Triangle Area Calculator (Heron's Formula)
Written by Thierno Sadou Diallo, formula verified per our methodology • Last checked on 9/6/2026
Heron's formula calculates a triangle's area from its three sides alone, with A = √(s(s−a)(s−b)(s−c)), where s is the semi-perimeter. For a triangle with sides 3, 4 and 5 cm, the area is exactly 6 cm².
Explanation
The site's standard triangle area calculator uses the formula base × height ÷ 2, but that approach assumes you know the triangle's height — a measurement that isn't always directly available, especially for an arbitrary triangle where only the three sides have been measured (a plot of land, a piece of fabric, a map). Heron's formula solves exactly this problem: it calculates the area purely from the lengths a, b and c of the three sides, without ever needing to know or construct a height. The principle rests on the semi-perimeter s = (a+b+c) ÷ 2, then on the product s(s−a)(s−b)(s−c), whose square root directly gives the area. This result isn't an approximation: it's an exact mathematical identity, provable through several independent methods (including the law of cosines, the same trigonometric relationship used by our circumscribed circle radius calculator to work with a triangle's sides and angles). One condition must be met, though: the three lengths must form a valid triangle, meaning each side must be strictly less than the sum of the other two (the triangle inequality) — otherwise the term under the square root becomes negative and the calculation no longer has geometric meaning.
Example: a triangle with sides 3 cm, 4 cm and 5 cm
Inputs
Side a: 3 cm. Side b: 4 cm. Side c: 5 cm.
Calculation
s = (3+4+5) ÷ 2 = 6. A = √(6×(6−3)×(6−4)×(6−5)) = √(6×3×2×1) = √36 = 6.
Result
This triangle's area is 6 cm² — a result that matches the base × height ÷ 2 calculation (3 × 4 ÷ 2 = 6), since this 3-4-5 triangle is also a right triangle.
Frequently asked questions
When should I use Heron's formula instead of base × height ÷ 2?
Use Heron's formula whenever you know a triangle's three sides but not its height — the most common situation for an arbitrary triangle measured in the field. If you already have a base and its corresponding perpendicular height, the base × height ÷ 2 calculation from our standard calculator remains more direct.
What happens if the three sides cannot form a triangle?
If one side is larger than the sum of the other two (for example 10, 2 and 3), the triangle inequality isn't satisfied: these three lengths simply cannot form a closed triangle in a plane. The term under the square root then becomes negative, which has no real solution — this calculator shows an undefined result rather than a misleading number.
Does Heron's formula work for all types of triangles?
Yes, without exception: equilateral, isosceles, right, or scalene, as long as the three sides satisfy the triangle inequality. That's actually the whole appeal of this formula compared to base × height ÷ 2: it requires no prior identification of a right angle or a particular height, only the three lengths.