Law of Cosines Calculator

About this calculator

The law of cosines, c² = a² + b² − 2ab·cos(C), generalizes the Pythagorean theorem to any triangle by relating the length of one side to the other two sides and the angle between them. Given two sides and the included angle (the SAS case), this calculator solves directly for the third side, then finds the remaining two angles using the law of sines (or a second application of the law of cosines) and computes the triangle's area from two sides and the included angle: Area = ½ab·sin(C).

Because it works for any triangle — not just right triangles — the law of cosines is a core tool in surveying (triangulating distances from measured angles), navigation (dead-reckoning position from bearing and distance), structural and truss analysis, and any physics problem involving vector addition where the angle between two vectors is known but their resultant isn't.