Heron's Formula Calculator
The area of a triangle from the lengths of its three sides.
Result
Area
6.00
Perimeter
12.00
How it works
Heron's formula finds a triangle's area from its three sides alone — no height, no angles. Compute the semi-perimeter s = (a+b+c)/2, then the area is √(s(s−a)(s−b)(s−c)). For sides 3, 4 and 5: s = 6, and the area is √(6×3×2×1) = 6. That independence from height is what makes it so useful in the field. Surveyors, carpenters and builders can measure three distances with a tape, and the area follows — no need to establish a perpendicular, which is awkward on real terrain. A built-in sanity check: if the three lengths cannot form a triangle, one bracket turns negative and the formula fails. That happens whenever one side is longer than the other two combined — the triangle inequality. A result of exactly zero means the three points are collinear: a flat, degenerate triangle.
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Frequently asked questions
When do I use Heron’s formula?
When you know the three side lengths but not the height — it gives the area directly.
Why might it return zero?
If the sides cannot form a triangle (one side ≥ the sum of the other two), there is no valid area.
Why does it sometimes return zero or an error?
Because the three lengths cannot form a triangle. If one side equals the sum of the other two the area is zero (flat), and if it exceeds that sum the formula has no real solution — that is the triangle inequality.
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