Heron's formula (the area of a triangle from its three sides)

Dependencies: Triangle area = ½ × base × height, Pythagorean theorem.

Statement

Let the three side lengths of $\triangle ABC$ be $a = BC$, $b = CA$, $c = AB$, and write the semi-perimeter

Then the area $S$ of $\triangle ABC$ is uniquely determined by the three sides:

This formula expresses the area of a triangle as a closed-form algebraic expression depending only on the three side lengths — no angles, no altitude lengths, just arithmetic on the three sides plus a single square root.