilings and polyhedra ==
{{multiple image
|image1=AnimSun2k.gif|caption1=Recursive construction of the kite and dart Penrose tiling
|image2=Fractal Penrose kite rosette.svg|caption2=Fractal rosette of Penrose kites
|total_width=540}}
All kites [[tessellation|tile the plane]] by repeated [[point reflection]] around the midpoints of their edges, as do more generally all quadrilaterals.{{r|schattschneider}} Kites and darts with angles 72°, 72°, 72°, 144° and 36°, 72°, 36°, 216°, respectively, form the [[prototile]]s of one version of the [[Penrose tiling]], an [[aperiodic tiling]] of the plane discovered by mathematical physicist [[Roger Penrose]].{{r|gardner}} When a kite has angles that, at its apex and one side, sum to <math>\pi(1-\tfrac1n)</math> for some positive integer {{tmath|n}}, then scaled copies of that kite can be used to tile the plane in a [[fractal]] rosette in which successively larger rings of <math>n</math> kites surround a central point.{{r|fathauer}} These rosettes can be used to study the phenomenon of inelastic collapse, in which a system of moving particles meeting in [[inelastic collision]]s all coalesce at a common point.{{r|chazelle-karntikoon-zheng}}

A kite with angles 60°, 90°, 120°, 90° can also tile the plane by repeated reflection across its edges; the resulting tessellation, the [[deltoidal trihexagonal tiling]], superposes a tessellation of the plane by regular hexagons and isosceles triangles.{{r|alsina-nelson}} The [[deltoidal icositetrahedron]], [[deltoidal hexecontahedron]], and [[trapezohedron]] are [[polyhedra]] with congruent kite-shaped [[face (geometry)|faces]],{{r|grunbaum}} which can alternatively be thought of as tilings of the sphere by congruent spherical kites.{{r|sakano-akama}} There are infinitely many [[isohedral figure|face-symmetric tilings]] of the [[hyperbolic plane]] by kites.{{r|dunham-lindgren-witte}} These polyhedra (equivalently, spherical tilings), the square and deltoidal trihexagonal tilings of the Euclidean plane, and some tilings of the hyperbolic plane are shown in the table below, labeled by [[face configuration]] (the numbers of neighbors of each of the four vertices of each tile). Some polyhedra and tilings appear twice, under two different face configurations.
{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
|-
!colspan=3|Polyhedra
!Euclidean
|- style="text-align:center;vertical-align:top;"
|[[Image:Rhombicdodecahedron.jpg|120px]]{{br}}[[rhombic dodecahedron|V4.3.4.3]]
|[[Image:deltoidalicositetrahedron.jpg|120px]]{{br}}[[Deltoidal icositetrahedron|V4.3.4.4]]
|[[Image:deltoidalhexecontahedron.jpg|120px]]{{br}}[[Deltoidal hexecontahedron|V4.3.4.5]]
|[[Image:Tiling Dual Semiregular V3-4-6-4 Deltoidal Trihexagonal.svg|120px]]{{br}}[[Deltoidal trihexagonal tiling|V4.3.4.6]]
|-
!Polyhedra
!Euclidean
!colspan=2|Hyperbolic tilings
|- style="text-align:center;vertical-align:top;"
|[[File:Deltoidalicositetrahedron.jpg|120px]]{{br}}[[deltoidal icositetrahedron|V4.4.4.3]]
|[[File:Square tiling uniform coloring 1.svg|120px]]{{br}}[[Square tiling|V4.4.4.4]]
|[[File:H2-5-4-deltoidal.svg|120px]]{{br}}[[Deltoidal tetrapentagonal tiling|V4.4.4.5]] 
|[[File:H2chess 246d.png|120px]]{{br}}[[Deltoidal tetrahexagonal tiling|V4.4.4.6]]
|-
!Polyhedra
!colspan=3|Hyperbolic tilings
|- style="text-align:center;vertical-align:top;"
|[[Image:deltoidalhexecontahedron.jpg|120px]]{{br}}[[Deltoidal hexecontahedron|V4.3.4.5]]
|[[File:H2-5-4-deltoidal.svg|120px]]{{br}}[[Deltoidal tetrapentagonal tiling|V4.4.4.5]] 
|[[File:H2-5-4-rhombic.svg|120px]]{{br}}[[Rhombic pentapentagonal tiling|V4.5.4.5]] 
|[[File:Deltoidal_pentahexagonal_tiling.png|120px]]{{br}}[[Deltoidal pentahexagonal tiling|V4.6.4.5]] 
|-
!Euclidean
!colspan=3|Hyperbolic tilings
|- style="text-align:center;vertical-align:top;"
|[[Image:Tiling Dual Semiregular V3-4-6-4 Deltoidal Trihexagonal.svg|120px]]{{br}}[[Deltoidal trihexagonal tiling|V4.3.4.6]]
|[[File:H2chess_246d.png|120px]]{{br}}[[Deltoidal tetrahexagonal tiling|V4.4.4.6]]
|[[File:Deltoidal pentahexagonal tiling.png|120px]]{{br}}[[Deltoidal pentahexagonal tiling|V4.5.4.6]]
|[[File:H2chess 266d.png|120px]]{{br}}[[Rhombic hexahexagonal tiling|V4.6.4.6]]
|}

[[File:2 10-sided dice.jpg|thumb|upright|Ten-sided dice]]
The [[trapezohedron|trapezohedra]] are another family of polyhedra that have congruent kite-shaped faces. In these polyhedra, the edges of one of the two side lengths of the kite meet at two "pole" vertices, while the edges of the other length form an equatorial zigzag path around the polyhedron. They are the [[dual polyhedron|dual polyhedra]] of the uniform [[antiprism]]s.{{r|grunbaum}} A commonly seen example is the [[pentagonal trapezohedron]], used for ten-sided [[dice]].{{r|alsina-nelson}}

== Outer billiards ==
Mathematician [[Richard Schwartz (mathematician)|Richard Schwartz]] has studied [[outer billiard]]s on kites. Outer billiards is a [[dynamical system]] in which, from a point outside a given [[compact set|compact]] [[convex set]] in the plane, one draws a tangent line to the convex set, travels from the starting point along this line to another point equally far from the point of tangency, and then repeats the same process. It had been open since the 1950s whether any system defined in this way could produce paths that get arbitrarily far from their starting point, and in a 2007 paper Schwartz solved this problem by finding unbounded billiards paths for the kite with angles 72°, 72°, 72°, 144°, the same as the one used in the Penrose tiling.{{r|schwartz-unbounded}} He later wrote a [[monograph]] analyzing outer billiards for kite shapes more generally. For this problem, any [[affine transformation]] of a kite preserves the dynamical properties of outer billiards on it, and it is possible to transform any kite into a shape where three vertices are at the points {{tmath|(-1,0)}} and {{tmath|(0,\pm1)}}, with the fourth at {{tmath|(\alpha,0)}} with {{tmath|\alpha}} in the open unit interval {{tmath|(0,1)}}. The behavior of outer billiards on any kite depends strongly on the parameter <math>\alpha</math> and in particular whether it is [[rational number|rational]]. For the case of the Penrose kite, {{tmath|1=\textstyle \alpha=1/\varphi^3}}, an irrational number, where {{tmath|1= \varphi =\tfrac12\bigl(1+\sqrt5~\!\bigr)}} is the [[golden ratio]].{{r|schwartz-monograph}}

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== External links ==
{{Commons category|Deltoids}}
* {{MathWorld |urlname=Kite |title=Kite|mode=cs2}}
* [http://www.mathopenref.com/kitearea.html area formulae] with interactive animation at Mathopenref.com

{{Polygons}}

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