# Difference between revisions of "Brahmagupta's Formula"

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A similar formula which Brahmagupta derived for the area of a general quadrilateral is | A similar formula which Brahmagupta derived for the area of a general quadrilateral is | ||

<cmath>[ABCD]^2=(s-a)(s-b)(s-c)(s-d)-abcd\cos^2\left({\frac{B+D}{2}}\right)</cmath> | <cmath>[ABCD]^2=(s-a)(s-b)(s-c)(s-d)-abcd\cos^2\left({\frac{B+D}{2}}\right)</cmath> | ||

− | <cmath>[ABCD]=\sqrt{(s-a)(s-b)(s-c)(s-d)-abcd\cos^2\left({\frac{B+D}{2 | + | <cmath>[ABCD]=\sqrt{(s-a)(s-b)(s-c)(s-d)-abcd\cos^2\left({\frac{B+D}{2}}\right)}</cmath> |

where <math>s=\frac{a+b+c+d}{2}</math> is the [[semiperimeter]] of the quadrilateral. What happens when the quadrilateral is cyclic? | where <math>s=\frac{a+b+c+d}{2}</math> is the [[semiperimeter]] of the quadrilateral. What happens when the quadrilateral is cyclic? | ||

## Revision as of 07:03, 5 July 2015

**Brahmagupta's Formula** is a formula for determining the area of a cyclic quadrilateral given only the four side lengths.

## Definition

Given a cyclic quadrilateral with side lengths , , , , the area can be found as:

where is the semiperimeter of the quadrilateral.

### Proof

If we draw , we find that . Since , . Hence, . Multiplying by 2 and squaring, we get: Substituting results in By the Law of Cosines, . , so a little rearranging gives

## Similar formulas

Bretschneider's formula gives a formula for the area of a non-cyclic quadrilateral given only the side lengths; applying Ptolemy's Theorem to Bretschneider's formula reduces it to Brahmagupta's formula.

Brahmagupta's formula reduces to Heron's formula by setting the side length .

A similar formula which Brahmagupta derived for the area of a general quadrilateral is where is the semiperimeter of the quadrilateral. What happens when the quadrilateral is cyclic?

## Problems

### Intermediate

- is a cyclic quadrilateral that has an inscribed circle. The diagonals of intersect at . If and then the area of the inscribed circle of can be expressed as , where and are relatively prime positive integers. Determine . (Source)