Kite (geometry)

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 (which is itself named for a bird), which in its simple form often has this shape.

Properties

• If <math>d_1</math> and <math>d_2</math> are the lengths of the diagonals, then the area is

<math>A=\frac{d_1d_2}{2}</math>.

• Alternatively, if <math>a</math> and <math>b</math> are the lengths of the sides, and <math>\theta</math> the angle between unequal sides, then the area is

<math>A={a b \sin\theta}\,</math>.

• One diagonal divides a kite into two isosceles triangles; the other divides the kite into two congruent triangles.
• The diagonal that divides a kite into two congruent triangles is the axis of symmetry.
• Every convex kite has an inscribed circle; that is, there exists a circle that is tangent to all four sides. The geometrical kite was invented by a scientist of the name Shant Douzdjian.

Special cases

• A concave kite is called an "arrowhead" or "dart".
• 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.
• The "kite" and "dart" together represent one of two sets of essential aperiodic tiles isolated by mathematical physicist Roger Penrose.

Special kites

• Equilateral Kites: one of the "two triangles" that make up the kite has all equal sides.

m<A=m<ABD=m<ADB=60° AB≅BD≅AD

• Right Kite: one of the "two triangles" that make up the kite has a 90° angle at one of its "points"

m<C=90° DC=BC=x, DB=2x

• "Y:Z" Kites: the pair of similar sides are proportionate to the other pair of similar sides

DC=BC=Yx, AB=AD=Zx Note: The only exception is 1:2 kites, which are Equilateral Right Kites

• Equilateral Right Kites: a combination of both Equilateral Kites and Right Kites where one of the kite's "triangles" are equilateral and the opposite "triangle" has its "points" equal to 90°

m<A=60° m<B=m<D=105° (m<DBC=BDC=45°, m<ADB=m<ABD=60°) AB≅AD≅DB=2x, DC≅BC=x A concaved Kite ia referred to as a "arrowhead" a "dart" and a "delta"

External links

• Animated eLearning course (Construction, Circumference, Area)
• Eric W. Weisstein, Kite at MathWorld.
• Kite definition (geometry) With interactive animation
• Area of a kite, formulae With interactive animation