Law of Cosines Calculator
Solve any triangle using the law of cosines. Enter two sides and the included angle to calculate the third side, the remaining angles, and the area of the triangle instantly.
The law of cosines generalizes the Pythagorean theorem to all triangles, not just right triangles. The formula states: c^2 = a^2 + b^2 - 2ab x cos(C), where C is the angle between sides a and b, and c is the side opposite angle C.
When angle C is 90 degrees, cos(90) = 0, and the formula reduces to c^2 = a^2 + b^2 (the Pythagorean theorem). When C is acute (less than 90), the cosine term subtracts from the sum, making c shorter. When C is obtuse (greater than 90), the cosine is negative, making c longer.
Once side c is found, the remaining angles can be calculated using the law of cosines in reverse: cos(A) = (b^2 + c^2 - a^2) / (2bc). The area is computed using A = 0.5 x a x b x sin(C). The law of cosines is essential in surveying, navigation, physics (resolving vector components), and any application involving non-right triangles.