A kite showing its equal sides and its inscribed circle.

In geometry a kite, or deltoid, is a quadrilateral with two disjoint pairs of congruent adjacent sides, in contrast to a parallelogram, where the congruent sides are opposite. The geometric object is named for the wind-blown, flying kite (itself named for a bird), which in its simple form often has this shape. For other uses, see Kite (disambiguation). ... Image File history File links GeometricKite. ... Image File history File links GeometricKite. ... For other uses, see Geometry (disambiguation). ... This article is about the geometric shape. ... In mathematics, two sets are said to be disjoint if they have no element in common. ... See also: congruence relation In geometry, two shapes are called congruent if one can be transformed into the other by a series of translations, rotations and reflections. ... Look up adjacent in Wiktionary, the free dictionary. ... A parallelogram. ... ... For other uses, see Kite (disambiguation). ... Genera Milvinae Harpagus Ictinia Rostrhamus Haliastur Milvus Lophoictinia Hamirostra Elaninae Elanus Chelictinia Machaerhamphus Gampsonyx Elanoides Kites are raptors with long wings and weak legs which spend a great deal of time soaring. ...

Equivalently, a kite is a quadrilateral with an axis of symmetry along one of its diagonals. A quadrilateral that has an axis of symmetry must be either a kite or an isosceles trapezoid. Kites and isosceles trapezoids are dual: the polar figure of a kite is an isosceles trapezoid, and vice versa.^[1] The axis of symmetry of a two-dimensional figure is a line such that, if a perpendicular is constructed, any two points lying on the perpendicular at equal distances from the axis of symmetry are identical. ... A diagonal can refer to a line joining two nonadjacent vertices of a polygon or polyhedron, or in contexts any upward or downward sloping line. ... An isosceles trapezoid and its axis of symmetry. ... In geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. ...

A kite may be either convex or concave; a concave kite is sometimes called a "dart", and is a type of pseudotriangle. A convex pentagon In geometry, a convex polygon is a simple polygon whose interior is a convex set. ... In geometry, concavity is a property of certain geometric figures, and in calculus, a property of certain graphs of functions. ...

Properties

• The two diagonals of a kite are perpendicular.
• Two opposite vertices of a kite are equal. However, not every quadrilateral in which two opposite vertices have equal angles is a kite.
• The area of a kite is half the product of the lengths of its diagonals: A = d₁d₂/2. Alternatively, if a and b are the lengths of the sides, and θ the angle between unequal sides, then the area is ab sin θ.
• One diagonal divides a (convex) kite into two isosceles triangles; the other (the axis of symmetry) divides the kite into two congruent triangles.
• Every convex kite has an inscribed circle; that is, there exists a circle that is tangent to all four sides. Therefore, every convex kite is a tangential quadrilateral. Additionally, if a convex kite is not a rhombus, there is another circle, outside the kite, tangent to all four lines through the kite's sides. For every concave kite there exist two circles tangent to the lines through all four sides: one is interior to the kite and touches the two sides opposite from the concave angle, while the other circle is exterior to the kite and touches the kite on the two edges incident to the concave angle.^[2]
• A kite is a cyclic quadrilateral, that is, can be inscribed in a circle, if and only if it is formed from two congruent right triangles.^[3]

Fig. ... This article is about angles in geometry. ... A diagonal can refer to a line joining two nonadjacent vertices of a polygon or polyhedron, or in contexts any upward or downward sloping line. ... For alternate meanings, such as the musical instrument, see triangle (disambiguation). ... An example of congruence. ... In Euclidean geometry, a circle is the set of all points at a fixed distance, called the radius, from a fixed point, called the centre (center). ... For other uses, see tangent (disambiguation). ... Tangential quadrilateral. ... In geometry, a cyclic quadrilateral is a quadrilateral whose vertices all lie on a single circle. ... For alternate meanings, such as the musical instrument, see triangle (disambiguation). ...

Special cases

• If all four sides of a kite are the same length (that is, if the kite is equilateral), it is a rhombus.
• If a kite is equiangular, it must also be equilateral and thus a square.
• Kites and darts in which the two isosceles triangles forming the kite have apex angles of 2π/5 and 4π/5 represent one of two sets of essential aperiodic tiles isolated by mathematical physicist Roger Penrose.

• The quadrilateral maximizing the ratio of its perimeter to its width is a kite with angles π/3, 5π/12, 5π/6, 5π/12.^[4]

• All kites tile the plane by repeated inversion around the midpoints of their edges, as do more generally all quadrilaterals. A kite with angles π/3, π/2, 2π/3, π/2 can also tile the plane by repeated reflection across its edges; the resulting tesselation superposes a tesselation of the plane by regular hexagons and isosceles triangles.^[5]

• The deltoidal icositetrahedron, deltoidal hexecontahedron, and trapezohedra are polyhedra with congruent kite-shaped facets.

In geometry, an equilateral polygon has all sides of the same length. ... For other uses, see Rhombus (disambiguation). ... Look up Polygon on Wiktionary, the free dictionary For other use please see Polygon (disambiguation) A polygon (literally many angle, see Wiktionary for the etymology) is a closed planar path composed of a finite number of sequential line segments. ... For other uses, see Square. ... A Penrose tiling is pattern of tiles, discovered by Roger Penrose, which could completely cover an infinite surface, but only in a pattern which is non-repeating (aperiodic). ... Sir Roger Penrose, OM, FRS (born 8 August 1931) is an English mathematical physicist and Emeritus Rouse Ball Professor of Mathematics at the Mathematical Institute, University of Oxford and Emeritus Fellow of Wadham College. ... This article is about the distance around an object. ... In general English usage, length (symbol: l) is but one particular instance of distance – an objects length is how long the object is – but in the physical sciences and engineering, the word length is in some contexts used synonymously with distance. Height is vertical distance; width (or breadth) is... A tessellated plane A tessellation of the plane is a collection of plane figures that fill the plane with no overlaps and no gaps. ... A deltoidal icositetrahedron (or trapezoidal icositetrahedron) is a catalan solid which looks a bit like an overinflated cube. ... A deltoidal hexecontahedron is a catalan solid which looks a bit like an overinflated dodecahedron. ... The trapezohedron is the dual polyhedron of the corresponding antiprism. ... In mathematics, there are three related meanings of the term polyhedron: in the traditional meaning it is a 3-dimensional polytope, and in a newer meaning that exists alongside the older one it is a bounded or unbounded generalization of a polytope of any dimension. ... A facet of an n-dimensional simplex is its (n-1)-dimensional face. ...

References

Eric W. Weisstein (born March 18, 1969, in Bloomington, Indiana) is an encyclopedist who created and maintains MathWorld and Eric Weissteins World of Science (ScienceWorld). ... MathWorld is an online mathematics reference work, sponsored by Wolfram Research Inc. ...

External links

• Eric W. Weisstein, Kite at MathWorld.
• Kite definition and area formulae with interactive animations.