Brahmagupta's Formula

For a Quadrilateral with sides of length $a$, $b$, $c$, and $d$, the Area $K$ is given by

$$K=\sqrt{(s-a)(s-b)(s-c)(s-d)-abcd\cos^2[{\textstyle{1\over 2}}(A+B)]},$$ (1)

where

$$s\equiv {\textstyle{1\over 2}}(a+b+c+d)$$ (2)

is the Semiperimeter, $A$ is the Angle between $a$ and $d$, and $B$ is the Angle between $b$ and $c$. For a Cyclic Quadrilateral (i.e., a Quadrilateral inscribed in a Circle), $A+B=\pi$, so

$$K = \sqrt{(s-a)(s-b)(s-c)(s-d)}$$ (3)

$$= {\sqrt{(bc+ad)(ac+bd)(ab+cd)}\over 4R},$$ (4)

where $R$ is the Radius of the Circumcircle. If the Quadrilateral is Inscribed in one Circle and Circumscribed on another, then the Area Formula simplifies to

$$K=\sqrt{abcd}.$$ (5)

See also Bretschneider's Formula, Heron's Formula

References

Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 56-60, 1967.

Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 81-82, 1929.