Law of Cosines Calculator
You do not need a right angle — three sides are enough. The cosine of an angle is the sum of the other two sides squared, minus the opposite side squared, over twice their product.
Your numbers
Angle (°)
48.19°
Interior angle
58.412°
Exterior angle
73.398°
Formula
Angle (°) = acos((Side b² + Side c² − Side a²) ÷ (2 × Side b × Side c))
The angle facing the longest side is the largest; if it exceeds 90° the triangle is obtuse.
What to enter
| Input | Default | Accepted range |
|---|---|---|
| Side a (cm)The length of the first side. | 7 | 0 and up |
| Side b (cm)The length of the second side. | 8 | 0 and up |
| Side c (cm)The third side; it must be shorter than the other two combined. | 9 | 0 and up |
Step by step
Quick reference table
Results when only Side a (cm) changes and everything else stays put.
| Side a (cm) | Angle (°) (°) | Interior angle (°) | Exterior angle (°) |
|---|---|---|---|
| 3.5 | 22.798 | 62.336 | 94.866 |
| 5.25 | 35.359 | 61.864 | 82.776 |
| 7 | 48.19 | 58.412 | 73.398 |
| 10.5 | 76.036 | 47.679 | 56.285 |
| 14 | 110.742 | 32.303 | 36.955 |
What each result means
| Result | At default values |
|---|---|
| Angle (°) (°)An angle in degrees; a full turn is 360°. | 48.19 |
| Interior angle (°)The interior angle at one corner of a regular polygon. | 58.412 |
| Exterior angle (°)How far you turn at a corner; they always add to 360°. | 73.398 |
Common mistakes
The angle facing the longest side is the largest; if it exceeds 90° the triangle is obtuse.
Glossary
- Side a
- The length of the first side.
- Side b
- The length of the second side.
- Side c
- The third side; it must be shorter than the other two combined.
- Angle (°)
- An angle in degrees; a full turn is 360°.
- Interior angle
- The interior angle at one corner of a regular polygon.
- Exterior angle
- How far you turn at a corner; they always add to 360°.
Frequently asked questions
Q. How is Law of Cosines Calculator calculated?
Angle (°) = acos((Side b² + Side c² − Side a²) ÷ (2 × Side b × Side c)) — You do not need a right angle — three sides are enough. The cosine of an angle is the sum of the other two sides squared, minus the opposite side squared, over twice their product.
Q. Can you walk through an example?
With Side a 7cm, Side b 8cm, Side c 9cm, the answer is Angle (°) 48.19°.
Q. What do I need to enter?
Enter Side a, Side b, Side c. The result recalculates as you type, and an empty box counts as zero.
Q. How much does the answer move if I change a number?
Changing only Side a (cm) moves the answer to Side a (cm) 3.5 → Angle (°) (°) 22.798 and Side a (cm) 14 → Angle (°) (°) 110.742. The table below lays out five steps.
Q. How are the numbers rounded?
Money is shown to the nearest whole unit, percentages to one decimal place and everything else to two. What you see is rounded; the calculation itself carries the unrounded value forward.
Q. Anything to watch out for?
The angle facing the longest side is the largest; if it exceeds 90° the triangle is obtuse.
Related calculators
Keep your length units consistent across inputs — mixing centimetres and metres throws results far off.