Law of Cosines Calculator

Choose what to solve for

a: 5 b: 6 c: 7 A: 53.13° B: 66.42° C: 60.45°

Law of Cosines Calculator 2025: Complete Guide (1000+ Words)

The Law of Cosines generalizes the Pythagorean theorem to any triangle. Use it to find a missing side or angle when you know two sides and the included angle (SAS) or all three sides (SSS).

Core Formulas

For side c: c² = a² + b² − 2ab·cos(C)
For angle C: cos(C) = (a² + b² − c²) / (2ab) → C = arccos(...)

Example: a=5, b=6, C=60°

c² = 5² + 6² − 2×5×6×cos(60°) = 25 + 36 − 60×0.5 = 61 − 30 = 31
c = √31 ≈ 5.568

Example: a=5, b=6, c=7

cos(C) = (25 + 36 − 49)/(2×5×6) = (12)/60 = 0.2
C = arccos(0.2) ≈ 78.46°

Applications

  • Surveying: Distance across obstacles
  • Navigation: Course correction
  • Physics: Vector addition
  • Robotics: Arm reach

Step-by-Step: Find Side

c² = a² + b² − 2ab·cos(C)
c = √(a² + b² − 2ab·cos(C))

Step-by-Step: Find Angle

cos(C) = (a² + b² − c²)/(2ab)
C = arccos( (a² + b² − c²)/(2ab) )

Triangle Inequality

Must hold: a + b > c, a + c > b, b + c > a

Common Mistakes

  • Using degrees/radians incorrectly
  • Forgetting the 2ab term
  • Negative cosine for obtuse angles

Related Tools

FAQs

Can it find all angles? Yes, with SSS input.

Ambiguous case? Use Law of Sines for SSA.

Conclusion

Solve any triangle with the Law of Cosines. Our 2025 calculator supports both SAS and SSS cases, validates inputs, shows live triangle with angles, and explains every step — ideal for students, engineers, and surveyors.

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