Difference between revisions of "Heron's Formula"
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<math>=\sqrt{s(s-a)(s-b)(s-c)}</math> | <math>=\sqrt{s(s-a)(s-b)(s-c)}</math> | ||
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| + | ==Isosceles Triangle Simplification== | ||
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| + | <math>A=\sqrt{s(s-a)(s-b)(s-c)}</math> for all triangles | ||
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| + | <math>b=c</math> for all isosceles triangles | ||
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| + | <math>A=\sqrt{s(s-a)(s-b)(s-b)}</math> simplifies to <math>A=(s-b)\sqrt{s(s-a)}</math> <math>\blacksquare</math> | ||
==Example== | ==Example== | ||
Revision as of 18:18, 8 March 2017
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)
Isosceles Triangle Simplification
for all triangles
for all isosceles triangles
simplifies to 
Example
Let's say that you have a right triangle with the sides 3,4, and 5. Your semi- perimeter would be 6.
Then you have 6-3=3, 6-4=2, 6-5=1.
1+2+3= 6
The square root of 36 is 6. The area of your triangle is 6.
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.
