# Difference between revisions of "Heron's Formula"

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where the [[semi-perimeter]] <math>s=\frac{a+b+c}{2}</math>. | where the [[semi-perimeter]] <math>s=\frac{a+b+c}{2}</math>. | ||

+ | |||

+ | === Proof === | ||

+ | |||

+ | <math>[ABC]=\frac{ab}{2}\sin C</math> | ||

+ | |||

+ | <math>=\frac{ab}{2}\sqrt{1-\cos^2 C}</math> | ||

+ | |||

+ | <math>=\frac{ab}{2}\sqrt{1-(\frac{a^2+b^2-c^2}{2ab})^2}</math> | ||

+ | |||

+ | <math>=\sqrt{\frac{a^2b^2}{4}(1-\frac{(a^2+b^2-c^2)^2}{4a^2b^2})}</math> | ||

+ | |||

+ | <math>=\sqrt{\frac{4a^2b^2-(a^2+b^2-c^2)^2}{16}}</math> | ||

+ | |||

+ | <math>=\sqrt{\frac{(2ab+a^2+b^2-c^2)(2ab-a^2-b^2+c^2)}{16}}</math> | ||

+ | |||

+ | <math>=\sqrt{\frac{((a+b)^2-c^2)(c^2-(a-b)^2)}{16}}</math> | ||

+ | |||

+ | <math>=\sqrt{\frac{(a+b+c)(a+b-c)(b+c-a)(a+c-b)}{16}}</math> | ||

+ | |||

+ | <math>=\sqrt{s(s-a)(s-b)(s-c)}</math> | ||

=== See Also === | === See Also === | ||

* [[Brahmagupta's formula]] | * [[Brahmagupta's formula]] |

## Revision as of 21:27, 30 June 2006

**Heron's formula** (sometimes called Hero's formula) is a method for finding the area of a triangle given only the three side lengths.

### Definition

For any triangle with side lengths , the area can be found using the following formula:

where the semi-perimeter .

### Proof