# Brahmagupta's formula

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In geometry, **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.

## Basic form

In its basic and easiest-to-remember form, **Brahmagupta's formula** gives the area of a cyclic quadrilateral whose sides have lengths as

where , the semiperimeter, is

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.

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.

## Proof of Brahmagupta's theorem

Area of the cyclic quadrilateral is

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 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

Introducing

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 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 . Since , we have .

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 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 . 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 .

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 links

ar:معادلة براهماغوبتاit:Formula di Brahmaguptakm:រូបមន្តប្រាម៉ាហ្គឹបតាsr:Формула Брамагупте