Sine Rule Calculator (Law of Sines)
Apply the sine rule to find a side or an angle, including the ambiguous case.
The sine rule
The sine rule says that in any triangle, each side divided by the sine of its opposite angle is the same:
a ÷ sin A = b ÷ sin B = c ÷ sin C
Use it when you know two angles and a side (to find the other sides), or two sides and an angle opposite one of them (to find another angle).
Worked example: two angles and a side
A = 40°, B = 70° and a = 8. The third angle is C = 180° − 40° − 70° = 70°. Then b = a × sin B ÷ sin A = 8 × sin 70° ÷ sin 40° = 11.6952, and c is the same because the two angles are equal.
The ambiguous case
When you know two sides and an angle opposite one of them (SSA), the sine rule gives sin B = b × sin A ÷ a, and there are three possibilities:
| Result | Number of triangles |
|---|---|
| sin B is more than 1 | None: the side is too short |
| sin B equals 1 | One, with a right angle at B |
| sin B is less than 1 and a ≥ b | One |
| sin B is less than 1 and a < b | Two: B and 180° − B, when both fit |
With a = 6, b = 8 and A = 30°, sin B = 0.6667, so B = 41.81° or 138.19°. Both give valid triangles, with c = 11.40 or c = 2.46. The solver shows both, and draws them.
The drawing has Download PNG and Download SVG buttons, so you can save it for homework, a report or a slide.
Tips
- Use the sine rule when you know a side and its opposite angle. Use the cosine rule when you know three sides, or two sides and the included angle.
- To avoid the ambiguous case, find the largest angle first with the cosine rule.
- Angles in a triangle add up to 180°, so the third angle is easy.
How to cite this page
CalcPulse. (2026). Sine Rule Calculator. https://calcpulse.com/sine-rule-calculator/
Frequently asked questions
What is the sine rule?
a ÷ sin A = b ÷ sin B = c ÷ sin C. It links each side of a triangle to the sine of the opposite angle.
When do I use the sine rule?
When you know two angles and a side, or two sides and an angle opposite one of them.
What is the ambiguous case of the sine rule?
With two sides and a non-included angle (SSA), there can be two triangles. The second uses 180° minus the first angle you find.
How do I find a side with the sine rule?
Rearrange it: b = a × sin B ÷ sin A. Use the 'Two angles and a side' tab.
What if sin B is greater than 1?
Then no triangle exists, because the side is too short to reach the other side. The solver tells you.
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