Triangle Calculator

Named after Hero of Alexandria (circa 60 CE), Heron's formula computes a triangle's area using only its three side lengths — no height, no angles, no coordinate geometry required. This is remarkable because computing area traditionally requires knowing the perpendicular height from a vertex to the opposite side, which is often difficult or impossible to measure directly for irregular or scalene triangles in field conditions. The formula first computes the semi-perimeter s = (a+b+c)/2, then uses it in a symmetric expression involving all three sides: Area = √(s(s−a)(s−b)(s−c)).

The symmetry is elegant — each side appears identically in the formula, so it doesn't matter which side you call "a" versus "b" versus "c." The derivation (first published in Hero's Metrica) relies on the Pythagorean theorem and clever algebra to eliminate the height variable entirely. Modern applications in surveying, CAD software, GIS mapping, and land measurement use Heron's formula extensively because field measurements typically capture side lengths via GPS or laser rangefinder more easily than perpendicular heights.

Heron's formula is ideal when you know all three sides but not the height or any angle — common in surveying scenarios, CAD drawings with only side dimensions, and competition math problems that specify three sides. For right triangles where the two legs meeting at the 90° angle are your base and height, the simpler formula A = ½ × base × height is faster and less computationally expensive. For triangles defined by two sides and the included angle between them (SAS configuration), the trigonometric formula A = ½ × a × b × sin(C) is more direct than converting to three-side format.

Choose the formula that matches your available data. If you have coordinates of the three vertices, the shoelace formula (determinant-based) is most efficient. If you have two angles and one side (AAS or ASA), use the law of sines to find the missing sides, then apply Heron's or the SAS formula. When inputs are approximate (field measurements), Heron's formula is more numerically stable than the law of cosines for angle computation on near-degenerate triangles where one angle approaches 0° or 180°.

The core formula is A = √(s(s−a)(s−b)(s−c)), where s is the semi-perimeter (a+b+c)/2. The Triangle Calculator applies this formula to your three side inputs and returns the exact area value without requiring any angle or height measurement. Behind the result, the calculator also validates triangle inequality — every pair of sides must sum to more than the third side, or the configuration doesn't form a valid triangle (the formula would produce a negative value under the square root).

Small changes in input produce proportional changes in output, which is why double-checking each measurement matters when the result feeds downstream calculations (material quantities, land values, fabrication specs). For triangles with one very small angle and two near-equal long sides, Heron's formula can lose precision due to catastrophic cancellation in the subtraction steps — in those edge cases, a numerically stable variant by Kahan (which sorts the sides first) preserves accuracy. For typical triangles with well-separated side lengths, the standard formula is accurate to the full precision of the input measurements. Measurements in feet produce square feet; meters produce square meters — the formula is dimensionally consistent across any unit system.