, Brahmagupta's formula
finds the area
of any quadrilateral
given the lengths of the sides and some of their angles. In its most common form, it yields the area of quadrilaterals that can be inscribed in a circle
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
This formula generalizes Heron's formula for the area of a triangle.
The area of a cyclic quadrilateral is the maximum possible area for any quadrilateral with the given side lengths.
Proof of Brahmagupta's formula
Area of the cyclic quadrilateral = Area of + Area of
But since is a cyclic quadrilateral, Hence Therefore
Applying law of cosines for and and equating the expressions for side we have
Substituting (since angles and are supplementary) and rearranging, we have
Substituting this in the equation for area,
which is of the form and hence can be written in the form as
Taking square root, we get
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.)
This more general formula is sometimes known as Bretschneider's formula, but according to MathWorld is apparently due to Coolidge in this form, Bretschneider's expression having been
where p and q are the lengths of the diagonals of the quadrilateral.
It is a property of cyclic quadrilaterals (and ultimately of inscribed angles) that opposite angles of a quadrilateral sum to 180°. Consequently, in the case of an inscribed quadrilateral, θ = 90°, whence the term
giving the basic form of Brahmagupta's formula.
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.