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Brahmagupta's formula
In geometry, Brahmagupta's formula formula finds the area of any quadrilateral. In its most common form, it yields the area of quadrilaterals that can be inscribed in a circle.
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Basic form
In its basic and easiest-to-remember form, Brahmagupta's formula gives the area of a cyclic quadrilateral whose sides have lengths a, b, c, d as:
where s, the semiperimeter, is determined by
Extension to non-cyclic quadrilaterals
In the case of non-cyclic quadrilaterals, Brahmagupta's formula can be extended by considering the measures of two opposite angles of the quadrilateral:
where θ is half the sum of two opposite angles. (The pair is irrelevant: if the other two angles are taken, half their sum is the supplement of θ. Since cos(180 - θ) = - cosθ, we have cos2(180 - θ) = cos2θ.)
It is a property of cyclic quadrilaterals (and ultimately of inscribed angles) that opposite angles of a quadrilateral sum to 
. Consequently, in the case of an inscribed quadrilateral, 
, whence the term 
, giving the basic form of Brahmagupta's formula.
Related theorems
Heron's formula for the area of a triangle is the special case obtained by taking d=0.
The relationship between the general and extended form of Brahmagupta's formula is similar to how the law of cosines extends the Pythagorean theorem.
External link
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