Law of Cosines Calculator

The Law of Cosines is a formula that relates the three sides of a triangle to one of its angles. It states: c² = a² + b² − 2ab·cos(C). It generalizes the Pythagorean theorem to any triangle, not just right triangles, making it useful for solving oblique (non-right) triangles.

Use the Law of Cosines when you have either SSS (all three sides known) or SAS (two sides and the included angle known). For ASA, AAS, or most SSA configurations, the Law of Sines is typically the better choice.

Use the Pythagorean theorem only when you have a right triangle and need to find a missing side. The Law of Cosines works for any triangle — when the included angle is 90°, it simplifies directly to the Pythagorean theorem since cos(90°) = 0.

No — the Law of Cosines works for any triangle, including acute, obtuse, and right triangles. It is specifically designed for cases where the Law of Sines is insufficient, particularly the SSS and SAS cases.

Using the Law of Cosines: c² = 5² + 6² − 2(5)(6)cos(60°) = 25 + 36 − 30 = 31, so c = √31 ≈ 5.568. Enter these values in the calculator above using SAS mode to verify.

Rearrange the Law of Cosines formula to: A = arccos((b² + c² − a²) / (2bc)). Enter all three side lengths in the calculator with SSS mode selected, and it will compute all three angles automatically.

You can enter angles in either degrees or radians. Select your preferred unit from the Angle Units dropdown before entering values. The results will be displayed in the same unit you selected.

For a valid triangle, all sides must be positive, all angles must be positive, and the sum of any two sides must exceed the third side (triangle inequality). The sum of all interior angles must equal 180° (or π radians). This calculator checks these conditions and warns you if the inputs do not form a valid triangle.