InEuclidean geometry, aright kite is akite (aquadrilateral whose four sides can be grouped into two pairs of equal-length sides that are adjacent to each other) that can be inscribed in a circle.^[1] That is, it is a kite with acircumcircle (i.e., acyclic kite). Thus the right kite is aconvex quadrilateral and has two oppositeright angles.^[2] If there are exactly two right angles, each must be between sides of different lengths. All right kites arebicentric quadrilaterals (quadrilaterals with both a circumcircle and an incircle), since all kites have anincircle. One of the diagonals (the one that is a line ofsymmetry) divides the right kite into tworight triangles and is also adiameter of the circumcircle. All right kites areharmonic quadrilaterals since they have a circumcircle and each pair of opposite sides has the same two lengths.

In atangential quadrilateral (one with an incircle), the four line segments between the center of the incircle and the points where it is tangent to the quadrilateral partition the quadrilateral into four right kites.

A special case of right kites aresquares, where the diagonals have equal lengths, and the incircle and circumcircle areconcentric.

A kite is a right kiteif and only if it has a circumcircle (by definition). This is equivalent to its being a kite with two opposite right angles.

Since a right kite can be divided into two right triangles, the following metric formulas easily follow from well known properties of right triangles. In a right kiteABCD where the opposite anglesB andD are right angles, the other two angles can be calculated from

${\displaystyle \tan \frac{A}{2}=\frac{b}{a},\tan \frac{C}{2}=\frac{a}{b}}$ 

wherea =AB =AD andb =BC =CD. Thearea of a right kite is

${\displaystyle {\displaystyle K=ab.}}$ 

ThediagonalAC that is a line of symmetry has the length

${\displaystyle p=\sqrt{a^2+b^2}}$ 

and, since the diagonals areperpendicular (so a right kite is anorthodiagonal quadrilateral with area${\displaystyle K=\frac{pq}{2}}$ ), the other diagonalBD has the length

${\displaystyle q=\frac{2ab}{\sqrt{a^2+b^2}}.}$ 

Theradius of the circumcircle is (according to thePythagorean theorem)

${\displaystyle R=\frac{1}{2}\sqrt{a^2+b^2}}$ 

and, since all kites aretangential quadrilaterals, the radius of the incircle is given by

${\displaystyle r=\frac{K}{s}=\frac{ab}{a+b}}$ 

wheres is the semiperimeter.

The area is given in terms of the circumradiusR and the inradiusr as^[3]

${\displaystyle K=r(r+\sqrt{4R^2+r^2}).}$ 

If we take the segments extending from the intersection of the diagonals to the vertices in clockwise order to be${\displaystyle d_1}$ ,${\displaystyle d_2}$ ,${\displaystyle d_3}$ , and${\displaystyle d_4}$ , then,

${\displaystyle d_1{d}_3=d_2{d}_4}$ 

This is a direct result of thegeometric mean theorem.

Thedual polygon to a right kite is anisosceles tangential trapezoid.^[1]

Sometimes a right kite is defined as a kite with at least one right angle.^[4] If there is only one right angle, it must be between two sides of equal length; in this case, the formulas given above do not apply.