Kite
A kite, showing its pairs of equal-length sides and its inscribed circle.
Type	Quadrilateral
Edges andvertices	4
Symmetry group	D1 (*)
Dual polygon	Isosceles trapezoid

InEuclidean geometry, akite is aquadrilateral withreflection symmetry across adiagonal. Because of this symmetry, a kite has two equal angles and two pairs of adjacent equal-length sides. Kites are also known asdeltoids,^[1] but the worddeltoid may also refer to adeltoid curve, an unrelated geometric object sometimes studied in connection with quadrilaterals.^[2][3] A kite may also be called adart,^[4] particularly if it is not convex.^[5][6]

Every kite is anorthodiagonal quadrilateral (its diagonals are at right angles) and, when convex, atangential 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 theright kites, with two opposite right angles; therhombi, with two diagonal axes of symmetry; and thesquares, which are also special cases of both right kites and rhombi.

The quadrilateral with the greatest ratio ofperimeter todiameter is a kite, with 60°, 75°, and 150° angles. Kites of two shapes (one convex and one non-convex) form theprototiles of one of the forms of thePenrose tiling. Kites also form the faces of severalface-symmetric polyhedra andtessellations, and have been studied in connection withouter billiards, a problem in the advanced mathematics ofdynamical systems.

A kite is aquadrilateral withreflection 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.^[1][7] A kite can be constructed from the centers and crossing points of any two intersectingcircles.^[8] Kites as described here may be eitherconvex orconcave, although some sources restrictkite to mean only convex kites. A quadrilateral is a kiteif 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.^[7]
• One diagonal crosses the midpoint of the other diagonal at a right angle, forming itsperpendicular bisector.^[9] (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.^[7]
• One diagonalbisects both of the angles at its two ends.^[7]

Kite quadrilaterals are named for the wind-blown, flyingkites, which often have this shape^[10][11] and which are in turn named fora hovering bird and the sound it makes.^[12][13] According toOlaus Henrici, the name "kite" was given to these shapes byJames Joseph Sylvester.^[14]

Quadrilaterals can be classifiedhierarchically, meaning that some classes of quadrilaterals include other classes, orpartitionally, meaning that each quadrilateral is in only one class. Classified hierarchically, kites include therhombi (quadrilaterals with four equal sides),squares,^[15] andApollonius quadrilaterals (in which the products of opposite sides are equal).^[16] Allequilateral kites are rhombi, and allequiangular kites are squares. When classified partitionally, rhombi and squares would not be kites, because they belong to a different class of quadrilaterals; similarly, theright kites 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.^[15]

Like kites, aparallelogram also has two pairs of equal-length sides, but they are opposite to each other rather than adjacent. Anynon-self-crossing quadrilateral that has an axis of symmetry must be either a kite, with a diagonal axis of symmetry; or anisosceles trapezoid, with an axis of symmetry through the midpoints of two sides. These include as special cases therhombus and therectangle respectively, and the square, which is a special case of both.^[1] The self-crossing quadrilaterals include another class of symmetric quadrilaterals, theantiparallelograms.^[17]

Theright kites have two oppositeright angles.^[15][17] The right kites are exactly the kites that arecyclic quadrilaterals, meaning that there is a circle that passes through all their vertices.^[18] The cyclic quadrilaterals may equivalently defined as the quadrilaterals in which two opposite angles aresupplementary (they add to 180°); if one pair is supplementary the other is as well.^[9] 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 arebicentric quadrilaterals (actually tricentric, as they also have a third circle externally tangent to theextensions of their sides).^[17] 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.^[19]

Among all quadrilaterals, the shape that has the greatest ratio of itsperimeter to itsdiameter (maximum distance between any two points) is anequidiagonal kite with angles 60°, 75°, 150°, 75°. Its four vertices lie at the three corners and one of the side midpoints of theReuleaux triangle.^[20][21] An equidiagonal kite is a special case of amidsquare 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 thegreatest ratio of area to diameter.^[22]

A kite with three 108° angles and one 36° angle forms theconvex hull of thelute of Pythagoras, afractal made of nestedpentagrams.^[23] The four sides of this kite lie on four of the sides of aregular pentagon, with agolden triangle glued onto the fifth side.^[17]

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 anedge tessellation. One of them is a tiling by a right kite, with 60°, 90°, and 120° angles. It produces thedeltoidal trihexagonal tiling (see§ Tilings and polyhedra).^[24] Aprototile made by eight of these kites tiles the plane onlyaperiodically, key to a claimed solution of theeinstein problem.^[25]

Innon-Euclidean geometry, a kite can have three right angles and one non-right angle, forming a special case of aLambert quadrilateral. The fourth angle is acute inhyperbolic geometry and obtuse inspherical geometry.^[26]

Every kite is anorthodiagonal quadrilateral, meaning that its two diagonals areat right angles to each other. Moreover, one of the two diagonals (the symmetry axis) is theperpendicular bisector of the other, and is also theangle bisector of the two angles it meets.^[1] Because of its symmetry, the other two angles of the kite must be equal.^[10][11] The diagonal symmetry axis of a convex kite divides it into twocongruent triangles; the other diagonal divides it into twoisosceles triangles.^[1]

As is true more generally for any orthodiagonal quadrilateral, the area⁠${\displaystyle A}$ ⁠ of a kite may be calculated as half the product of the lengths of the diagonals⁠${\displaystyle p}$ ⁠ and⁠${\displaystyle q}$ ⁠:^[10]$${\displaystyle A=\frac{p\cdot q}{2}.}$$ Alternatively, the area can be calculated by dividing the kite into two congruent triangles and applying theSAS formula for their area. If${\displaystyle a}$  and${\displaystyle b}$  are the lengths of two sides of the kite, and${\displaystyle \theta }$  is theangle between, then the area is^[27]$${\displaystyle {\displaystyle A=ab\cdot \sin \theta .}}$$ 

Everyconvex kite is also atangential quadrilateral, a quadrilateral that has aninscribed circle. That is, there exists a circle that istangent 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 anex-tangential quadrilateral. The convex kites that are not rhombi are exactly the quadrilaterals that are both tangential and ex-tangential.^[17] For everyconcave 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.^[28]

For a convex kite with diagonal lengths⁠${\displaystyle p}$ ⁠ and⁠${\displaystyle q}$ ⁠ and side lengths⁠${\displaystyle a}$ ⁠ and⁠${\displaystyle b}$ ⁠, the radius⁠${\displaystyle r}$ ⁠ of the inscribed circle is$${\displaystyle r=\frac{pq}{2(a+b)},}$$ and the radius⁠${\displaystyle \rho }$ ⁠ of the ex-tangential circle is^[17]$${\displaystyle \rho =\frac{pq}{2|a-b|}.}$$ 

A tangential quadrilateral is also a kiteif and only if any one of the following conditions is true:^[29]

• The area is one half the product of thediagonals.
• The diagonals areperpendicular. (Thus the kites are exactly the quadrilaterals that are both tangential andorthodiagonal.)
• The two line segments connecting opposite points of tangency have equal length.
• Thetangent 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 twobimedians, 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${\displaystyle ABCD}$ ⁠ intersect at⁠${\displaystyle P}$ ⁠, and theincircles of triangles⁠${\displaystyle ABP}$ ⁠},⁠${\displaystyle BCP}$ ⁠,⁠${\displaystyle CDP}$ ⁠,⁠${\displaystyle DAP}$ ⁠ have radii⁠${\displaystyle r_1}$ ⁠,⁠${\displaystyle r_2}$ ⁠,⁠${\displaystyle r_3}$ ⁠, and⁠${\displaystyle r_4}$ ⁠ respectively, then the quadrilateral is a kite if and only if^[29]$${\displaystyle r_1+r_3=r_2+r_4.}$$ If theexcircles to the same four triangles opposite the vertex⁠${\displaystyle P}$ ⁠ have radii⁠${\displaystyle R_1}$ ⁠,⁠${\displaystyle R_2}$ ⁠,⁠${\displaystyle R_3}$ ⁠, and⁠${\displaystyle R_4}$ ⁠ respectively, then the quadrilateral is a kite if and only if^[29]$${\displaystyle R_1+R_3=R_2+R_4.}$$ 

Kites andisosceles trapezoids 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 ofpolar 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.^[30] The features of kites and isosceles trapezoids that correspond to each other under this duality are compared in the table below.^[7]

Theequidissection problem concerns the subdivision of polygons into triangles that all have equal areas. In this context, thespectrum of a polygon is the set of numbers${\displaystyle n}$  such that the polygon has an equidissection into${\displaystyle n}$  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.^[31][32]

Every triangle can be subdivided into three right kites meeting at the center of its inscribed circle. More generally, a method based oncircle packing can be used to subdivide any polygon with${\displaystyle n}$  sides into${\displaystyle O(n)}$  kites, meeting edge-to-edge.^[33]

All kitestile the plane by repeatedpoint reflection around the midpoints of their edges, as do more generally all quadrilaterals.^[34] Kites and darts with angles 72°, 72°, 72°, 144° and 36°, 72°, 36°, 216°, respectively, form theprototiles of one version of thePenrose tiling, anaperiodic tiling of the plane discovered by mathematical physicistRoger Penrose.^[5] When a kite has angles that, at its apex and one side, sum to${\displaystyle \pi (1-\frac{1}{n})}$  for some positive integer⁠${\displaystyle n}$ ⁠, then scaled copies of that kite can be used to tile the plane in afractal rosette in which successively larger rings of${\displaystyle n}$  kites surround a central point.^[35] These rosettes can be used to study the phenomenon of inelastic collapse, in which a system of moving particles meeting ininelastic collisions all coalesce at a common point.^[36]

A kite with angles 60°, 90°, 120°, 90° can also tile the plane by repeated reflection across its edges; the resulting tessellation, thedeltoidal trihexagonal tiling, superposes a tessellation of the plane by regular hexagons and isosceles triangles.^[17] Thedeltoidal icositetrahedron,deltoidal hexecontahedron, andtrapezohedron arepolyhedra with congruent kite-shapedfaces,^[37] which can alternatively be thought of as tilings of the sphere by congruent spherical kites.^[38] There are infinitely manyface-symmetric tilings of thehyperbolic plane by kites.^[39] 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 byface 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.

Thetrapezohedra 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 thedual polyhedra of the uniformantiprisms.^[37] A commonly seen example is thepentagonal trapezohedron, used for ten-sideddice.^[17]

MathematicianRichard Schwartz has studiedouter billiards on kites. Outer billiards is adynamical system in which, from a point outside a givencompactconvex 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.^[40] He later wrote amonograph analyzing outer billiards for kite shapes more generally. For this problem, anyaffine 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${\displaystyle (-1,0)}$ ⁠ and⁠${\displaystyle (0,\pm 1)}$ ⁠, with the fourth at⁠${\displaystyle (\alpha ,0)}$ ⁠ with⁠${\displaystyle \alpha }$ ⁠ in the open unit interval⁠${\displaystyle (0,1)}$ ⁠. The behavior of outer billiards on any kite depends strongly on the parameter${\displaystyle \alpha }$  and in particular whether it isrational. For the case of the Penrose kite,⁠${\displaystyle \alpha =1/{\phi }^3}$ ⁠, an irrational number, where⁠${\displaystyle \phi =\frac{1}{2}(1+\sqrt{5})}$ ⁠ is thegolden ratio.^[41]

• Weisstein, Eric W.,"Kite",MathWorld
• area formulae with interactive animation at Mathopenref.com