Compute the area of any triangle from its three side lengths using A = √(s(s−a)(s−b)(s−c)) where s = (a+b+c)/2. No angle measurement required — just three sides. Shows full step-by-step work, all three angles, circumradius, inradius, altitudes, triangle classification, and a labeled canvas diagram.
Side Lengths
Formula: s = (a+b+c)/2 → A = √(s·(s−a)·(s−b)·(s−c))
Examples
Results
Who Was Heron — and Where Does the Formula Come From?
A = √(s(s−a)(s−b)(s−c)) where s = (a+b+c)/2
Heron of Alexandria (c. 10–70 AD) was a Greek mathematician and engineer who catalogued the formula in his work Metrica — though it may have been known to Archimedes a century earlier. The formula expresses area purely in terms of the three side lengths, without needing to know any angle.
Derivation sketch: drop an altitude from vertex A to side a (length = ha). Using the law of cosines to express cos A in terms of sides, then sin A = √(1 − cos²A), and substituting into A = ½bc·sin A produces the factored Heron form after algebraic simplification. Equivalently, one can verify that A = ½bc·sin A when you know all three sides — so Heron's formula is exactly the SAS formula in disguise.
Why Heron's Formula Is Useful
No angle needed — just a, b, c
No angle measurement required. If you can measure or are given all three side lengths — for instance, in surveying, architecture, or a geometry problem — Heron's formula gives area directly, without any trigonometry lookup.
Works for any triangle shape. Scalene, isosceles, equilateral, right, acute, or obtuse — the formula is universal. The triangle inequality (each side must be shorter than the sum of the other two) is the only constraint.
Numerical stability note. For very flat (nearly degenerate) triangles, the standard form can lose precision. A numerically stable variant reorders sides so a ≥ b ≥ c before computing: A = ¼·√((a+(b+c))(c−(a−b))(c+(a−b))(a+(b−c))).
Our tutors break down Heron's formula, triangle properties, the Law of Cosines, and more — step by step, at your pace.