Kite (geometry)

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, but the word deltoid may also refer to a deltoid curve, an unrelated geometric object sometimes studied in connection with quadrilaterals. A kite may also be called a dart, particularly if it is not convex.

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 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 of two shapes (one convex and one non-convex) form the prototiles of one of the forms of the Penrose tiling. 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.

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. A kite can be constructed from the centers and crossing points of any two intersecting circles. Kites as described here may be either convex or 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:

Kite quadrilaterals are named for the wind-blown, flying kites, which often have this shape and which are in turn named for a hovering bird and the sound it makes. According to Olaus Henrici, the name "kite" was given to these shapes by James Joseph Sylvester.

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 rhombi (quadrilaterals with four equal sides), squares, and Apollonius quadrilaterals (in which the products of opposite sides are equal). All equilateral kites are rhombi, and all 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 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.

Like kites, a parallelogram also has two pairs of equal-length sides, but they are opposite to each other rather than adjacent. Any 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. The self-crossing quadrilaterals include another class of symmetric quadrilaterals, the antiparallelograms.

The right kites have two opposite right angles. The right kites are exactly the kites that are cyclic quadrilaterals, meaning that there is a circle that passes through all their vertices. The cyclic quadrilaterals may equivalently defined as the quadrilaterals in which two opposite angles are supplementary (they add to 180°); if one pair is supplementary the other is as well. 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 quadrilaterals (actually tricentric, as they also have a third circle externally tangent to the extensions of their sides). 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.