# 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:

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

### Corollary

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

- $\dfrac {\sqrt {\paren {a^2 + b^2 + c^2 + d^2}^2 + 8 a b c d - 2 \paren {a^4 + b^4 + c^4 + d^4} } } 4$

## Proof

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

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

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

\(\ds \triangle ABC\) | \(=\) | \(\ds \frac 1 2 a b \sin \angle ABC\) | ||||||||||||

\(\ds \triangle ADC\) | \(=\) | \(\ds \frac 1 2 c d \sin \angle ADC\) |

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:

\(\ds \sin \angle ABC\) | \(=\) | \(\ds \sin \angle ADC\) | Sine and Cosine of Supplementary Angles | |||||||||||

\(\ds \cos \angle ABC\) | \(=\) | \(\ds -\cos \angle ADC\) | Sine and Cosine of Supplementary Angles |

This leads to:

\(\ds \Area\) | \(=\) | \(\ds \frac 1 2 a b \sin \angle ABC + \frac 1 2 c d \sin \angle ABC\) | ||||||||||||

\(\ds \leadsto \ \ \) | \(\ds \paren {\Area}^2\) | \(=\) | \(\ds \frac 1 4 \paren {a b + c d}^2 \sin^2 \angle ABC\) | |||||||||||

\(\ds \leadsto \ \ \) | \(\ds 4 \paren {\Area}^2\) | \(=\) | \(\ds \paren {a b + c d}^2 \paren {1 - \cos^2 \angle ABC}\) | Sum of Squares of Sine and Cosine | ||||||||||

\(\ds \) | \(=\) | \(\ds \paren {a b + c d}^2 - \cos^2 \angle ABC \paren {a b + c d}^2\) |

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:

\(\ds 4 \paren {\Area}^2\) | \(=\) | \(\ds \paren {a b + c d}^2 - \frac 1 4 \paren {a^2 + b^2 - c^2 - d^2}^2\) | ||||||||||||

\(\ds \leadsto \ \ \) | \(\ds 16 \paren {\Area}^2\) | \(=\) | \(\ds 4 \paren {a b + c d}^2 - \paren {a^2 + b^2 - c^2 - d^2}^2\) |

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:

\(\ds \) | \(\) | \(\ds \paren {2 \paren {a b + c d} + a^2 + b^2 - c^2 - d^2} \paren {2 \paren {a b + c d} - a^2 - b^2 + c^2 + d^2}\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \paren {\paren {a + b}^2 - \paren {c - d}^2} \paren {\paren {c + d}^2 - \paren {a - b}^2}\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \paren {a + b + c - d} \paren {a + b + d - c} \paren {a + c + d - b} \paren {b + c + d - a}\) |

When we introduce the expression for the semiperimeter:

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

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:

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

$\blacksquare$

## Also known as

Some sources refer to **Brahmagupta's Formula** as the **Archimedes-Heron-Brahmagupta Formula**, for Archimedes of Syracuse and Heron of Alexandria as well as Brahmagupta.

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

The reference to Archimedes of Syracuse comes from the possibility that (despite Heron 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.

## Source of Name

This entry was named for Brahmagupta.

## Historical Note

While **Brahmagupta's Formula** bears the name of Brahmagupta, it was apparently known by Archimedes of Syracuse.

## Sources

- 1992: John Hadley/2 and David Singmaster:
*Problems to Sharpen the Young*(*Math. Gazette***Vol. 76**,*no. 475*: pp. 102 – 126) www.jstor.org/stable/3620384 - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {B}.1$: The Pythagorean Theorem: Appendix: The Formulas of Heron and Brahmagupta - 1992: David Wells:
*Curious and Interesting Puzzles*... (previous) ... (next): The Area Enclosed Against The Seashore: $31$ - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**Brahmagupta's formula** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**Brahmagupta's formula**