# Brahmagupta's formula

In Euclidean geometry, **Brahmagupta's formula** finds the area of any cyclic quadrilateral (one that can be inscribed in a circle) given the lengths of the sides.

## Formula

Brahmagupta's formula gives the area *K* 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

## Proof

### Trigonometric proof

Here the notations in the figure to the right are used. The area *K* of the cyclic quadrilateral equals the sum of the areas of △*ADB* and △*BDC*:

But since *ABCD* is a cyclic quadrilateral, ∠*DAB* = 180° − ∠*DCB*. Hence sin *A* = sin *C*. Therefore,

Solving for common side *DB*, in △*ADB* and △*BDC*, the law of cosines gives

Substituting cos *C* = −cos *A* (since angles *A* and *C* are supplementary) and rearranging, we have

Substituting this in the equation for the area,

The right-hand side is of the form *a*^{2} − *b*^{2} = (*a* − *b*)(*a* + *b*) and hence can be written as

which, upon rearranging the terms in the square brackets, yields

Introducing the semiperimeter *S* = *p* + *q* + *r* + *s*/2,

Taking the square root, we get

### Non-trigonometric proof

An alternative, non-trigonometric proof utilizes two applications of Heron's triangle area formula on similar triangles.^{[1]}

## 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 any two opposite angles. (The choice of which pair of opposite angles is irrelevant: if the other two angles are taken, half their sum is 180° − *θ*. Since cos(180° − *θ*) = −cos *θ*, we have cos^{2}(180° − *θ*) = cos^{2} *θ*.) 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* is 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.

A related formula, which was proved by Coolidge, also gives the area of a general convex quadrilateral. It is^{[2]}

where *p* and *q* are the lengths of the diagonals of the quadrilateral. In a cyclic quadrilateral, *pq* = *ac* + *bd* according to Ptolemy's theorem, and the formula of Coolidge reduces to 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.
- Increasingly complicated closed-form formulas exist for the area of general polygons on circles, as described by Maley et al.
^{[3]}

## References

- ↑ 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. - ↑ Maley, F. Miller; Robbins, David P.; Roskies, Julie (2005). "On the areas of cyclic and semicyclic polygons" (PDF).
*Advances in Applied Mathematics*.**34**(4): 669–689. doi:10.1016/j.aam.2004.09.008.

## External links

*This article incorporates material from proof of Brahmagupta's formula on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.*