A surveyor measures the three sides of a triangular lot — 50, 60, and 70 feet — but the lot's odd angle makes base-times-height useless. Enter Hero of Alexandria, whose formula finds any triangle's area from its three sides alone, no height required. First compute the semiperimeter s = (a + b + c)/2 — half the perimeter — then the area is √(s(s−a)(s−b)(s−c)). For the 50-60-70 lot: s = 90, and the area is √(90 × 40 × 30 × 20) ≈ 1,469 square feet. Landscapers pricing sod, farmers estimating seed, and real-estate agents describing lot sizes all reach for it.

The formula's elegance hides a practical warning: it is sensitive to measurement error, because four multiplied terms amplify small mistakes — measure twice. It also doubles as a triangle validity check: if s(s−a)(s−b)(s−c) comes out negative, your "triangle" violates the triangle inequality and can't exist. This calculator evaluates the formula's pieces: enter two of your side values and confirm the computed result against your hand calculation. Students usually meet Heron's formula as the payoff after mastering the Pythagorean theorem — same triangle, no right angle required.

With the pre-filled defaults — Variable A 10, Variable B 5 — the Herons Formula Calculator returns Result 50.0000. Change any input and those results update instantly, so you can see exactly how each number you enter moves the answer.