Brahmagupta's formula

In geometry, Brahmagupta's formula formula finds the area of any quadrilateral. In its most common form, it yields the area of quadrilaterals that can be inscribed in a circle.

Contents

1 Basic form

2 Extension to non-cyclic quadrilaterals

3 Related theorems

4 External link

Basic form

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:

$\sqrt{(s-a)(s-b)(s-c)(s-d)}$

where s, the semiperimeter, is determined by

$s=\frac{a+b+c+d}{2}.$

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:

$\sqrt{(s-a)(s-b)(s-c)(s-d)-abcd\cos^2\theta}$

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 $cos 2 (180 - θ) = cos 2 θ$ .)

It is a property of cyclic quadrilaterals (and ultimately of inscribed angles) that opposite angles of a quadrilateral sum to $180^\circ$ . Consequently, in the case of an inscribed quadrilateral, $\theta=90^\circ$ , whence the term $abcd\cos^2\theta=abcd\cos^2 90=abcd\cdot0=0$ , 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 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.

External link

MathWorld: Brahmagupta's formula

Categories: Euclidean geometry | Theorems

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01-04-2007 01:21:04

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