Law of Sines Calculator
Enter two angles and a side (AAS or ASA), or two sides and an angle opposite one of them (SSA). The calculator uses a/sin A = b/sin B = c/sin C to find every missing side and angle, plus the area and perimeter. For SSA it also checks whether 0, 1 or 2 triangles fit.
Enter angles A and B and the side a opposite angle A.
How to use the Law of Sines Calculator
- Pick the case that matches your data. AAS is two angles and a side opposite one of them, ASA is two angles and the side between them, and SSA is two sides and an angle opposite one of those sides.
- Label the triangle so side a is opposite angle A, side b opposite angle B and side c opposite angle C. Enter angles in degrees.
- Press Calculate. You get all three sides, all three angles, the area and the perimeter. In SSA mode the result can be no triangle, one, or two, and each one is solved in full.
The law of sines formula
a / sin(A) = b / sin(B) = c / sin(C)
Divide any side by the sine of the angle opposite it and you get the same number for all three pairs. That shared value is the diameter of the circle through the three corners. To use the rule you need one full pair, an angle and the side across from it. AAS hands you that pair. ASA gives it after one subtraction, because C = 180° − A − B and side c is opposite C. SSA gives it directly, but the unknown angle comes out of an inverse sine, and that is why two answers are possible.
Worked example (AAS)
Given A = 40°, B = 60° and a = 10. The third angle is C = 180° − 40° − 60° = 80°. The shared ratio is a / sin A = 10 / 0.6428 = 15.557. Multiply it by the other two sines: b = 15.557 × sin 60° = 13.47 and c = 15.557 × sin 80° = 15.32. Enter the same values in AAS mode above to check.
Which triangles the law of sines can solve
The law of sines works whenever you know at least one angle together with the side opposite it. The table lists the five ways a triangle can be given and which law to start with.
| You know | Case | Start with | Triangles |
|---|---|---|---|
| Two angles and a side opposite one of them | AAS | Law of sines | 1 |
| Two angles and the side between them | ASA | Angle sum, then law of sines | 1 |
| Two sides and an angle opposite one of them | SSA | Law of sines | 0, 1 or 2 |
| Two sides and the angle between them | SAS | Law of cosines | 1 |
| Three sides | SSS | Law of cosines | 0 or 1 |
Example 2: ASA (A = 50°, B = 70°, c = 12)
Side c sits between the two known angles, so it is opposite the unknown angle C. Find C first: 180° − 50° − 70° = 60°. Now you have a full pair, and c / sin C = 12 / 0.8660 = 13.856.
a = 13.856 × sin 50° = 10.61 and b = 13.856 × sin 70° = 13.02.
Example 3: SSA with two answers (A = 35°, a = 9, b = 12)
sin B = 12 × sin 35° / 9 = 0.7648. Two angles have that sine, B = 49.89° and B = 130.11°, and both leave room for angle C. The height test agrees. h = 12 × sin 35° = 6.88, which is shorter than a = 9, and a is shorter than b = 12.
Triangle 1 has B = 49.89°, C = 95.11° and c = 15.63.
Triangle 2 has B = 130.11°, C = 14.89° and c = 4.03.
The calculator lists both. The SSA guide on the triangle calculator page explains when you get 0, 1 or 2 triangles.
Frequently asked questions
When do you use the law of sines instead of the law of cosines?
Use the law of sines when you know an angle and the side opposite it, which covers AAS, ASA and SSA. If you know two sides and the angle between them (SAS), or all three sides (SSS), no such pair exists yet, so start with the law of cosines.
Why does the law of sines sometimes give two triangles?
In SSA you solve for an angle from its sine, and every sine below 1 belongs to two angles, B and 180° − B. For A = 35°, a = 9 and b = 12, sin B = 0.7648 fits both 49.89° and 130.11°. Both leave a positive angle C, so two triangles exist. When you solve for a side, as in AAS and ASA, this never comes up.
Does the law of sines work for right triangles?
Yes. If C = 90°, then sin C = 1 and the rule becomes a / sin A = c, so sin A = a / c. That is the opposite-over-hypotenuse definition of sine. The right triangle calculator handles that case with fewer inputs.
Can the law of sines find the angles from three sides?
No. Each ratio in the formula needs an angle, and three sides give you none. Use the law of cosines for SSS, for example cos A = (b² + c² − a²) / 2bc.