Kite (geometry)

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Kite (geometry)

A kite is a four-sided figure (quadrilateral) that has a line of symmetry along one of its diagonals. Its four sides come in two pairs of equal-length, adjacent sides. This symmetry makes two of its angles equal.

Key properties
- The diagonals are perpendicular (they cross at right angles).
- One diagonal is the axis of symmetry and bisects the angles it touches; the other diagonal is bisected by that axis.
- A convex kite always has an inscribed circle (it is tangential). If it’s not a rhombus, there is also a circle tangent to the extensions of its sides (an excircle).
- A concave kite is sometimes called a dart.
- If a kite has two opposite right angles, it is a right kite. Right kites are cyclic (they fit on a circle) and have special circle properties.

Area
- If the diagonals have lengths p and q, the area is A = p q / 2.
- If two adjacent sides are a and b with included angle θ, the area is A = a b sin(θ).

Special relationships
- Kites are dual to isosceles trapezoids: the kite’s incircle touches the sides at the four vertices of an isosceles trapezoid, and vice versa.

Tiling and shapes
- Kites can tile the plane by reflecting them across their edges.
- A famous Penrose tiling uses kites with angles 72°, 72°, 72°, 144° (and a related 36° kite).
- Kites also appear as faces of certain polyhedra and in various tilings of the plane and other geometries.

Other notes
- A kite can be constructed from the centers and intersection points of two intersecting circles.
- In hyperbolic and spherical geometry, kites can have interesting angle configurations, such as three right angles in some cases.

Origin and naming
- The name comes from the shape of flying kites; the term is linked to Sylvester in the history of geometry.