Heron's Formula

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 $6$ since $(3+4+5)$ ÷ $2$ is $6$. Then you have $6-3=3$, $6-4=2$, $6-5=1$. $1\cdot 2\cdot 3=6.$ $6\cdot 6 = 36$ 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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