Heron's Formula Calculator
Heron’s formula computes the area of any triangle directly from its three side lengths, with no need for the height or any angle.
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From Heron’s formula
- 1
Semi-perimeter s = (a + b + c) ÷ 2
(5 + 6 + 7) ÷ 2 = 9 - 2
Inner product s(s−a)(s−b)(s−c)
9 × 4 × 3 × 2 = 216The value under the square root in Heron's formula. - 3
Area = √(inner product)
√216 = 14.70
How does this calculator work?
Heron’s formula finds a triangle’s area from its three sides a, b, and c. Compute the semi-perimeter s = (a+b+c)/2, then Area = √(s(s−a)(s−b)(s−c)). No height or angle is needed, but the sides must satisfy the triangle inequality for a real area to exist.
Formula
How this is calculated
Enter the three side lengths a, b, and c in the same unit. The calculator first computes the semi-perimeter s = (a + b + c) / 2, which is half of the full perimeter. The area is then Area = √(s(s−a)(s−b)(s−c)). This works for any triangle and uses only the sides, so you never need the height or an interior angle.
For the three sides to actually form a triangle they must satisfy the triangle inequality: the sum of any two sides must be strictly greater than the third (a + b > c, a + c > b, and b + c > a). If this fails, the value inside the square root becomes zero or negative and no real triangle exists, so the calculator asks for valid inputs.
The diagram places vertex A at the origin, B at (c, 0) along the x-axis, and C at (b·cos A, b·sin A), where cos A = (b² + c² − a²) / (2bc) from the law of cosines. The area is returned in square units matching whatever unit you used for the sides (for example, cm gives cm²).
Frequently asked questions
No. Heron’s formula needs only the three side lengths. The semi-perimeter and a square root give the area without any height or angle measurement.
The sides must be positive and satisfy the triangle inequality: each side must be shorter than the sum of the other two. Otherwise the three lengths cannot close into a triangle.
The area is in squared units of your input. If sides are in metres the area is in square metres; if in inches it is in square inches.
Also known as
TG we-Calculate Editorial Team. (2026). Heron's Formula Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/herons-formula-calculator
TG we-Calculate Editorial Team. "Heron's Formula Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/herons-formula-calculator.
TG we-Calculate Editorial Team, "Heron's Formula Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/herons-formula-calculator
@misc{wecalculate_herons_formula_calculator, title = {Heron's Formula Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/herons-formula-calculator}}, year = {2026}, note = {TG we-Calculate} }
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