Law of Cosines (and Heron's Formula)

The Law of Cosines is an equation used to solve oblique triangles when given three sides or two sides and their included angle.

The equation can be derived in the following way.















This equation can be used to solve oblique triangles when the law of sines can not be used. For example, here is how the following triangle could be solved. (A triangle can also be solved using Law of Cosines when given two sides and their included angle.)

From the Law of Cosines, Heron's Area Formula is established. This formula can be used to find the area of an oblique triangle when only given sides of the triangle. To do this, the semiperimeter must be found. The semiperimeter is the sum of the three sides of the triangle divided by two, as shown in the picture below. The formula is as follows.










This formula can be used to find the area of an oblique triangle when given the side lengths. The side lengths can also be found using the Law of Sines or Law of Cosines. An example of how to find the area of a triangle is as followed, with the side lengths given.