Law of Cosines Triangle Calculator

What this calculator does

The law of cosines is useful when you know two sides and the included angle, or when all three sides are known and you need the angles.

When to use the Law of Cosines

The law of cosines is most useful for two common triangle cases: SAS and SSS. Use it when you know two sides and the included angle and need the third side, or when you know three sides and need the angles.

Formula reference

For side c opposite angle C, the law of cosines is:

c² = a² + b² − 2ab cos(C)

To find angle C from three sides, rearrange it as:

cos(C) = (a² + b² − c²) / (2ab)

The calculator uses degrees for the included angle.

Step-by-step SAS example

Suppose side a = 8, side b = 11, and included angle C = 40°. The missing side c is found by substituting into c² = a² + b² − 2ab cos(C). After side c is known, the remaining angles can be solved from the completed side lengths.

1. Square the two known sides.
2. Multiply 2 × a × b × cos(C).
3. Subtract that product from a² + b².
4. Take the square root to get the missing side.

Common Law of Cosines mistakes

• The angle must be the included angle between the two known sides for SAS.
• Use degree mode if your angle is in degrees.
• Do not use the law of sines first for an SSS case; there is no known angle yet.
• Check whether the three sides form a valid triangle before calculating angles.

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