Heron Triangle Area Calculator

Triangle area — Heron's formula

Heron's formula (Hero of Alexandria, ~60 AD) gives you the area of a triangle straight from its three side lengths a, b, c. You don't need the height, you don't need an angle, and you don't need coordinates: A = √(s·(s−a)·(s−b)·(s−c)), where s = (a+b+c)/2 is the semiperimeter. Take the familiar 3-4-5 right triangle: s = (3+4+5)/2 = 6, so A = √(6·3·2·1) = √36 = 6. One catch — the sides have to obey the triangle inequality, meaning each one stays smaller than the sum of the other two. If they don't, the radicand goes negative and there's no triangle to speak of.

Applications

This is the go-to method in topography and land surveying when all you have are the distances between markers (from a theodolite, total station, or measuring tape) and no angle or altitude at all. It shows up in GPS triangulation and trilateration for area between known points, in INCRA rural plot measurement where polygons get split into triangles, in computer graphics for triangulated meshes, and in the classroom as the classic stand-in for base × height ÷ 2 whenever the height is awkward to measure.

FAQ

Why use Heron instead of base × height ÷ 2? Out in the field, dropping a perpendicular height across rough ground or around obstacles is usually a pain. Sides, on the other hand, are quick to get with a tape or laser, and Heron asks for nothing else.

What if the result is "not a number"? Your three lengths break the triangle inequality, with one side landing at or above the sum of the other two, so they can't close up into a real triangle.

Does it work for any triangle? It does. Equilateral, isosceles, scalene, right, acute, obtuse — the formula handles every valid triangle.

Is there a numerically stable variant? There is. For very thin triangles, Kahan's reformulation with the sides sorted cuts the floating-point loss you'd get from the textbook expression.