# Difference between revisions of "Brahmagupta's Formula"

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===Proof=== | ===Proof=== | ||

− | {{ | + | If we draw <math>AC</math>, we find that <math>[ABCD]=\frac{ab\sin B}{2}+\frac{cd\sin D}{2}=\frac{ab\sin B+cd\sin D}{2}</math>. Since <math>B+D=180^\circ</math>, <math>\sin B=\sin D</math>. Hence, <math>[ABCD]=\frac{\sin B(ab+cd)}{2}</math>. Multiplying by 2 and squaring, we get: |

− | + | <cmath>4[ABCD]}^2=\sin^2 B(ab+cd)^2</cmath> | |

+ | Substituting <math>\sin^2B=1-\cos^2B</math> results in | ||

+ | <cmath>4[ABCD]^2=(1-\cos^2B)(ab+cd)^2=(ab+cd)^2-\cos^2B(ab+cd)^2</cmath> | ||

+ | By the Law of Cosines, <math>a^2+b^2-2ab\cos B=c^2+d^2-2cd\cos D</math>. <math>\cos B=-\cos D</math>, so a little rearranging gives | ||

+ | <cmath>2\cos B(ab+cd)=a^2+b^2-c^2-d^2</cmath> | ||

+ | <cmath>4[ABCD]^2=(ab+cd)^2-\frac{1}{4}(a^2+b^2-c^2-d^2)^2</cmath> | ||

+ | <cmath>16[ABCD]^2=4(ab+cd)^2-(a^2+b^2-c^2-d^2)^2</cmath> | ||

+ | <cmath>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))</cmath> | ||

+ | <cmath>16[ABCD]^2=(a^2+2ab+b^2-c^2+2cd-d^2)(-a^2+2ab-b^2+c^2+2cd+d^2)</cmath> | ||

+ | <cmath>16[ABCD]^2=((a+b)^2-(c-d)^2)((c+d)^2-(a-b)^2)</cmath> | ||

+ | <cmath>16[ABCD]^2=(a+b+c-d)(a+b-c+d)(c+d+a-b)(c+d-b+a)</cmath> | ||

+ | <cmath>16[ABCD]^2=16(s-a)(s-b)(s-c)(s-d)</cmath> | ||

+ | <cmath>[ABCD]=\sqrt{(s-a)(s-b)(s-c)(s-d)}</cmath> | ||

== Similar formulas == | == Similar formulas == |

## Revision as of 22:52, 24 December 2008

**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 , , , , the area can be found as:

where is the semiperimeter of the quadrilateral.

### Proof

If we draw , we find that . Since , . Hence, . Multiplying by 2 and squaring, we get:

\[4[ABCD]}^2=\sin^2 B(ab+cd)^2\] (Error compiling LaTeX. ! Extra }, or forgotten $.)

Substituting results in By the Law of Cosines, . , so a little rearranging gives

## 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 .
*This article is a stub. Help us out by expanding it.*