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Brahmagupta's Formula

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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 {a}, {b}, {c}, {d}, the area {K} can be found as:

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

where s=\frac{a+b+c+d}{2} is the semiperimeter of the quadrilateral.


Proof

If we draw AC, we find that [ABCD]=\frac{ab\sin B}{2}+\frac{cd\sin D}{2}=\frac{ab\sin B+cd\sin D}{2}. Since B+D=180^\circ, \sin B=\sin D. Hence, [ABCD]=\frac{\sin B(ab+cd)}{2}. Multiplying by 2 and squaring, we get: 4[ABCD]}^2=\sin^2 B(ab+cd)^2 Substituting \sin^2B=1-\cos^2B results in 4[ABCD]^2=(1-\cos^2B)(ab+cd)^2=(ab+cd)^2-\cos^2B(ab+cd)^2 By the Law of Cosines, a^2+b^2-2ab\cos B=c^2+d^2-2cd\cos D. \cos B=-\cos D, so a little rearranging gives 2\cos B(ab+cd)=a^2+b^2-c^2-d^2 4[ABCD]^2=(ab+cd)^2-\frac{1}{4}(a^2+b^2-c^2-d^2)^2 16[ABCD]^2=4(ab+cd)^2-(a^2+b^2-c^2-d^2)^2 16[ABCD]^2=(2(ab+cd)+(a^2+b^2-c^2-d^2))(2(ab+cd)-(a^2+b^2-c^2-d^2)) 16[ABCD]^2=(a^2+2ab+b^2-c^2+2cd-d^2)(-a^2+2ab-b^2+c^2+2cd+d^2) 16[ABCD]^2=((a+b)^2-(c-d)^2)((c+d)^2-(a-b)^2) 16[ABCD]^2=(a+b+c-d)(a+b-c+d)(c+d+a-b)(c+d-b+a) 16[ABCD]^2=16(s-a)(s-b)(s-c)(s-d) [ABCD]=\sqrt{(s-a)(s-b)(s-c)(s-d)}

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 {d}=0.

A similar formula which Brahmagupta derived for the area of a general quadrilateral is [ABCD]^2=(s-a)(s-b)(s-c)(s-d)-abcd\cos{\frac{B+D}{2}} [ABCD]=\sqrt{(s-a)(s-b)(s-c)(s-d)-abcd\cos{\frac{B+D}{2}}} where s=\frac{a+b+c+d}{2} is the semiperimeter of the quadrilateral. What happens when the quadrilateral is cyclic? This article is a stub. Help us out by expanding it.

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Looking for a challenging geometry text? Preparing for MATHCOUNTS or the AMC exams? Check out Art of Problem Solving's Introduction to Geometry by Richard Rusczyk.
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