Law of Sines Calculator: Solve AAS, ASA & SSA Triangles
Solve any triangle from two angles and a side (AAS/ASA) or two sides and a non-included angle (SSA) using the law of sines — includes the ambiguous SSA case.
Law of sines formula
The law of sines relates every side of a triangle to the sine of its opposite angle:
Any complete side/angle pair — one known side together with its OPPOSITE angle — lets you solve for anything else, as long as you have one more piece of information:
| Known | Case | Solve for |
|---|---|---|
| Two angles + one side | AAS or ASA | The remaining side(s) — always exactly one triangle |
| Two sides + a non-included angle | SSA | The remaining angle — 0, 1, or 2 triangles (the ambiguous case) |
Solving AAS / ASA (two angles + one side)
Example: angle A = 40°, angle B = 60°, side a = 10 (opposite A)
First, the third angle: . Then solve for side using the ratio anchored on the known pair :
Side follows the same way, and with all three sides known the area comes from Heron's formula. This case is always exactly one triangle — knowing two angles fixes the triangle's shape completely, and one side fixes its size.
Solving SSA — the ambiguous case
Example: side a = 6, side b = 8, angle A = 40° (opposite side a, NOT between a and b)
Solve for the other side's opposite angle: , so or . Both keep under 180°, so both are valid triangles:
- Solution 1: B ≈ 58.99°, C ≈ 81.01°, c ≈ 9.22
- Solution 2: B ≈ 121.01°, C ≈ 18.99°, c ≈ 3.04
Change just the numbers and the case flips entirely: a = 15, b = 10, angle A = 40° gives exactly ONE triangle (a ≥ b, so the supplementary angle would overshoot 180°), while a = 3, b = 10, angle A = 40° gives NO triangle at all — the ratio makes , which no real angle can have.
Common mistakes
- Pairing a side with the wrong angle. The law of sines ratio only works with a side and its OPPOSITE angle — side a pairs with angle A, not with B or C.
- Dropping the second SSA solution. When the ambiguous case applies, reporting only one triangle silently throws away a mathematically valid answer.
- Assuming SSA always has a solution. If the known side is too short relative to the other side and angle, no triangle can close — check that before trusting the result.
Where this shows up
- Surveying and navigation: fixing a distant point's position from a measured baseline and two angles (triangulation).
- Astronomy: parallax measurements that resolve to exactly a law-of-sines triangle.
- Engineering: truss and frame analysis where two angles and one member length are known.
The law of sines is the natural partner to the law of cosines — together they solve every possible triangle case (SSS, SAS, ASA, AAS, SSA). Its sine-based ratio also underlies the right triangle calculator's own trigonometric side-solving, and the isosceles triangle case is a special instance where two of the three side-angle pairs are already equal.
Frequently asked questions
- What is the law of sines?
- A relationship between a triangle's sides and the sines of their opposite angles: . This ratio is the same for all three side-angle pairs, and it equals the diameter of the triangle's circumscribed circle.
- When do I use the law of sines instead of the law of cosines?
- Use the law of sines whenever you have at least one complete side-angle pair (a side and its OPPOSITE angle) plus one more piece of information — that covers AAS, ASA, and SSA. The law of cosines instead handles SSS and SAS, where no side/angle pair is available.
- What is the SSA ambiguous case?
- Given two sides and a non-included angle (the angle NOT between them), there can be zero, one, or two valid triangles — because sine gives the same value for an angle θ and its supplement 180°−θ. This calculator checks both candidate angles and shows every triangle that actually closes (angle sum exactly 180°).
- How many triangles does SSA with a = 6, b = 8, angle A = 40° produce?
- Two. Solving sinB = (b·sinA)/a gives B ≈ 58.99°, but 180°−58.99° = 121.01° also keeps A+B under 180°, so both are valid — Solution 1 has C ≈ 81.01° and Solution 2 has C ≈ 18.99°, each a genuinely different triangle.
- How do I find a missing side from two angles and one side (AAS/ASA)?
- First find the third angle: . Then use the ratio to solve for any remaining side. For , , : .
- Why does SSA sometimes have NO solution?
- If the known angle's opposite side is too short to ever reach the other known side, sinB from the ratio comes out greater than 1 — no real angle has that sine, so no triangle exists. For a=3, b=10, angle A=40°, sinB ≈ 2.14, which is impossible.
- Is there ever exactly ONE SSA solution instead of zero or two?
- Yes — when the known side (a) is at least as long as the other side (b), only the smaller candidate angle keeps the triangle valid (the supplement would push the angle sum past 180°). It also happens at the exact boundary a = b·sinA, which gives a single right triangle.
- Does the law of sines also give the triangle's area?
- Yes — once every side and angle is solved, the area follows from Heron's formula using the three sides, or equivalently using any two sides and their included angle.