Brahmagupta's formula
From Academic Kids
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
