Triangle Area Calculator (Heron's Formula)

Heron's Formula (also called Hero's Formula) is a method to calculate the area of a triangle when the lengths of all three sides are known. The formula is A = √[s(s−a)(s−b)(s−c)], where s is the semi-perimeter: s = (a + b + c) / 2. It was described by the Greek mathematician Heron of Alexandria around 60 AD.

Simply enter the three side lengths — Side A, Side B, and Side C — into the input fields. The calculator automatically computes the triangle area, semi-perimeter, and perimeter using Heron's Formula. No angles or height measurements are needed.

's' is the semi-perimeter of the triangle, which is half of the total perimeter. You calculate it as s = (a + b + c) / 2. For example, if the sides are 3, 4, and 5, then s = (3 + 4 + 5) / 2 = 6.

For a 3-4-5 right triangle, the semi-perimeter s = (3 + 4 + 5) / 2 = 6. Applying Heron's Formula: A = √[6 × (6−3) × (6−4) × (6−5)] = √[6 × 3 × 2 × 1] = √36 = 6 square units.

Heron's Formula works for any valid triangle — scalene, isosceles, or equilateral. However, the three sides must satisfy the triangle inequality: the sum of any two sides must be greater than the third side. If this condition is not met, the triangle is invalid and no real area exists.

The calculator is unit-agnostic — you can enter side lengths in any unit (centimeters, meters, inches, feet, etc.). The resulting area will be in the square of whatever unit you used. For example, sides in centimeters produce an area in square centimeters.

Yes, Heron's Formula works perfectly for right triangles. However, for right triangles you can also use the simpler formula A = (1/2) × base × height, since the two legs serve as base and height. Both methods give the same result.

If the sides violate the triangle inequality (e.g., sides 1, 2, and 10), the value inside the square root becomes negative, which means no real triangle exists. The calculator will alert you that the entered side lengths do not form a valid triangle.