About the Law of Sines Calculator
This law of sines calculator solves a triangle when you know a side and its opposite angle plus one more piece of information. Choose AAS (two angles and a non-included side), ASA (two angles and the side between them) or SSA (two sides and an angle that is not between them), and it returns all three sides, all three angles and the area.
SSA is the famous ambiguous case: the same three measurements can describe no triangle, exactly one triangle, or two different triangles. The calculator checks which situation you are in and, when there are two valid triangles, reports both so you can pick the one that matches your drawing or problem statement.
Angles are in degrees, and sides can be in any unit as long as you use the same unit throughout. Standard labelling is used: side a is opposite angle A, side b opposite B and side c opposite C.
How to use the law of sines calculator
- 1Pick the case that matches what you know: AAS, ASA or SSA.
- 2Enter the angles in degrees and the side lengths in one consistent unit.
- 3Read the missing sides and angles, plus the area.
- 4For SSA, check the number of triangles — if it is 2, both solutions are listed.
Formula and method
The law of sines says the ratio of each side to the sine of its opposite angle is the same for all three pairs (and equals the diameter of the circumcircle). Once you know one complete side–angle pair, any other side or angle follows: b = a·sin B ÷ sin A, and the third angle is always 180° minus the other two.
For SSA, sin B = b·sin A ÷ a. If that value is greater than 1 there is no triangle. Otherwise B can be the acute angle arcsin(…) or its supplement 180° − arcsin(…); each is valid only if it leaves a positive angle C. That is why SSA can produce zero, one or two triangles. Area is ½·a·b·sin C.
- a, b, c
- Side lengths
- A, B, C
- Angles opposite those sides (degrees)
Worked examples
AAS: A = 40°, B = 60°, a = 10
C = 180° − 40° − 60° = 80°. The common ratio is 10 ÷ sin 40° ≈ 15.557, so b = 15.557 × sin 60° ≈ 13.47 and c = 15.557 × sin 80° ≈ 15.32. The area is about 66.34.
SSA ambiguous case: a = 6, b = 8, A = 35°
sin B = 8 × sin 35° ÷ 6 ≈ 0.7648, so B ≈ 49.89° or 130.11°. Both leave a positive third angle, so there are two triangles: one with C ≈ 95.11° and c ≈ 10.42, and a narrow one with C ≈ 14.89° and c ≈ 2.69.
ASA: A = 50°, B = 70°, c = 12
C = 60°, so the ratio is 12 ÷ sin 60° ≈ 13.856. Then a = 13.856 × sin 50° ≈ 10.61 and b = 13.856 × sin 70° ≈ 13.02.
SSA with no solution
sin B = 10 × sin 40° ÷ 3 ≈ 2.14, which is impossible because a sine can never exceed 1. Side a is too short to close the triangle.
Frequently asked questions
What is the ambiguous case of the law of sines?+
When you know two sides and a non-included angle (SSA), the unknown angle comes from arcsin, which has two possible values between 0° and 180°. Depending on the lengths, zero, one or two triangles fit the data.
How do I know if SSA gives two triangles?+
If angle A is acute and side a is longer than the height b·sin A but shorter than side b, there are two triangles. If a is at least b, or a exactly equals b·sin A (a right triangle), there is exactly one. If A is 90° or more, you need a > b for any triangle at all.
When should I use the law of cosines instead?+
If you know three sides (SSS) or two sides with the angle between them (SAS), you have no side–angle pair, so start with the law of cosines. After one angle is found, the law of sines can finish the job.
What does the common ratio a/sin A mean?+
It equals 2R, the diameter of the circle that passes through all three vertices of the triangle. This is why the ratio is the same for every side–angle pair.
Can the law of sines be used for right triangles?+
Yes. With C = 90°, sin C = 1, so c = a ÷ sin A — the familiar hypotenuse relationship. For right triangles the right triangle calculator is usually quicker.