Law of Cosines Calculator

The law of cosines relates the three sides of a triangle to one of its angles: c^2 = a^2 + b^2 - 2ab cos(C), where C is the angle opposite side c. It is a generalization of the Pythagorean theorem, since cos(90 degrees) is 0 and the formula reduces to c^2 = a^2 + b^2 for a right triangle. You can rotate the labels to write the same relation for sides a and b.

Use the law of cosines when you know two sides and the included angle (SAS) or all three sides (SSS), because those cases have a single, unambiguous solution. The law of sines is better when you know two angles and a side (AAS or ASA). The law of cosines also avoids the ambiguous case that can produce two valid triangles with the law of sines.

In SAS mode you enter two sides (a and b) and the angle between them (C), and the calculator finds the third side, the other two angles, and the area. In SSS mode you enter all three sides and the calculator finds all three angles and the area. Both modes use the same law of cosines formula, just rearranged to solve for a side or for an angle.

Three lengths only form a triangle when each side is shorter than the sum of the other two, which is the triangle inequality. For example 1, 2, and 3 cannot form a triangle because 1 plus 2 equals 3, so the shape collapses into a straight line. When the inequality fails, the calculator returns no result instead of an impossible triangle.

In SAS mode it uses the formula area = (1/2) a b sin(C), since two sides and the included angle are known directly. In SSS mode it uses Heron's formula, area = the square root of s(s - a)(s - b)(s - c), where s is the semiperimeter (a + b + c) divided by 2. Both give the same area for the same triangle.

It looks at the largest angle. If the largest angle is exactly 90 degrees the triangle is right, if it is below 90 degrees the triangle is acute, and if it is above 90 degrees the triangle is obtuse. The largest angle always sits opposite the longest side.