Brahmagupta's Formula

Theorem
The area of a cyclic quadrilateral with sides of lengths $a, b, c, d$ is:


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

where $s$ is the semiperimeter:


 * $s = \dfrac{a + b + c + d} 2$

Equivalently, the area of the cyclic quadrilateral is equal to:


 * $\dfrac{\sqrt{\left({a^2 + b^2 + c^2 + d^2}\right)^2 + 8 a b c d - 2 \left({a^4 + b^4 + c^4 + d^4}\right)}} 4$

which can be seen by making the substitutions:

Proof
Let $ABCD$ be a cyclic quadrilateral with sides $a, b, c, d$.


 * BrahmaguptasFormula.png

Area of $ABCD$ = Area of $\triangle ABC$ + Area of $\triangle ADC$

From Area of Triangle in Terms of Two Sides and Angle:

From Opposite Angles of Cyclic Quadrilateral, $\angle ABC + \angle ADC$ equals two right angles, that is, are supplementary.

Hence we have:

This leads to:

Applying the Law of Cosines for $\triangle ABC$ and $\triangle ADC$ and equating the expressions for side $AC$:


 * $a^2 + b^2 - 2a b \cos \angle ABC = c^2 + d^2 - 2cd \cos \angle ADC$

From the above:
 * $\cos \angle ABC = -\cos \angle ADC$

Hence:
 * $2 \cos \angle ABC \left({ab + cd}\right) = a^2 + b^2 - c^2 - d^2$

Substituting this in the above equation for the area:

This is of the form $x^2 - y^2$.

Hence, by Difference of Two Squares, it can be written in the form $(x+y)(x-y)$ as:

When we introduce the expression for the semiperimeter:
 * $\displaystyle s = \frac {a + b + c + d} 2$

the above converts to:
 * $16 \left({\text {Area}}\right)^2 = 16 \left({s - a}\right) \left({s - b}\right) \left({s - c}\right) \left({s - d}\right)$

Taking the square root:


 * $\text {Area} = \sqrt{\left({s - a}\right) \left({s - b}\right) \left({s - c}\right) \left({s - d}\right)}$

Also see

 * This formula is a generalization of Heron's Formula for the area of a triangle, which can be obtained from this by setting $d = 0$.


 * The relationship between the general and extended form of Brahmagupta's formula' is similar to how the Law of Cosines extends Pythagoras's Theorem.


 * Bretschneider's Formula, which extends this result to the general quadrilateral.


 * Area of Quadrilateral with Given Sides is Greatest when Quadrilateral is Cyclic