Kite (geometry)

Did You Know?

• Mathematician Richard Schwartz solved a long-standing problem by finding unbounded outer billiards paths for the kite with angles 72°, 72°, 72°, 144°, the same as used in the Penrose tiling.
• The behavior of outer billiards on any kite depends strongly on the parameter a and in particular whether it is rational.

Definition and classification

A kite is defined as a quadrilateral with reflection symmetry across one of its diagonals, or equivalently as a quadrilateral whose four sides can be grouped into two pairs of adjacent equal-length sides. It can be constructed from the centers and crossing points of any two intersecting circles. Kites may be convex or concave, though some sources restrict the term to convex ones. A quadrilateral is a kite if and only if any of these conditions holds: the four sides split into two pairs of adjacent equal-length sides; one diagonal crosses the midpoint of the other at a right angle, forming its perpendicular bisector (in the concave case, the line through one diagonal bisects the other); one diagonal is a line of symmetry dividing the quadrilateral into two congruent mirror-image triangles; or one diagonal bisects both angles at its ends. The name derives from wind-blown flying kites, themselves named for a hovering bird and its sound; Olaus Henrici attributed the geometric term to James Joseph Sylvester. Quadrilaterals can be classified hierarchically (where kites include rhombi, squares, and Apollonius quadrilaterals) or partitionally (where rhombi and squares are not kites). Under hierarchical classification, all equilateral kites are rhombi, all equiangular kites are squares, and right kites are considered kites, simplifying certain facts.

Special cases

Right kites have two opposite right angles and are exactly the kites that are cyclic quadrilaterals, meaning a circle passes through all vertices; they are also bicentric (and tricentric) quadrilaterals. Among all quadrilaterals, the shape with the greatest ratio of perimeter to diameter is an equidiagonal kite with angles 60°, 75°, 150°, 75°, whose vertices lie at three corners and one side midpoint of a Reuleaux triangle. A kite with three 108° angles and one 36° angle forms the convex hull of the lute of Pythagoras, a fractal made of nested pentagrams, with its sides lying on four sides of a regular pentagon and a golden triangle glued onto the fifth side. Only eight polygons can tile the plane in an edge tessellation; one is a tiling by a right kite with 60°, 90°, and 120° angles, producing the deltoidal trihexagonal tiling. A prototile made of eight such kites tiles the plane only aperiodically, key to a claimed solution of the einstein problem. In non-Euclidean geometry, a kite can have three right angles and one non-right angle, forming a special Lambert quadrilateral; the fourth angle is acute in hyperbolic geometry and obtuse in spherical geometry.

Properties

Every kite is orthodiagonal, with diagonals at right angles; the symmetry axis is the perpendicular bisector of the other diagonal and bisects the two angles it meets. The other two angles of a kite are equal. In a convex kite, the symmetry axis divides it into two congruent triangles, while the other diagonal divides it into two isosceles triangles. The area of a kite can be calculated as half the product of the diagonal lengths, or by dividing it into two congruent triangles using the SAS formula. Every convex kite is a tangential quadrilateral with an inscribed circle; if it is not a rhombus, there is also an ex-tangential circle tangent to the extensions of the sides. For a convex kite with diagonals p and q and sides a and b, the inscribed circle radius r = (pq)/(2√(a² + b²)) and the ex-tangential circle radius R = (pq)/(2|a² - b²|). A tangential quadrilateral is a kite if and only if its area equals half the product of the diagonals, the diagonals are perpendicular, the two line segments connecting opposite points of tangency have equal length, or the tangent lengths at two opposite vertices are equal.

Tilings and polyhedra

Kites can tile the plane through point reflection around the midpoints of their edges, a property shared by all quadrilaterals. In Penrose tilings, aperiodic tilings discovered by Roger Penrose, the prototiles are kites and darts with specific angle sets: 72°, 72°, 72°, 144° for the kite and 36°, 72°, 36°, 216° for the dart. A kite with angles 60°, 90°, 120°, 90° tiles the plane via edge reflections, forming the deltoidal trihexagonal tiling, which overlays regular hexagons and isosceles triangles. Polyhedra with congruent kite faces, such as the deltoidal icositetrahedron, deltoidal hexecontahedron, and trapezohedra, can be viewed as spherical tilings. Trapezohedra, dual to uniform antiprisms, have kite faces with two edge lengths; the pentagonal trapezohedron is a common ten-sided die. Infinitely many face-symmetric hyperbolic tilings by kites also exist.