Calculadora da Lei dos Cossenos
- Created by
- Renato Passos, Eng. de Software
- Reviewed by
- Renato Passos, Eng. de Software
Last updated: Apr 18, 2026
Formula
c² = a² + b² − 2·a·b·cos(C)
About this calculator
The Law of Cosines Calculator finds the third side of a triangle when you know two sides and the included angle. Unlike the Law of Sines, which relates sides and opposite angles, this formula is ideal for non-right triangles. The calculation uses the equation c² = a² + b² − 2·a·b·cos(C), where a and b are the known sides and C is the angle between them.
To use the tool, enter the values of the two sides and the angle in degrees. The result is the length of the third side, rounded as needed. The formula derives from a generalized Pythagorean theorem, useful in situations like navigation, engineering, and physics where oblique triangles appear.
An important caution: the angle must be between the two given sides. If you input the angle opposite one of the sides, the result will be incorrect. Also, ensure the angle is in degrees (not radians) unless the calculator converts automatically. The law of cosines can also find angles when all sides are known, but this calculator focuses on the side.
Common use cases include calculating distances on maps (triangulation), determining lengths in construction projects, or solving vector problems. For example, if you know two forces and the angle between them, the resultant can be found using this law. Remember: the result is the side opposite angle C, so label sides correctly.
Frequently asked questions
Can I use this calculator if the angle is greater than 180 degrees?
No, the angle between two sides of a triangle must be between 0 and 180 degrees (exclusive). Values outside this range do not form a valid triangle.
What if I want to find an angle instead of the side?
This calculator is specific for finding the third side. To find an angle, use the Law of Cosines calculator for angles or the Law of Sines.
Is the result always accurate?
The calculator uses standard rounding. For higher precision, use more decimal places in the input values.
Can I use radians instead of degrees?
No, this version expects angles in degrees. Convert radians to degrees before entering (multiply by 180/π).
What is the difference between Law of Cosines and Law of Sines?
The Law of Cosines is used when you know two sides and the included angle (SAS) or all three sides (SSS). The Law of Sines is used when you know two angles and one side (ASA or AAS).