For a cyclic quadrilateral with given sides a, b, c, and d, the formula for the area is given by
Where s = (a + b + c + d)/2 known as the semi-perimeter.
Derivation of Formula
Cosine law for triangle ABC
Cosine law for triangle ADC
note that cos (180° - x) = -cos x
Equate the two x2
Area of ABCD
note that sin (180° - x) = sin x
Square both sides of the equation
Note that a + b + c + d = P, the perimeter. Thus,
Substitute 1 - cos2 θ to the equation of A2 above.
Recall that , the semi-perimeter. Thus,