Solve a triangle completely from its three side lengths — finding all three angles and the area.
How it works
Each angle is found with the law of cosines: angle A = arccos((b² + c² − a²) ÷ (2bc)), cycled for the other two angles, with area from Heron’s formula. A 3-4-5 triangle gives angles of 36.87°, 53.13°, and exactly 90°, with an area of 6.
What this does not include
This is distinct from this site’s Heron’s formula calculator, which reports area only — this page also solves for all three angles 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: angle A = 36.8699°, angle B = 53.1301°, angle C = 90°, area = 6.
Sides 7, 8, 9: angle A = 48.1897°, angle B = 58.4119°, angle C = 73.3985°, area = 26.8328.
What the variables mean
| Variable | Meaning |
|---|---|
| a, b, c | The triangle’s three side lengths |
Edge cases worth knowing
Not every three lengths form a valid 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 a result.
All three angles are found using the law of cosines, and always sum to exactly 180° — a useful way to sanity-check the result.
Why does the law of cosines work for any triangle, not just right triangles?
Unlike basic right-triangle trigonometry, the law of cosines is a general relationship that holds for any triangle shape, reducing to the familiar Pythagorean theorem exactly when one angle is 90°.
Do the three angles always add up to 180°?
Yes — that’s true of every triangle, and this calculator’s third angle is found by subtracting the other two from 180° directly, guaranteeing it.
What if the three sides can’t form a valid triangle?
The triangle inequality requires any two sides added together to exceed the third — if that fails, this calculator returns no result rather than an invalid angle.
