Difference between revisions of "Heron's Formula"
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<math>=\frac{ab}{2}\sqrt{1-\cos^2 C}</math> | <math>=\frac{ab}{2}\sqrt{1-\cos^2 C}</math> | ||
| − | <math>=\frac{ab}{ | + | <math>=\frac{ab}{2}\sqrt{1-\left(\frac{a^2+b^2-c^2}{2ab}\right)^2}</math> |
<math>=\sqrt{\frac{a^2b^2}{4}\left[1-\frac{(a^2+b^2-c^2)^2}{4a^2b^2}\right]}</math> | <math>=\sqrt{\frac{a^2b^2}{4}\left[1-\frac{(a^2+b^2-c^2)^2}{4a^2b^2}\right]}</math> | ||
Revision as of 20:23, 8 February 2016
Heron's Formula (sometimes called Hero's formula) is a formula for finding the area of a triangle given only the three side lengths.
Contents
Theorem
For any triangle with side lengths 
, the area 
can be found using the following formula:
where the semi-perimeter 
.
Proof
![$[ABC]=\frac{ab}{2}\sin C$](http://latex.artofproblemsolving.com/a/c/4/ac4b71b27e69d9e101968239d56c96a437257a1d.png)
![$=\sqrt{\frac{a^2b^2}{4}\left[1-\frac{(a^2+b^2-c^2)^2}{4a^2b^2}\right]}$](http://latex.artofproblemsolving.com/6/d/1/6d165f658a8a212657a849e8c47d47d0700ce7aa.png)
See Also
External Links
In general, it is a good advice not to use Heron's formula in computer programs whenever we can avoid it. For example, whenever vertex coordinates are known, vector product is a much better alternative. Main reasons:
- Computing the square root is much slower than multiplication.
- For triangles with area close to zero Heron's formula computed using floating point variables suffers from precision problems.
