Law of Sines
In any triangle the ratio of a side to the sine of its opposite angle is constant, and that ratio equals the circumcircle’s diameter. One side with its opposite angle therefore fixes the whole triangle, and the third angle follows from the 180° sum.
Your numbers
Side b
14.1cm
Side c
15.027cm
Angle C
75°
Circumradius
7.779cm
Formula
Side b = Side a × sin Angle B ÷ sin Angle A
Two sides and a non-included angle can yield two valid triangles. Taking one side and two angles, as here, avoids that ambiguity.
What to enter
| Input | Default | Accepted range |
|---|---|---|
| Side a (cm)The length of the first side. | 10 | 0.01 and up |
| Angle A (°)The angle opposite side a. | 40 | 0.1 ~ 179 |
| Angle B (°)The angle opposite side b. | 65 | 0.1 ~ 179 |
Step by step
Quick reference table
Results when only Side a (cm) changes and everything else stays put.
| Side a (cm) | Side b (cm) | Side c (cm) | Angle C (°) |
|---|---|---|---|
| 5 | 7.05 | 7.514 | 75 |
| 7.5 | 10.575 | 11.27 | 75 |
| 10 | 14.1 | 15.027 | 75 |
| 15 | 21.149 | 22.541 | 75 |
| 20 | 28.199 | 30.054 | 75 |
What each result means
| Result | At default values |
|---|---|
| Side b (cm)The length of the second side. | 14.1 |
| Side c (cm)The third side; it must be shorter than the other two combined. | 15.027 |
| Angle C (°)The angle opposite side c; the three add to 180°. | 75 |
| Circumradius (cm)Radius of the circle through every vertex, wrapping the shape. | 7.779 |
Common mistakes
Two sides and a non-included angle can yield two valid triangles. Taking one side and two angles, as here, avoids that ambiguity.
Glossary
- Side a
- The length of the first side.
- Angle A
- The angle opposite side a.
- Angle B
- The angle opposite side b.
- Side b
- The length of the second side.
- Side c
- The third side; it must be shorter than the other two combined.
- Angle C
- The angle opposite side c; the three add to 180°.
- Circumradius
- Radius of the circle through every vertex, wrapping the shape.
Frequently asked questions
Q. How is Law of Sines calculated?
Side b = Side a × sin Angle B ÷ sin Angle A — In any triangle the ratio of a side to the sine of its opposite angle is constant, and that ratio equals the circumcircle’s diameter. One side with its opposite angle therefore fixes the whole triangle, and the third angle follows from the 180° sum.
Q. Can you walk through an example?
With Side a 10cm, Angle A 40°, Angle B 65°, the answer is Side b 14.1cm.
Q. What do I need to enter?
Enter Side a, Angle A, Angle B. 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) 5 → Side b (cm) 7.05 and Side a (cm) 20 → Side b (cm) 28.199. 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?
Two sides and a non-included angle can yield two valid triangles. Taking one side and two angles, as here, avoids that ambiguity.
Related calculators
Keep your length units consistent across inputs — mixing centimetres and metres throws results far off.