Law of Sines Calculator (Solve a Triangle, AAS & ASA)
By Santaji GadepatilinUpdated November 4, 2026
The law of sines says that each side of a triangle divided by the sine of its opposite angle gives the same ratio. This law of sines calculator uses that rule to solve a triangle from one side and two angles (AAS or ASA), returning the third angle, both missing sides and the area.
How to use the law of sines calculator
Enter one side a and the two angles A (opposite a) and B. The calculator finds the third angle C = 180° − A − B, then sides b and c from the law of sines, and the area. For the SSA case, clear angle B and enter side b instead — the calculator solves sin B and returns the acute solution (see the ambiguous case below).
Law of sines formula
a ÷ sin A = b ÷ sin B = c ÷ sin C · Area = ½ a b sin C
Example
Side a = 10, angle A = 30°, angle B = 45°:
- C = 180° − 30° − 45° = 105°
- b = 10 × sin 45° ÷ sin 30° = 10 × 0.7071 ÷ 0.5 = 14.142
- c = 10 × sin 105° ÷ sin 30° = 10 × 0.9659 ÷ 0.5 = 19.319
- Area = ½ × 10 × 14.142 × sin 105° = 68.30
Same triangle, different angle B (a = 10, A = 30°)
| Angle B | Angle C | Side b | Side c | Area |
|---|---|---|---|---|
| 15° | 135° | 5.176 | 14.142 | 18.30 |
| 30° | 120° | 10.000 | 17.321 | 43.30 |
| 45° | 105° | 14.142 | 19.319 | 68.30 |
| 60° | 90° | 17.321 | 20.000 | 86.60 |
| 75° | 75° | 19.319 | 19.319 | 93.30 |
| 90° | 60° | 20.000 | 17.321 | 86.60 |
| 105° | 45° | 19.319 | 14.142 | 68.30 |
| 120° | 30° | 17.321 | 10.000 | 43.30 |
The ambiguous case (SSA)
When you know a, b and A, sin B = b sin A ÷ a. If that value is above 1 there is no triangle; if it equals 1 there is one right triangle; if it is below 1, B can be acute or 180° minus that angle, and both give a valid triangle whenever A + B stays below 180°. For three known sides use the law of cosines calculator.
Frequently asked questions
What is the law of sines?
a ÷ sin A = b ÷ sin B = c ÷ sin C. In any triangle, the ratio of a side to the sine of the angle opposite it is the same for all three sides (and equals the diameter of the circumscribed circle).
When should I use the law of sines instead of the law of cosines?
Use the law of sines when you know two angles and a side (AAS, ASA) or two sides and a non-included angle (SSA). Use the law of cosines when you know three sides (SSS) or two sides and the included angle (SAS).
What is the ambiguous case of the law of sines?
With two sides and a non-included angle (SSA) there can be two different triangles, because sin B = sin(180° − B). Check whether 180° − B also leaves a positive third angle; if it does, both triangles are valid.
Do the sides need units?
Any length unit works — the missing sides come out in the same unit you used for side a, and the area in that unit squared.