Brahmagupta's Formula

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


 * $\sqrt {\paren {s - a} \paren {s - b} \paren {s - c} \paren {s - d} }$

where $s$ is the semiperimeter:


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

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 sum to Two Right Angles, $\angle ABC + \angle ADC$ 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 $AC$:


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

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

Hence:
 * $2 \cos \angle ABC \paren {a b + c d} = a^2 + b^2 - c^2 - d^2$

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 $\paren {x + y} \paren {x - y}$ as:

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

the above converts to:
 * $16 \paren {\Area}^2 = 16 \paren {s - a} \paren {s - b} \paren {s - c} \paren {s - d}$

Taking the square root:


 * $\Area = \sqrt {\paren {s - a} \paren {s - b} \paren {s - c} \paren {s - d} }$

Also known as
Some sources refer to this as the Archimedes-Heron-Brahmagupta formula, for and  as well as.

The connection comes from the application of this to the triangle to obtain Heron's Formula.

The reference to comes from the possibility that (despite  being the one to publish) he may have been the one to first come up with Heron's Formula.

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