Heron's Formula Area

If a triangle has sides a, b, c and semi-perimeter $s = \frac{a+b+c}{2}$, its area is $A = \sqrt{s(s-a)(s-b)(s-c)}$. The formula needs no height; just the three side lengths are enough.

Slide the three sides and the lab shows s and the area instantly. Try a = 13, b = 14, c = 15 to get the classic area 84 square units. Make sides equal to see how an equilateral triangle's area emerges.

Students often forget to halve the perimeter to get s, plugging in the full perimeter instead. Also, square-rooting at the end is a must - many leave the answer as $s(s-a)(s-b)(s-c)$ and lose marks.

Farmers with triangular plots use Heron's formula because measuring the height is hard on uneven ground. A surveyor can just chain the three boundary sides and compute the area for land records.