Area
The area K of a cyclic quadrilateral with sides a, b, c, d is given by Brahmagupta's formula
where s, the semiperimeter, is . It is a corollary to Bretschneider's formula since opposite angles are supplementary. If also d = 0, the cyclic quadrilateral becomes a triangle and the formula is reduced to Heron's formula.
The cyclic quadrilateral has maximal area among all quadrilaterals having the same sequence of side lengths. This is another corollary to Bretschneider's formula. It can also be proved using calculus.
Four unequal lengths, each less than then sum of the other three, are the sides of each of three non-congruent cyclic quadrilaterals, which by Brahmagupta's formula all have the same area. Specifically, for sides a, b, c, and d, side a could be opposite any of side b, side c, or side d.
The area of a cyclic quadrilateral with successive sides a, b, c, d and angle B between sides a and b can be expressed as
or
where θ is the angle between the diagonals. Provided A is not a right angle, the area can also be expressed as
Another formula is
where R is the radius in the circumcircle. As a direct consequence,
where there is equality if and only if the quadrilateral is a square.
Read more about this topic: Cyclic Quadrilateral
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