Heron's Formula Calculator
Heron's formula is a classical method for calculating the area of a triangle when all three side lengths are known, without requiring any angle measurements. Named after Hero of Alexandria, the formula uses the semi-perimeter (half the perimeter) to compute area: first calculate s = (a + b + c) / 2, then area = sqrt(s(s - a)(s - b)(s - c)). This calculator accepts the three side lengths as inputs and automatically checks whether they form a valid triangle using the triangle inequality test (the sum of any two sides must exceed the third). The calculator displays the semi-perimeter, area, perimeter, and validity of the triangle. It also shows the calculation step by step. Heron's formula is indispensable for surveyors, architects, and engineers who measure distances in the field but lack angle measurements. It works for any triangle shape: acute, right-angled, or obtuse.
Heron's formula
s = (a + b + c) / 2
area = sqrt(s(s - a)(s - b)(s - c))
Triangle inequality test
a + b > c
b + c > a
a + c > b
Worked example
Three sides: a = 3, b = 4, c = 5 (a right triangle).
- Check triangle inequality: 3 + 4 = 7 > 5 (yes), 4 + 5 = 9 > 3 (yes), 3 + 5 = 8 > 4 (yes). Valid triangle.
- Calculate semi-perimeter: s = (3 + 4 + 5) / 2 = 12 / 2 = 6
- Apply Heron's formula: area = sqrt(6(6-3)(6-4)(6-5)) = sqrt(6 * 3 * 2 * 1) = sqrt(36) = 6 square units
- Perimeter = 3 + 4 + 5 = 12
Heron's formula calculator: frequently asked questions
What is Heron's formula?
Heron's formula calculates the area of a triangle when you know all three side lengths. First, calculate the semi-perimeter s = (a + b + c) / 2. Then, area = sqrt(s(s - a)(s - b)(s - c)). It does not require knowing any angles, making it useful when angle measurements are unavailable.
Why is Heron's formula useful?
Heron's formula lets you find the area of a triangle using only side lengths, which is often what surveyors, engineers, and architects have available in the field. You do not need to measure or calculate angles. It works for any triangle: acute, obtuse, or right-angled.
What is the triangle inequality?
The triangle inequality states that the sum of any two sides must be greater than the third side. For sides a, b, c: a + b > c, b + c > a, and a + c > b. If any of these fail, the three lengths cannot form a valid triangle. This calculator checks this condition automatically.
Can Heron's formula be used for right triangles?
Yes. For a right triangle with legs a and b and hypotenuse c, Heron's formula gives the same area as the simple formula (a * b) / 2. Heron's formula is more general and works for any triangle.
What is the semi-perimeter?
The semi-perimeter s is half the perimeter of the triangle: s = (a + b + c) / 2. It is used in Heron's formula and in other geometric calculations. For a triangle with sides 3, 4, 5, the semi-perimeter is (3 + 4 + 5) / 2 = 6.
Official sources
- Geometry fundamentals: National Institute of Standards and Technology.
Reviewed by the CalculatorHub team, edited by James Graham, 14 June 2026. See our methodology.