Sine Rule Calculator
Given one side, its opposite angle, and a second angle, find the side opposite the second angle (Law of Sines).
Solving Triangles That Aren't Right-Angled
Most real triangles don't include a convenient 90° angle, so the ordinary right-triangle trig ratios don't apply directly. The sine rule (law of sines) fixes that: it links every side of a triangle to the sine of its opposite angle, letting you find a missing side once you know one side and two angles.
The Formula
a / sin(A) = b / sin(B)
b = a × sin(B) / sin(A)
b = a × sin(B) / sin(A)
The third angle follows from the angle sum of a triangle: C = 180° − A − B.
Worked Example
Given side a = 10, angle A = 30°, and angle B = 45°:
| Step | Calculation | Result |
|---|---|---|
| sin(A), sin(B) | sin(30°), sin(45°) | 0.5, 0.7071 |
| Side b | 10 × 0.7071 / 0.5 | 14.1421 |
| Angle C | 180 − 30 − 45 | 105° |
Where This Matters
- Surveying and triangulation — finding a distance to an inaccessible point using two measured angles and one known baseline.
- Navigation — determining distances between ships, aircraft, or landmarks from bearing angles.
- Engineering trusses — solving member lengths in a non-right-angled triangular frame.
Note: The sine rule can produce an ambiguous case (two possible triangles) when solving for an angle from two sides and a non-included angle. This calculator's side-from-two-angles setup avoids that ambiguity entirely.
How to Use This Calculator
- Enter Side a, the length of the known side.
- Enter Angle A, the angle opposite side a, in degrees.
- Enter Angle B, the angle opposite the side you want to find, in degrees.
- Select Calculate to get side b and the remaining angle C.
Related Calculations
Working from two sides and an included angle instead? Use the Cosine Rule Calculator, or solve every side and angle at once with the Triangle Solver.