A Look at Brahmagupta's Formula
by Amy Benson
We begin this investigation by recalling Heron's formula for finding the area of a triangle:
where s is the semiperimeter of the triangle and a, b, and c are the lengths of the sides of the triangle as shown.
Can Heron's formula be applied more generally to a quadrilateral so that
where s is the semiperimeter of the quadrilateral? We know that if d = 0, then the formula is indeed correct. Alas, this conjecture is not always true. The conjecture is true in the instance when the quadrilateral is inscribed in a circle and this is Brahmagupta's formula.
So, let us examine the situation visually.
First we insert the diagonal e of the quadrilateral. We know that m<G + m<H = 180 degrees because they are inscribed angles that intercept a major and minor arc that together measure 360 degrees. From this, we know that cos G = - cos H and sin G = sin H. Using the law of cosines,
Substituting cosH for - cos G,
and this leads to
The area A of a quadrilateral is given by
Taking the two starred equations, adding and squaring them, we see that
At this point, we will focus on factoring the left side of this equation using
Finally, let x = a+b and y = c- d. So,
Recall that s = 0.5 (a + b + c + d), so 2s = a + b + c + d.
proving Brahmagupta's formula for the area of a quadrilateral inscribed in a circle.
Reference: Precalculus Mathematics in a Nutshell: Geometry, Algebra, Trignometry by George F. Simmons. Jansen Publications, Inc. 1987.
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