The law of cosines generalises the Pythagorean theorem to triangles without a right angle.
How to use the Law of Cosines Calculator
- Side a
- Side b
- Angle C (between a and b)
Results update as you type, and every field accepts the units shown in its menu.
How it works
Plug two sides and the included angle into the formula to get the third side, then use it again for the other angles.
Inputs explained
| Field | What to enter |
|---|---|
| Side a | A number |
| Side b | A number |
| Angle C (between a and b) | In ° |
What each result means
- Side c = sqrt([Side a]^2 + [Side b]^2 – 2×[Side a]×[Side b]×cos(rad([Angle C (between a and b)])))
- Angle A = deg(acos(([Side b]^2 + [Side c]^2 – [Side a]^2)÷(2×[Side b]×[Side c])))
- Angle B = 180 – [Angle A] – [Angle C (between a and b)]
- Area = [Side a]×[Side b]×sin(rad([Angle C (between a and b)]))÷2
Example values
How the main result changes when Side a changes and everything else stays at the example values above:
| Side a | Side c |
|---|---|
| 3.5 | 8.2009 |
| 5.25 | 7.7505 |
| 7 | 7.6818 |
| 8.75 | 8.0047 |
| 10.5 | 8.6755 |
| 14 | 10.7712 |
Formula
c² = a² + b² − 2ab cos(C)Worked example
Inputs — Side a: 7; Side b: 10; Angle C (between a and b): 50 °.
Results — Side c = 7.681779; Angle A = 44.271 °; Angle B = 85.729 °; Area = 26.811556.
Frequently asked questions
When do I use the law of sines instead?
When you know two angles and a side, or two sides and a non-included angle.
What if C is 90°?
The formula reduces to Pythagoras.
What formula does the Law of Cosines Calculator use?
c² = a² + b² − 2ab cos(C).