In any triangle, each side is proportional to the sine of its opposite angle.
How to use the Law of Sines Calculator
- Side a
- Angle A (opposite a)
- Angle B
Results update as you type, and every field accepts the units shown in its menu.
How it works
Find the third angle from 180°, then scale the known side by the ratio of sines.
Inputs explained
| Field | What to enter |
|---|---|
| Side a | A number |
| Angle A (opposite a) | In ° |
| Angle B | In ° |
What each result means
- Angle C = 180 – [Angle A (opposite a)] – [Angle B]
- Side b = [Side a] × sin(rad([Angle B])) ÷ sin(rad([Angle A (opposite a)]))
- Side c = [Side a] × sin(rad([Angle C])) ÷ sin(rad([Angle A (opposite a)]))
- Circumradius = [Side a] ÷ (2×sin(rad([Angle A (opposite a)])))
Example values
How the main result changes when Side a changes and everything else stays at the example values above:
| Side a | Side b |
|---|---|
| 5 | 7.5136 |
| 7.5 | 11.2704 |
| 10 | 15.0271 |
| 12.5 | 18.7839 |
| 15 | 22.5407 |
| 20 | 30.0543 |
Formula
a ÷ sin A = b ÷ sin B = c ÷ sin C = 2RWorked example
Inputs — Side a: 10; Angle A (opposite a): 40 °; Angle B: 75 °.
Results — Angle C = 65 °; Side b = 15.027138; Side c = 14.099646; Circumradius = 7.778619.
Frequently asked questions
What is the ambiguous case?
With two sides and a non-included angle, two different triangles may fit.
Do the angles need to add to 180°?
Yes; if A + B ≥ 180°, no triangle exists.
What formula does the Law of Sines Calculator use?
a ÷ sin A = b ÷ sin B = c ÷ sin C = 2R.