Editing Kite (geometry) (section)

== 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}}