Law of Cosines Calculator
Enter sides a and b and the included angle C to find side 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.
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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.
- 1Label the triangle sides a, b, c and opposite angles A, B, C
- 2To find side c: c² = a² + b² − 2ab·cos(C)
- 3To find angle C: cos(C) = (a² + b² − c²) / 2ab
a=8, b=11, C=37°=c ≈ 6.67c²=64+121−2(8)(11)cos(37°)
a=5, b=7, c=10=C ≈ 111.8°cos(C)=(25+49−100)/70=−0.37
| 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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