What this calculator does
The law of sines says each side of a triangle is proportional to the sine of the angle opposite it. Given two angles and any side, that single relationship determines the entire triangle.
Two angles fix the third immediately, since they sum to 180 degrees. Angles of 30 and 60 leave 90.00 degrees, which means the calculator has just solved a right triangle without being told it was one.
The formula
The third angle is what remains of 180 degrees. The ratio of the known side to the sine of its opposite angle is constant, and multiplying it by the sine of each other angle gives the other sides.
| Term | Meaning |
|---|---|
| Law of sines | Each side divided by the sine of its opposite angle gives the same constant. |
| AAS and ASA | Two angles and a side, which the law of sines solves unambiguously. |
| Ambiguous case | Two sides and a non-included angle, which can give two valid triangles. |
The inputs explained
| Field | What to enter |
|---|---|
| Angle A (°) | Angle A in degrees. |
| Angle B (°) | Angle B in degrees. The two angles must sum to less than 180. |
| Side a (opposite angle A) | The side opposite angle A. |
When to use it
Surveying and navigation
Two measured bearings and one measured distance determine everything else.
Solving a non-right triangle
Basic trigonometry only handles right triangles; this handles any of them.
Finding an inaccessible distance
Measuring two angles from a known baseline gives the unreachable side.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How does angle A reshape the triangle?
The same side and angle B at three values of angle A.
Questions
When does the law of sines fail?
It does not fail so much as become ambiguous. Given two sides and an angle not between them, two different triangles can satisfy the same data, which is the ambiguous SSA case. Two angles and a side, as here, is always unambiguous.
When should I use the law of cosines instead?
When you have three sides, or two sides and the angle between them. The law of sines needs an angle paired with its opposite side, and those cases do not provide one.
Why does the ratio stay constant?
Because each side over the sine of its opposite angle equals the diameter of the triangle's circumscribed circle. That is the extended law of sines, and it explains why a single ratio governs the whole triangle.
Do the angles have to be in degrees?
This calculator takes degrees. The underlying relationship is unit-agnostic, but mixing degrees and radians is a common source of wrong answers in hand calculations.
For general triangle measurements, see the triangle area calculator. For angle basics, see the trigonometry calculator.