Heron's Triangle Area Calculator

Heron’s formula calculates the area of a triangle when you know all three side lengths. Unlike the standard base-times-height formula, Heron’s formula does not require you to know the height of the triangle, making it extremely practical for real-world measurements.

The formula:

Area = √(s(s−a)(s−b)(s−c))

Where s is the semi-perimeter: s = (a + b + c) / 2

And a, b, c are the three side lengths.

Step-by-step example:

For a triangle with sides 5, 7, and 10:

1. Semi-perimeter: s = (5 + 7 + 10) / 2 = 11
2. Calculate each factor: s−a = 6, s−b = 4, s−c = 1
3. Product: 11 × 6 × 4 × 1 = 264
4. Area = √264 ≈ 16.25 square units

Historical background: This formula is named after Heron of Alexandria, a Greek mathematician and engineer who lived in the 1st century AD. He published the formula in his work Metrica, though some historians believe Archimedes may have known the formula earlier.

When is this useful?

Heron’s formula is particularly valuable in land surveying and construction. If you can measure the three distances between fence posts or property markers, you can immediately calculate the enclosed area without having to measure any heights or angles. Architects and engineers use it when working with irregular triangular plots.

Triangle validity check: Not every combination of three numbers forms a valid triangle. The triangle inequality requires that the sum of any two sides must be greater than the third side. This calculator automatically checks this condition for you.

Additional results: This calculator also shows the triangle’s perimeter, semi-perimeter, all three angles (from the Law of Cosines), the inradius (radius of the inscribed circle, which equals Area / s), the circumradius (abc / 4·Area), and the height on each side. The diagram is drawn to scale on a square grid, so the shape you see is the shape you entered.

Units: The area is in square units of whatever unit your side lengths are in. If sides are in meters, area is in square meters. If sides are in feet, area is in square feet.

A word about precision. Written exactly as it appears above, Heron’s formula quietly falls apart on long thin triangles. Take sides of 100000, 99999.99979 and 0.00029. The semi-perimeter is almost identical to the longest side, so s − a is a subtraction of two nearly equal numbers, and most of the significant digits cancel and vanish. What survives is dominated by rounding.

William Kahan published the fix in 1986. Sort the sides so a ≥ b ≥ c, then compute

Area = ¼√((a + (b + c))(c − (a − b))(c + (a − b))(a + (b − c)))

That is algebraically the same formula with the brackets arranged so the dangerous subtractions happen between numbers of similar size. This calculator uses Kahan’s version and shows the textbook result alongside it whenever the two disagree, which for ordinary triangles is never.