How to Calculate Heron's Formula
What is Heron's Formula?
Heron's formula calculates the area of a triangle from its three side lengths alone, without needing the height. It was discovered by the ancient Greek mathematician Hero of Alexandria.
Formula
s = (a+b+c)/2; A = √(s(s−a)(s−b)(s−c))
- a, b, c
- triangle side lengths (length)
- s
- semi-perimeter (length)
- A
- area (length²)
Step-by-Step Guide
- 1s = (a + b + c) / 2 (semi-perimeter)
- 2Area = √(s(s−a)(s−b)(s−c))
- 3Works for any triangle given three sides
- 4Triangle inequality must hold: each side < sum of other two
Worked Examples
Input
Sides 3, 4, 5
Result
s=6, Area = √(6×3×2×1) = 6
Input
Sides 5, 5, 6
Result
s=8, Area = √(8×3×3×2) = 12
Frequently Asked Questions
When is Heron's formula most useful?
When you know all three sides but not the height. It's ideal for surveying and triangulation problems.
What happens if the triangle inequality is violated?
The result under the square root becomes negative, which signals an impossible triangle.
Is Heron's formula always accurate?
Yes, but for very flat triangles (small area relative to sides), numerical precision issues can arise in computation.
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