Transparent method
The formula
Worked example
Example inputs
How to interpret the result
Heron's formula finds a triangle's area from its three side lengths alone, with no angle measurement needed at all: genuinely useful when a protractor isn't practical but a tape measure is. It works for any triangle, acute, obtuse or right, which is easy to forget after learning area formulas that only work for special cases.
Before you rely on it
What to check
Confirm the three lengths can actually form a triangle before trusting the result. Each side must be shorter than the sum of the other two.
The common error
Where people go wrong with triangle area (heron's formula) calculator
Assuming Heron's formula is a workaround only for right triangles. It's a general formula that works identically regardless of the triangle's angles; the familiar half-base-times-height formula is the special case, not this one.
Sensitivity evidence
How side a changes the area
| Side A | Area | Perimeter |
|---|---|---|
| 4 | 12 | 17 |
| 4.5 | 13.4 | 17.5 |
| 5worked example | 14.7 | 18 |
| 5.5 | 15.9 | 18.5 |
| 6 | 17.1 | 19 |
Every input, tested
Which input moves the area most
| Input | Tested from | To | Area at each end | Swing |
|---|---|---|---|---|
| Side A | 4.5 | 5.5 | 13.4 to 15.9 | 2.5 (17%) |
| Side B | 5.4 | 6.6 | 13.4 to 15.8 | 2.3 (16%) |
| Side C | 6.3 | 7.7 | 14 to 15 | 0.972 (6.6%) |
Two variables at once
Area by side a and side b
| Side A \ Side B | 4.8 | 6 | 7.2 |
|---|---|---|---|
| 4 | 9.3 | 12 | 13.6 |
| 4.5 | 10.7 | 13.4 | 15.1 |
| 5 | 12 | 14.7 | 16.6 |
| 5.5 | 13.2 | 15.9 | 18 |
| 6 | 14.2 | 17.1 | 19.3 |
Step by step
The worked example, input by input
| Input | Value used |
|---|---|
| Side A | 5 |
| Side B | 6 |
| Side C | 7 |
| Area | 14.7 |
| Perimeter | 18 |
Where the inputs come from
Enter the lengths of the three sides, in the same units. The sides must satisfy the triangle inequality: the sum of any two must exceed the third, or no triangle exists. Heron's formula uses the semi-perimeter s, half the perimeter, and computes the area as the square root of s times s minus a, s minus b and s minus c. Use it when you know all three sides but not the height.
When the result stops applying
Measurement errors in the sides can produce a large error in area for thin triangles, and rounding matters when the triangle is nearly flat. If the inequality fails the result is not a real triangle. The formula treats the shape as flat and does not handle curved surfaces such as a triangle drawn on a globe.
What to compare it with
See the right triangle, equilateral triangle and law of sines calculators for special cases, and the Pythagorean theorem page for right triangles. The triangle third side calculator finds a missing side, and the trapezoid and parallelogram pages cover other shapes. The land acreage calculator turns areas into acres.
People also ask
Frequently asked questions
What is Heron's formula?
A method to find a triangle's area from its three sides: the square root of s times s minus a, s minus b and s minus c, where s is half the perimeter.
When can three lengths form a triangle?
When the sum of any two is greater than the third. Otherwise the sides cannot meet.
How do I find the area if I know the base and height?
Multiply them and divide by two, which is simpler than Heron's formula when you have a height.