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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)]},
\end{displaymath} (1)

s\equiv {\textstyle{1\over 2}}(a+b+c+d)
\end{displaymath} (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
$\displaystyle K$ $\textstyle =$ $\displaystyle \sqrt{(s-a)(s-b)(s-c)(s-d)}$ (3)
  $\textstyle =$ $\displaystyle {\sqrt{(bc+ad)(ac+bd)(ab+cd)}\over 4R},$ (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
\end{displaymath} (5)

See also Bretschneider's Formula, Heron's Formula


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.

© 1996-9 Eric W. Weisstein