How does the Law of Cosines work?

Label the triangle sides a, b, c and opposite angles A, B, C Then: To find side c: c² = a² + b² − 2ab·cos(C) Then: To find angle C: cos(C) = (a² + b² − c²) / 2ab.

Can you give an example of how to use the Law of Cosines?

Example: Input a=8, b=11, C=37° gives a result of c ≈ 6.67 (c²=64+121−2(8)(11)cos(37°)).

Is the Law of Cosines free to use?

Yes — completely free with no registration, download, or subscription required. All calculations happen instantly in your browser.

How accurate is the Law of Cosines?

Our Law of Cosines uses verified mathematical formulas and is accurate to multiple decimal places. Results are calculated in real-time using the same methods used by professionals.

Mathematics

Law of Cosines

Law of Cosines Calculator

Enter sides a and b and the included angle C to find side c.

Side a

Side b

Angle C (°)

The Law of Cosines relates the three sides of any triangle to one of its angles. It generalises the Pythagorean theorem: when the angle C = 90°, cos(C) = 0 and the formula reduces to a² + b² = c².

Find	Formula
Side c	c² = a² + b² − 2ab·cos(C)
Side b	b² = a² + c² − 2ac·cos(B)
Side a	a² = b² + c² − 2bc·cos(A)
Angle C	cos(C) = (a² + b² − c²) / 2ab

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Pro Tip

Use the Law of Cosines when you know: (1) two sides and the included angle (SAS), or (2) all three sides (SSS). For other cases, use the Law of Sines.

References

›Law of cosines — Wikipedia
›Law of Cosines — Khan Academy

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