Triangle Area Calculator (3 Sides)

If you know all three sides of a triangle, you can find its area without measuring a single angle or height. Heron's formula does exactly that, and it's the math behind this calculator. The reverse works too: enter the area along with two sides, and the calculator returns the missing third side.

What is Heron's formula?

Heron of Alexandria worked this out in the first century AD, and the math has held up. Start with the semi-perimeter,

s = \frac{a + b + c}{2}

which is just half the triangle's perimeter. The area is then

\text{Area} = \sqrt{s(s - a)(s - b)(s - c)}

Run it on a 3-4-5 triangle: s = 6:

\begin{aligned}s &= \frac{3 + 4 + 5}{2} = 6 \\\text{Area} &= \sqrt{6 \times (6 - 3) \times (6 - 4) \times (6 - 5)} \\&= \sqrt{6 \times 3 \times 2 \times 1} \\&= \sqrt{36} \\&= 6 \text{ square units}\end{aligned}

The formula works on any valid triangle, whether equilateral, isosceles, or scalene, as long as the three sides can actually close into one.

How to use this calculator

Enter any three of the four values: side A, side B, side C, or area. Pick units for each input independently; mixing centimeters for one side and inches for another is fine, since everything is converted to a common base before the math runs. If the three sides violate the triangle inequality (the sum of any two sides has to exceed the third), you'll see a validation error before the area is computed.

Where this comes up

The three-sides method is what you reach for when you've got lengths but not angles. A surveyor working an irregular property line walks the perimeter with a tape or GPS rover and feeds the three legs in. Roofers and flooring installers estimate material for triangular sections this way, since pulling out a protractor on a job site isn't always practical. Trusses and bracing are another natural fit: every member has a recorded length, but the joint angles are rarely written down anywhere. Geometry students also use it to check textbook answers that only give sides.

Tips for accurate results

A few things worth checking before you trust the result:

Small measurement errors compound in the area calculation. If one side feels slightly off, re-measure before relying on the result.
Verify the triangle inequality before entering anything. When the sum of two sides is less than or equal to the third, there is no triangle to compute.
The calculator handles unit conversion, but taking all three measurements in the same unit cuts down on input mistakes.

Frequently asked questions

Can I calculate a side if I know the area?

That's exactly what the reverse mode is for. Enter the area and any two sides, and the calculator solves for the third using the inverted form of Heron's formula. There can be more than one geometrically valid answer in edge cases, but the calculator returns the standard positive root.

Why does it say my triangle is invalid?

The triangle inequality failed for your inputs. The sum of any two sides has to exceed the third, otherwise the three lengths can't close into a triangle. Double-check the measurements, since this error usually points to a typo or a unit mismatch.

Does this work for all triangle types?

Equilateral, isosceles, and scalene triangles all work the same way under Heron's formula. The shape doesn't matter as long as the three lengths can form a valid triangle. The only triangles it can't handle are degenerate ones, where the three points are collinear and the area collapses to zero.

Do I need to know the height of the triangle?

Skipping the height is the whole point of Heron's formula. The base-times-height-over-two approach you learn in school works when you have a clean perpendicular to measure against, but on a job site or out in a field that perpendicular is often the hardest number to pin down. Three side lengths are usually much easier to record.