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Kite (geometry)

 

 
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Kite (geometry)
 
 
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In geometry
Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
 a kite, or deltoid, is a quadrilateral
Quadrilateral

In geometry, a quadrilateral is a polygon with four 'sides' or edges and four vertices or corners. Sometimes, the term quadrangle is used, for analogy with triangle, and sometimes tetragon for consistency with pentagon , hexagon and so on....
 with two disjoint
Disjoint sets

In mathematics, two Set are said to be disjoint if they have no element in common. For example, and are disjoint sets....
 pairs of congruent adjacent
Adjacent

Adjacent is an adjective meaning contiguous, adjoining or abutting.In geometry, adjacent is when sides meet to make an angle....
 sides, in contrast to a parallelogram
Parallelogram

In geometry, a parallelogram is a quadrilateral with two sets of parallel sides. The opposite or facing sides of a parallelogram are of equal length, and the opposite angles of a parallelogram are of equal size....
, where the congruent sides are opposite
Opposite

Opposite may refer to:* Antonym, a word that means the opposite of a word* a kind of Leaf#Arrangement on the stem* Additive inverse, in mathematics, taking the negative of a number...
. The geometric object is named for the wind-blown, flying kite
Kite

A kite is a flying tethered aircraft that depends upon the tension of a tethering system. The necessary Lift that makes the kite wing fly is generated when air flows over and under the kite's wing, producing low pressure above the wing and high pressure below it....
 (itself named for a bird
Kite (bird)

Kites are Bird of preys with long wings and weak legs which spend a great deal of time soaring. Most feed mostly on carrion but some take various amounts of live prey....
), which in its simple form often has this shape.

Equivalently, a kite is a quadrilateral with an axis of symmetry along one of its diagonal
Diagonal

A diagonal can refer to a line joining two nonconsecutive vertices of a polygon or polyhedron, or in informal contexts any upward or downward sloping line....
s.






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Geometrickite
In geometry
Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
 a kite, or deltoid, is a quadrilateral
Quadrilateral

In geometry, a quadrilateral is a polygon with four 'sides' or edges and four vertices or corners. Sometimes, the term quadrangle is used, for analogy with triangle, and sometimes tetragon for consistency with pentagon , hexagon and so on....
 with two disjoint
Disjoint sets

In mathematics, two Set are said to be disjoint if they have no element in common. For example, and are disjoint sets....
 pairs of congruent adjacent
Adjacent

Adjacent is an adjective meaning contiguous, adjoining or abutting.In geometry, adjacent is when sides meet to make an angle....
 sides, in contrast to a parallelogram
Parallelogram

In geometry, a parallelogram is a quadrilateral with two sets of parallel sides. The opposite or facing sides of a parallelogram are of equal length, and the opposite angles of a parallelogram are of equal size....
, where the congruent sides are opposite
Opposite

Opposite may refer to:* Antonym, a word that means the opposite of a word* a kind of Leaf#Arrangement on the stem* Additive inverse, in mathematics, taking the negative of a number...
. The geometric object is named for the wind-blown, flying kite
Kite

A kite is a flying tethered aircraft that depends upon the tension of a tethering system. The necessary Lift that makes the kite wing fly is generated when air flows over and under the kite's wing, producing low pressure above the wing and high pressure below it....
 (itself named for a bird
Kite (bird)

Kites are Bird of preys with long wings and weak legs which spend a great deal of time soaring. Most feed mostly on carrion but some take various amounts of live prey....
), which in its simple form often has this shape.

Equivalently, a kite is a quadrilateral with an axis of symmetry along one of its diagonal
Diagonal

A diagonal can refer to a line joining two nonconsecutive vertices of a polygon or polyhedron, or in informal contexts any upward or downward sloping line....
s. A quadrilateral that has an axis of symmetry must be either a kite or an isosceles trapezoid
Isosceles trapezoid

An isosceles trapezoid is a quadrilateral with a line of symmetry bisecting one pair of opposite sides, making it automatically a trapezoid. Two opposite sides are Parallel , the two other sides are of equal length....
. Kites and isosceles trapezoids are dual: the polar figure
Dual polyhedron

In geometry, polyhedron are associated into pairs called duals, where the wikt:vertex of one correspond to the face s of the other. The dual of the dual is the original polyhedron....
 of a kite is an isosceles trapezoid, and vice versa.

A kite may be either convex
Convex polygon

In geometry, a polygon can be either convex or concave....
 or concave; a concave kite is sometimes called a "dart", and is a type of pseudotriangle
Pseudotriangle

In Euclidean geometry, a pseudotriangle is the simply connected subset of the plane that lies between any three mutually tangent convex sets. A pseudotriangulation is a partition of a region of the plane into pseudotriangles, and a pointed pseudotriangulation is a pseudotriangulation of a convex polygon in which at each vertex th...
.

Properties

  • The two diagonals of a kite are perpendicular
    Perpendicular

    In geometry, two line or plane , are considered perpendicular to each other if they form congruence adjacent angles angles . The term may be used as a noun or adjective....
    .
  • Two interior angles at opposite vertices of a kite are equal.
  • 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 two unequal sides, and ? is the angle
    Angle

    In geometry and trigonometry, an angle is the figure formed by two Ray sharing a common endpoint, called the vertex of the angle . The magnitude of the angle is the "amount of rotation" that separates the two rays, and can be measured by considering the length of circular arc swept out when one ray is rotated about the vertex to coincide...
     between unequal sides, then the area is ab sin ?.
  • One diagonal
    Diagonal

    A diagonal can refer to a line joining two nonconsecutive vertices of a polygon or polyhedron, or in informal contexts any upward or downward sloping line....
     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
    Tangent

    In geometry, the tangent line to a curve at a given Point is the straight line that "just touches" the curve at that point . As it passes through the point of tangency, the tangent line is "going in the same direction" as the curve, and in this sense it is the best straight-line approximation to the curve at that point....
     to all four sides. Therefore, every convex kite is a tangential quadrilateral
    Tangential quadrilateral

    In geometry, a tangential quadrilateral is a convex polygon quadrilateral whose sides all lie tangent to a single circle inscribed within the quadrilateral....
    . Additionally, if a convex kite is not a rhombus, there is another circle, outside the kite, tangent to all four sides, suitably extended. For every concave kite there exist two circles tangent to all four (possibly extended) 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.


Special cases


  • A kite is a cyclic quadrilateral
    Cyclic quadrilateral

    In geometry, a cyclic quadrilateral is a quadrilateral whose vertex all lie on a single circle. The vertices are said to be concyclic.In a cyclic quadrilateral, opposite angles are supplementary angle ....
    , that is, can be inscribed in a circle, if and only if it is formed from two congruent right triangles.
  • If all four sides of a kite are the same length (that is, if the kite is equilateral
    Equilateral

    In geometry, an equilateral polygon is a polygon which has all sides of the same length.For instance, an equilateral triangle is a triangle of equal edge lengths....
    ), it is a rhombus
    Rhombus

    In geometry, a rhombus , or rhomb is an equilateral polygon parallelogram. In other words, it is a four-sided polygon in which every side has the same length....
    .
  • If a kite is equiangular
    Equiangular polygon

    File:Rectangle definition.svgIn Euclidean geometry, an equiangular polygon is a polygon whose vertex angles are equal. If the lengths of the sides are also equal then it is a regular polygon....
    , it must also be equilateral and thus a square
    Square (geometry)

    In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted ....
    .
  • Kites and darts in which the two isosceles triangles forming the kite have apex angles of 2p/5 and 4p/5 represent one of two sets of essential aperiodic tiles isolated by mathematical physicist Roger Penrose
    Roger Penrose

    Sir Roger Penrose, Order of Merit , Royal Society 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....
    .


  • The quadrilateral maximizing the ratio of its perimeter
    Perimeter

    A perimeter is a path that bounds an area. The word comes from the Greek peri and meter . The term may be used either for the path or its length....
     to its width is a kite with angles p/3, 5p/12, 5p/6, 5p/12.


  • All kites tile the plane
    Tessellation

    A tessellation or tiling of the plane is a collection of plane figures that fills the plane with no overlaps and no gaps. One may also speak of tessellations of the parts of the plane or of other surfaces....
     by repeated inversion around the midpoints of their edges, as do more generally all quadrilaterals. A kite with angles p/3, p/2, 2p/3, p/2 can also tile the plane by repeated reflection across its edges; the resulting tessellation, the deltoidal trihexagonal tiling
    Deltoidal trihexagonal tiling

    In geometry, the Deltoidal trihexagonal tiling is a dual of the semiregular tiling Small rhombitrihexagonal tiling. Its faces are deltoids or Kite ....
    , superposes a tessellation of the plane by regular hexagons and isosceles triangles.


  • The deltoidal icositetrahedron
    Deltoidal icositetrahedron

    A deltoidal icositetrahedron is a Catalan solid which looks a bit like an overinflated cube . Its dual polyhedron is the rhombicuboctahedron....
    , deltoidal hexecontahedron
    Deltoidal hexecontahedron

    A deltoidal hexecontahedron is a catalan solid which looks a bit like an overinflated dodecahedron. It is sometimes also called the trapezoidal hexecontahedron or strombic hexecontahedron....
    , and trapezohedra
    Trapezohedron

    The n-gonal trapezohedron, antidipyramid or deltohedron is the dual polyhedron of an n-gonal antiprism. Its 2n faces are congruent kite ....
     are polyhedra with congruent kite-shaped facets
    Facet (mathematics)

    A facet of a simplicial complex is a maximal simplex.In the general theory of polyhedra and polytopes, two conflicting meanings are currently jostling for acceptability:...
    .


Deltoidalicositetrahedron

Deltoidal icositetrahedron
Deltoidal icositetrahedron

A deltoidal icositetrahedron is a Catalan solid which looks a bit like an overinflated cube . Its dual polyhedron is the rhombicuboctahedron....
Deltoidalhexecontahedron

Deltoidal hexecontahedron
Deltoidal hexecontahedron

A deltoidal hexecontahedron is a catalan solid which looks a bit like an overinflated dodecahedron. It is sometimes also called the trapezoidal hexecontahedron or strombic hexecontahedron....

Deltoidal trihexagonal tiling
Deltoidal trihexagonal tiling

In geometry, the Deltoidal trihexagonal tiling is a dual of the semiregular tiling Small rhombitrihexagonal tiling. Its faces are deltoids or Kite ....

Deltoidal triheptagonal tiling
Uniform tilings in hyperbolic plane

There are an infinite number of uniform tilings on the hyperbolic plane based on the where p + q + r > 9 List_of_regular_polytopes#Hyperbolic_tilings....


External links

  • and with interactive animations.