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.

They include as special cases the right kites, with two opposite right angles; the rhombi, with two diagonal axes of symmetry; and the squares, 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 also form the faces of several face-symmetric polyhedra and tessellations, and have been studied in connection with outer billiards, a problem in the advanced mathematics of dynamical systems.

[1][7] A kite can be constructed from the centers and crossing points of any two intersecting circles.

[12][13] According to Olaus Henrici, the name "kite" was given to these shapes by James Joseph Sylvester.

By avoiding the need to consider special cases, this classification can simplify some facts about kites.

[15] Like kites, a parallelogram also has two pairs of equal-length sides, but they are opposite to each other rather than adjacent.

[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 its perimeter to its diameter (maximum distance between any two points) is an 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.

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 greatest ratio of area to diameter.

[24] A prototile made by eight of these kites tiles the plane only aperiodically, key to a claimed solution of the einstein problem.

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.

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] The equidissection problem concerns the subdivision of polygons into triangles that all have equal areas.

[31][32] 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

[33] All kites tile the plane by repeated point 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 the prototiles of one version of the Penrose tiling, an aperiodic tiling of the plane discovered by mathematical physicist Roger Penrose.

⁠, then scaled copies of that kite can be used to tile the plane in a fractal rosette in which successively larger rings of

[36] 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.

[17] The deltoidal icositetrahedron, deltoidal hexecontahedron, and trapezohedron are polyhedra with congruent kite-shaped faces,[37] which can alternatively be thought of as tilings of the sphere by congruent spherical kites.

[17] Mathematician Richard Schwartz has studied outer billiards on kites.

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 a monograph analyzing outer billiards for kite shapes more generally.

The behavior of outer billiards on any kite depends strongly on the parameter