Brahmagupta's formula gives the area A of a cyclic quadrilateral whose sides have lengths a, b, c, d as
where s, the semiperimeter, is defined to be
This formula generalizes Heron's formula for the area of a triangle. A triangle may be regarded as a quadrilateral with one side of length zero. From this perspective, as d approaches zero, a cyclic quadrilateral converges into a cyclic triangle (all triangles are cyclic), and Brahmagupta's formula simplifies to Heron's formula.
If the semiperimeter is not used, Brahmagupta's formula is
Another equivalent version is
Brahmagupta's formula may be seen as a formula in the half-lengths of the sides, but it also gives the area as a formula in the altitudes from the center to the sides, although if the quadrilateral does not contain the center, the altitude to the longest side must be taken as negative.
Here the notations in the figure to the right are used. Area K of the cyclic quadrilateral = Area of + Area of
But since is a cyclic quadrilateral, Hence Therefore,
Solving for common side DB, in ADB and BDC, the law of cosines gives
Substituting (since angles and are supplementary) and rearranging, we have
Substituting this in the equation for the area,
The right-hand side is of the form and hence can be written as
which, upon rearranging the terms in the square brackets, yields
Introducing the semiperimeter
Taking the square root, we get
An alternative, non-trigonometric proof utilizes two applications of Heron's triangle area formula on similar triangles.
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 choice of which pair of opposite angles 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 known as Bretschneider's formula.
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. It follows from the latter equation that the area of a cyclic quadrilateral is the maximum possible area for any quadrilateral with the given side lengths.
- 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.
- Hess, Albrecht, "A highway from Heron to Brahmagupta", Forum Geometricorum 12 (2012), 191–192.
- J. L. Coolidge, "A Historically Interesting Formula for the Area of a Quadrilateral", American Mathematical Monthly, 46 (1939) pp. 345-347.