Heron's formula

Heron’s formula computes the area of a triangle from its three side lengths, without needing the height. Named for Hero of Alexandria, the 1st-century mathematician who published it in his Metrica.

Given side lengths a, b, c, define the semi-perimeter s = (a + b + c) / 2. Then:

A = √(s(s−a)(s−b)(s−c))

The formula is elegant because it uses only the side lengths — no angles, no height. When all you have is three measurements of a triangular plot, three sides of a sail, or three edges of a tile, Heron’s formula is the most direct path to the area.

Edge case: if the three values don’t satisfy the triangle inequality (any side ≥ sum of the other two), the quantity under the square root is negative and the formula returns no real solution. Our area calculator surfaces this as a zero result rather than NaN.