The Law of Cosines

For any triangle with sides a, b, c opposite angles A, B, C, the law of cosines states c² = a² + b² − 2ab·cos C. It is the tool of choice when you know three sides (SSS) or two sides and the included angle (SAS) — the cases where the law of sines alone cannot start.

Solving SSS and SAS

With SAS, the law of cosines directly yields the third side, then the remaining angles follow. With SSS, rearrange the formula to isolate each angle's cosine. This calculator uses exact trigonometric inverses, so results are limited only by display precision.

Area and Classification

Area is computed with Heron's formula for SSS and the ½·b·c·sin A formula for SAS. Comparing the squares of the sides also classifies the triangle as acute, right, or obtuse — a built-in consistency check on your inputs.