Law of Sines Calculator
Solve an oblique triangle for a missing side or angle using the law of sines from one known side-and-opposite-angle pair.
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Solved with the law of sines
- 1
Common ratio R = side ÷ sin(angle)
8 ÷ sin(45°) = 8 ÷ 0.707107 = 11.3137This ratio equals a/sin A = b/sin B = c/sin C for every pair in the triangle. - 2
Missing side = R × sin(opposite angle)
11.3137 × sin(60°) = 11.3137 × 0.866025 = 9.798
How does this calculator work?
The law of sines states a/sin A = b/sin B = c/sin C. Enter one known side with its opposite angle to set the common ratio, then the calculator finds any missing side as ratio × sin(angle) or any missing angle as the inverse sine of side ÷ ratio, reporting results in degrees.
Formula
How this is calculated
Enter what you know for the three sides (a, b, c) and their opposite angles (A, B, C), in degrees. Leave any unknown box blank. The calculator needs at least one complete pair — a side together with the angle directly across from it — to establish the common ratio R = side / sin(angle), which is identical for all three pairs in any triangle.
Once R is known, every other quantity follows. A missing side equals R × sin(its opposite angle), and a missing angle equals asin(its opposite side ÷ R), converted back to degrees. Angles are handled in radians internally and reported in degrees. The first quantity solved is shown as the hero result, and the supporting StatGrid lists the common ratio along with each value found.
Keep in mind the ambiguous case (SSA): when solving for an angle from a side, asin returns only the acute solution, so a second obtuse triangle may also exist. The ratio must be between -1 and 1 for a valid angle, and any angle whose sine is zero (0° or 180°) is rejected because it would make the ratio undefined.
Frequently asked questions
At least one complete side-and-opposite-angle pair (for example side a and angle A), plus the side or angle you want to solve. The pair fixes the common ratio used for everything else.
When you solve for an angle from a known side (the SSA case), the inverse sine gives only the acute angle. A second, obtuse triangle can also satisfy the data, so check whether the obtuse supplement fits your situation.
Use the law of cosines when you know two sides and the included angle (SAS) or all three sides (SSS), since those cases lack a ready side-opposite-angle pair for the law of sines.
Also known as
TG we-Calculate Editorial Team. (2026). Law of Sines Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/law-of-sines-calculator
TG we-Calculate Editorial Team. "Law of Sines Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/law-of-sines-calculator.
TG we-Calculate Editorial Team, "Law of Sines Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/law-of-sines-calculator
@misc{wecalculate_law_of_sines_calculator, title = {Law of Sines Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/law-of-sines-calculator}}, year = {2026}, note = {TG we-Calculate} }
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