Heron's Formula Calculator
Find a triangle's area from its three sides.
Results update as you type.
How it works
Heron's formula is a classic result that finds a triangle's area from just the lengths of its three sides, with no need to know any angle or height, and this calculator applies it. You enter the three side lengths, and it returns the area, the perimeter, and the semi-perimeter. The method, attributed to Hero of Alexandria, first computes the semi-perimeter, which is half the sum of the sides, then takes the square root of the semi-perimeter times its difference with each side. This is especially useful when you know the sides but cannot easily measure a height, such as for an irregular plot of land, a triangular part in construction, or a surveying problem. The formula works for any valid triangle, including scalene triangles with three different sides. For a result to exist, the three sides must satisfy the triangle inequality, meaning the sum of any two sides must exceed the third; otherwise the lengths cannot form a closed triangle and the calculator returns no area. A well-known check is the 3-4-5 right triangle, which Heron's formula correctly gives an area of 6. Beyond practical measurement, Heron's formula is elegant for showing that area depends only on side lengths. Enter the three sides to compute the triangle's area, perimeter, and semi-perimeter.
Area = √(s(s−a)(s−b)(s−c)), where s = (a+b+c)/2
Heron's formula finds a triangle's area from its three side lengths alone. First compute the semi-perimeter s, half the perimeter. The area is the square root of s times (s − a) times (s − b) times (s − c). It works for any triangle whose sides satisfy the triangle inequality, needing no angle or height.
Source:Geometry (Heron of Alexandria)
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Worked example
How to use this calculator
- 1
Enter your values
Enter side a, side b and side c.
- 2
Adjust the options
Fine-tune any options to match your situation.
- 3
Read your result
Triangle area updates instantly as you type — no button to press.
Frequently asked questions
What is Heron's formula?
Heron's formula calculates the area of a triangle from the lengths of its three sides. You first find the semi-perimeter, half the sum of the sides, then take the square root of the semi-perimeter multiplied by its difference with each side. It is valuable because it needs no height or angle, only the three side lengths.
When should I use Heron's formula?
Use it whenever you know all three side lengths of a triangle but not its height or angles. This is common in surveying, land measurement, and construction, where measuring the sides is easy but finding a perpendicular height is not. It works for any triangle shape, including scalene triangles with three unequal sides.
What is the triangle inequality?
The triangle inequality states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. If this fails, the three lengths cannot close into a triangle, so no area exists. The calculator checks this and returns no result when the sides violate the inequality.
What is the semi-perimeter?
The semi-perimeter, usually written s, is half of a triangle's perimeter, that is, half the sum of its three sides. It appears in Heron's formula as the central quantity from which the area is built. The calculator reports the semi-perimeter alongside the area and perimeter so you can see each step of the calculation.
Does Heron's formula work for right triangles?
Yes. Heron's formula works for every valid triangle, including right triangles. For a 3-4-5 right triangle, it gives an area of 6, matching the simpler one-half base times height calculation. This consistency makes it a reliable general method regardless of whether the triangle is right, acute, or obtuse.