Heron's Formula

Revision as of 19:47, 27 May 2020 by Estthebest (talk | contribs) (Example)

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


For any triangle with side lengths ${a}, {b}, {c}$, the area ${A}$ can be found using the following formula:


where the semi-perimeter $s=\frac{a+b+c}{2}$.


$[ABC]=\frac{ab}{2}\sin C$

$=\frac{ab}{2}\sqrt{1-\cos^2 C}$








Isosceles Triangle Simplification

$A=\sqrt{s(s-a)(s-b)(s-c)}$ for all triangles

$b=c$ for all isosceles triangles

$A=\sqrt{s(s-a)(s-b)(s-b)}$ simplifies to $A=(s-b)\sqrt{s(s-a)}$ $\blacksquare$


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 $6\cdot 6 = 36$ 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.
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