@The-Darkin-Blade Hi again! The variable \(s\) is there just to make the expression look a little bit cleaner and to make Heron's Formula easier to memorize. See, \(s\) is just half of the triangle's perimeter, or the semiperimeter:
$$ s = \frac{ a + b + c}{2} $$
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!
🙂
