$$ s = \frac{ a + b + c}{2} $$

If we tried to remember Heron's Formula only in terms of \(a, b,\) and \(c,\) (without \(s\)), the expression would look a lot more ugly. It would look like this:

$$ \sqrt{(\frac{a + b + c}{2} )(\frac{-a + b + c}{2} )(\frac{a - b + c}{2} )(\frac{a + b -c }{2}} $$

There's another nice thing about \(s\): since it is equal to the perimeter divided by \(2,\) it takes us a long way toward the area formula for a triangle.

Remember the area formula is

$$ \frac{1}{2} \times \text{ base } \times \text{ height} $$

So all you have to do to get the area of the triangle is take \(s\) and multiply it to the sum of the heights of each triangle!

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