Brahmagupta's formula
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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.
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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:
- <math>\sqrt{(s-a)(s-b)(s-c)(s-d)}<math>
where s, the semiperimeter, is determined by
- <math>s=\frac{a+b+c+d}{2}.<math>
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:
- <math>\sqrt{(s-a)(s-b)(s-c)(s-d)-abcd\cos^2\theta}<math>
where <math>\theta<math> 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 <math>\theta<math>. Since <math>\cos(180-\theta)=-\cos\theta<math>, we have <math>\cos^2(180-\theta)=\cos^2\theta<math>.)
It is a property of cyclic quadrilaterals (and ultimately of inscribed angles) that opposite angles of a quadrilateral sum to <math>180^\circ<math>. Consequently, in the case of an inscribed quadrilateral, <math>\theta=90^\circ<math>, whence the term <math>abcd\cos^2\theta=abcd\cos^2 90=abcd\cdot0=0<math>, 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 (http://mathworld.wolfram.com/BrahmaguptasFormula.html)it:Formula di Brahmagupta