Right kite

A special case of right kites are squares, where the diagonals have equal lengths, and the incircle and circumcircle are concentric.

A kite is a right kite if 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 kite ABCD where the opposite angles B and D are right angles, the other two angles can be calculated from

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

where a = AB = AD and b = BC = CD. The area of a right kite is

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

The diagonal AC that is a line of symmetry has the length

${\displaystyle p={\sqrt {a^{2}+b^{2}}}}$

and, since the diagonals are perpendicular (so a right kite is an orthodiagonal quadrilateral with area ${\displaystyle K={\frac {pq}{2}}}$), the other diagonal BD has the length

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

The radius of the circumcircle is (according to the Pythagorean theorem)

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

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

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

where s is the semiperimeter.

The area is given in terms of the circumradius R and the inradius r 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 the geometric mean theorem.

The dual polygon to a right kite is an isosceles 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.