Heron's Formula Calculator
Find the area of a triangle from its three side lengths using Heron’s formula, with the perimeter and semi-perimeter.
How it works
s = (a + b + c) ÷ 2 · Area = √(s(s − a)(s − b)(s − c))
Heron’s formula finds a triangle’s area from just its three side lengths, without needing a height or any angle. First compute the semi-perimeter s, half the perimeter, then take the square root of s times each of the three (s − side) terms. If any side is longer than or equal to the sum of the other two, the triangle inequality fails and no real triangle exists — the value under the root would go negative.
Worked example
For sides a = 3, b = 4, c = 5: the semi-perimeter is s = (3 + 4 + 5) ÷ 2 = 6. Then area = √(6 × (6 − 3) × (6 − 4) × (6 − 5)) = √(6 × 3 × 2 × 1) = √36 = 6 square units — matching the ½ × 3 × 4 you get from this right triangle’s legs.
Sources: Heron's formula — Wikipedia
Frequently asked questions
When should I use Heron’s formula?
Use it whenever you know all three side lengths but no height or angle — the side-side-side case. It saves you from first finding an altitude or applying trigonometry to get an angle.
What is the triangle inequality?
For three lengths to close into a triangle, each must be shorter than the sum of the other two. If a + b ≤ c (or any similar case), the sides can’t meet, and the formula’s square root would take a negative argument.
Why is it called the semi-perimeter?
The semi-perimeter s is half the total perimeter, (a + b + c) ÷ 2. It appears four times in the formula and keeps the expression symmetric in the three sides, which is part of why Heron’s formula is so compact.
Related calculators
This calculator is for educational purposes. Double-check important results before acting on them.