Find a triangle’s area using only its three side lengths — no angle or height measurement needed.
How it works
First find the semi-perimeter: s = (a + b + c) ÷ 2. Then the area is √(s(s−a)(s−b)(s−c)). A 3-4-5 triangle (a right triangle) gives an area of exactly 6.
What this does not include
This only computes area. It doesn’t report the triangle’s angles — for those, the law of cosines would need to be applied separately using the same three side lengths.
How to use this calculator
- Enter the three side lengths.
A worked example
A triangle with sides 3, 4, 5: using Heron’s formula, area = 6.
Sides 7, 8, 9: area = 26.8328.
What the variables mean
| Variable | Meaning |
|---|---|
| a, b, c | The triangle’s three side lengths |
Edge cases worth knowing
Not every set of three lengths forms a real triangle. Sides 1, 1, and 10 fail the triangle inequality — the two shorter sides can’t reach across the longest one — so the calculator correctly declines to show an area.
This finds area from side lengths alone, with no angles needed — a useful shortcut compared to the more familiar ½ × base × height formula, which requires knowing a height.
What if the three sides can’t actually form a triangle?
The triangle inequality requires that any two sides added together be longer than the third — if that fails, no triangle exists, and this calculator returns no result rather than a nonsensical negative-square-root answer.
Who was Heron?
Heron of Alexandria was a 1st-century engineer and mathematician credited with this formula, which remains in standard use because it needs no angle measurement at all — just the three sides.
Does this work for any triangle, not just right triangles?
Yes — Heron’s formula works for any valid triangle, obtuse, acute, or right, using only the three side lengths.
