Find the third side of a triangle from two sides and the included angle, plus the remaining angles. Enter two sides and the angle between them.
Usage Tip
The law of cosines is the go-to for SAS and SSS triangles, where the law of sines alone is not enough.
A generalization of the Pythagorean theorem
When the angle is 90 degrees, the cosine term vanishes and it becomes the Pythagorean theorem.
The other two angles and the area are also computed.
The angle must be between 0 and 180 degrees.
The law of cosines finds the third side of a triangle when you know two sides and the angle between them. The formula is c squared equals a squared plus b squared minus two times a times b times the cosine of C. It is the Pythagorean theorem with a correction term, and it reduces to the Pythagorean theorem when the angle is 90 degrees.
Once the third side is known, the other angles can be found. The angle must be the one included between the two known sides, not one of the other angles. The law of cosines is also used the other way round to find an angle when all three sides are known. It is useful in surveying, navigation, and any triangle that is not a right triangle.