All Three Angles from Three Sides
Solving the cosine rule for the angle gives all three from the sides alone. It is how you find angles with a tape measure and no protractor, which is why carpenters and surveyors use it. Check that the three add to 180°.
Your numbers
Angle A
35.43°
Angle B
48.19°
Angle C
96.38°
What it means
The largest angle is 96.4° — an obtuse triangle.
Formula
Angle A = acos((Side b ² + Side c ² − Side a ²) ÷ (2 × Side b × Side c))
If one side exceeds the other two combined, no triangle exists and the angles collapse to 0 or 180°.
What to enter
| Input | Default | Accepted range |
|---|---|---|
| Side a (cm)The length of the first side. | 7 | 0.01 and up |
| Side b (cm)The length of the second side. | 9 | 0.01 and up |
| Side c (cm)The third side; it must be shorter than the other two combined. | 12 | 0.01 and up |
Step by step
Quick reference table
Results when only Side a (cm) changes and everything else stays put.
| Side a (cm) | Angle A (°) | Angle B (°) | Angle C (°) |
|---|---|---|---|
| 3.5 | 9.95 | 26.38 | 143.66 |
| 5.25 | 23.93 | 44.05 | 112.02 |
| 7 | 35.43 | 48.19 | 96.38 |
| 10.5 | 57.91 | 46.57 | 75.52 |
| 14 | 82.28 | 39.57 | 58.14 |
What each result means
| Result | At default values |
|---|---|
| Angle A (°)The angle opposite side a. | 35.43 |
| Angle B (°)The angle opposite side b. | 48.19 |
| Angle C (°)The angle opposite side c; the three add to 180°. | 96.38 |
Common mistakes
If one side exceeds the other two combined, no triangle exists and the angles collapse to 0 or 180°.
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 A
- The angle opposite side a.
- Angle B
- The angle opposite side b.
- Angle C
- The angle opposite side c; the three add to 180°.
Frequently asked questions
Q. How is All Three Angles from Three Sides calculated?
Angle A = acos((Side b ² + Side c ² − Side a ²) ÷ (2 × Side b × Side c)) — Solving the cosine rule for the angle gives all three from the sides alone. It is how you find angles with a tape measure and no protractor, which is why carpenters and surveyors use it. Check that the three add to 180°.
Q. Can you walk through an example?
With Side a 7cm, Side b 9cm, Side c 12cm, the answer is Angle A 35.43°.
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 A (°) 9.95 and Side a (cm) 14 → Angle A (°) 82.28. 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?
If one side exceeds the other two combined, no triangle exists and the angles collapse to 0 or 180°.
Related calculators
Keep your length units consistent across inputs — mixing centimetres and metres throws results far off.