Brahmagupta's Formula
Brahmagupta's Formula is a formula for determining the area of a cyclic quadrilateral given only the four side lengths.
Definition
Given a cyclic quadrilateral with side lengths 
, 
, 
, 
, the area 
can be found as:
![\[K = \sqrt{(s-a)(s-b)(s-c)(s-d)}\]](http://latex.artofproblemsolving.com/1/a/3/1a3a0e62995e8ed2d8e6e8a119206ee174dfdb18.png)
where 
is the semiperimeter of the quadrilateral.
Proof
Similar formulas
Bretschneider's formula gives a formula for the area of a non-cyclic quadrilateral given only the side lengths; applying Ptolemy's Theorem to Bretschneider's formula reduces it to Brahmagupta's formula.
Brahmagupta's formula reduces to Heron's formula by setting the side length 
.
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