The Triangle Law of Sines Calculator solves a triangle when two angles and a side are known (ASA or AAS), or when two sides and a non-included angle are known (SSA). Enter the known measurements, and the tool applies the law of sines to find the remaining sides and angles, flagging the ambiguous case when it arises.
The law of sines relates every side of a triangle to the sine of the angle directly opposite it, which makes it the direct method whenever a known angle and its opposite side appear together in a problem.
Apply the law of sines
Every side of a triangle divided by the sine of its opposite angle gives the same ratio across the whole triangle.
a / sin(A) = b / sin(B) = c / sin(C)Knowing one full side-angle pair sets that ratio for the entire triangle. Any other known angle then gives its opposite side directly, and vice versa.
Solve a triangle from two angles and a side (ASA or AAS)
When two angles are known, the third follows immediately from the angle sum, since all three interior angles of any triangle total 180 degrees. The law of sines then finds the two remaining sides. Worked example: a triangle with A = 50°, B = 60°, and side a = 10 (opposite angle A).
1. Find the third angle. C = 180 − 50 − 60 = 70°.
2. Set up the ratio using the known pair. a / sin(A) = 10 / sin(50°) ≈ 10 / 0.766 ≈ 13.05.
3. Find side b. b = 13.05 × sin(60°) ≈ 13.05 × 0.866 ≈ 11.30.
4. Find side c. c = 13.05 × sin(70°) ≈ 13.05 × 0.940 ≈ 12.26.
This method is direct and gives a single, unambiguous answer whenever the known pair includes both a side and its opposite angle.
Solve the ambiguous SSA case
When two sides and an angle not between them are known, the law of sines can produce two different valid triangles, one answer, or no valid triangle at all, depending on how the numbers compare. This is called the ambiguous case.
Worked example: side a = 8, side b = 10, and angle A = 40° (opposite side a, not between the two given sides).
1. Set up the ratio to find angle B. sin(B) = b × sin(A) / a = 10 × sin(40°) / 8 ≈ 10 × 0.643 / 8 ≈ 0.803.
2. Take the inverse sine. B ≈ 53.4° or, since sine is positive in both the first and second quadrant, B could also be 180 − 53.4 = 126.6°.
3. Check both possibilities against the angle sum. If B = 53.4°, then C = 180 − 40 − 53.4 = 86.6°, a valid triangle. If B = 126.6°, then C = 180 − 40 − 126.6 = 13.4°, which is still positive and also valid.
Both triangles are legitimate solutions to the same three given measurements, which is exactly why this case is called ambiguous: the given information alone does not pin down a single triangle.
Know when SSA has no solution
If the calculated value of sin(B) in the process above exceeds 1, no real angle exists, meaning no triangle can be formed from the given measurements at all. This happens when the side opposite the known angle is too short relative to the other given side to reach around and close the triangle.
Recognize which cases the law of sines solves directly
| Case | You know | Result |
|---|---|---|
| ASA | two angles, the side between them | single answer |
| AAS | two angles, a side not between them | single answer |
| SSA | two sides, an angle not between them | zero, one, or two answers |
ASA and AAS always resolve to exactly one triangle, since knowing two angles fixes the third immediately and leaves no room for a second interpretation. SSA is the one combination that needs the extra check described above.
Frequently asked questions
What is the law of sines formula?
a / sin(A) = b / sin(B) = c / sin(C), where each side is divided by the sine of the angle directly opposite it, and all three ratios are equal within one triangle.
Why is SSA called the ambiguous case?
Because knowing two sides and a non-included angle can produce two different valid triangles, since the inverse sine function has two possible angles (one acute, one obtuse) that share the same sine value.
How do you know if SSA has zero, one, or two solutions?
Calculate sin(B) using the law of sines ratio. If it exceeds 1, there is no solution. If exactly 1, there is exactly one right-triangle solution. Otherwise, check whether both the acute and obtuse angle options keep all three angles positive; if both do, there are two solutions.
When should I use the law of sines instead of the law of cosines?
Use the law of sines whenever a known angle and its opposite side appear together in the given information, which covers ASA, AAS, and SSA. Use the law of cosines for SSS and SAS instead.
Do ASA and AAS ever have an ambiguous result?
No. Once two angles are known, the third is fixed immediately by the 180-degree angle sum, leaving no ambiguity in either case.
Can the law of sines find a triangle's area?
Not directly, but once all three sides or two sides and the included angle are known from a law of sines solution, the standard area formulas, including ½ab·sin(C), can be applied.
What if my SSA triangle only has one valid solution instead of two?
This happens when only one of the two candidate angles (from the acute or obtuse inverse sine result) keeps the three interior angles summing to less than 180° with all positive values; the other candidate is discarded as geometrically invalid.
Summary
The law of sines, a / sin(A) = b / sin(B) = c / sin(C), solves ASA and AAS triangles directly and without ambiguity, such as finding sides of about 11.30 and 12.26 from angles 50°, 60° and a side of 10.
The SSA case is different: it can yield zero, one, or two valid triangles, since the inverse sine function returns both an acute and an obtuse candidate angle that must each be checked against the 180-degree angle sum.