Law of Cosines Calculator
Solve a triangle using the cosine rule: c² = a² + b² − 2ab·cos C.
About Law of Cosines Calculator
The Law of Cosines generalises the Pythagorean theorem to any triangle: c² = a² + b² − 2ab·cos C. Use it for SAS (two sides + included angle) or SSS (three sides) configurations.
Formula
Law of Cosines
c² = a² + b² − 2ab·cos C
- a
- First side
- b
- Second side
- c
- Side opposite angle C
- C
- Included angle
Worked example
With the default values below, Law of Cosines Calculator returns Side c of 6.244998. Change any input above and every figure here is recalculated in your browser.
Inputs used
- Known valuesone of Two sides + included angle (SAS), Three sides (SSS)
- Two sides + included angle (SAS)
- Side aminimum 0.001
- 5
- Side bminimum 0.001
- 7
- Angle C (degrees, between a and b)accepts 0.01–179.99
- 60
- Side cminimum 0.001
- 8
Results
- Side c
- 6.244998
- Angle A
- 43.897886 °
- Angle B
- 76.102114 °
Step by step
- 1Apply cosine rule= 39
- 2Side c= 6.244998
- 3Angle A= 43.89788625°
- 4Angle B= 76.10211375°
Assumptions and limits
- The result is derived only from the 5 inputs listed above. Anything not among them is not part of the calculation.
- In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
- Figures are rounded for display to 6 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.
How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.
Frequently Asked Questions
How does the Law of Cosines relate to the Pythagorean theorem?
When C = 90°, cos(90°) = 0, so it reduces to c² = a² + b², which is exactly the Pythagorean theorem.
Can I use Law of Cosines if I know two angles?
If you know two angles, you know the third (they sum to 180°). The Law of Sines is simpler in that case. Law of Cosines is best for SSS or SAS.