Law of Sines Calculator
Solve any triangle from ASA, AAS or SSA using a/sin A = b/sin B = c/sin C — with the ambiguous case handled explicitly and every solution drawn to scale.
What is the law of sines?
The law of sines — sometimes called the sine rule — says that in any triangle, the ratio of a side to the sine of its opposite angle is the same for all three sides:
That shared value equals 2R, the diameter of the triangle's circumscribed circle. The rule is the natural tool for the three cases where you already know a side paired with its opposite angle: ASA (two angles and the side between them), AAS (two angles and a side not between them), and the famous SSA ambiguous case (two sides and a non-included angle). For SAS or SSS — where no side is paired with its opposite angle — use the full Triangle Calculator, which applies the law of cosines to handle those cases.
Law of sines, case by case
ASA — two angles and the side between them
You know both angles at the ends of a side. Find the third angle by C = 180° − A − B, then read every other side off the shared ratio c / sin C = a / sin A = b / sin B.
AAS — two angles and a non-included side
Same idea as ASA, but the known side is opposite one of the angles instead of between them. Find the third angle first, then use the pair that includes your known side to build the common ratio.
SSA — two triangles (the classic ambiguous case)
You know two sides and an angle opposite one of them. When the given angle is acute and the opposite side is shorter than the adjacent one but longer than the altitude, sin B has two valid angles — the acute solution and its 180° − B supplement — so two real triangles fit.
SSA — exactly one triangle
When the supplement of B would push A + B past 180°, the obtuse companion is discarded and only the acute solution survives. This also happens when sin B works out to exactly 1, giving a right angle at B.
SSA — no triangle at all
If the computed sin B is greater than 1, no angle can satisfy it — side b is simply too long to close on side a at angle A. The calculator flags this instead of silently returning NaN.
Why the law works — a quick altitude proof
Drop the altitude h from vertex C to side AB. In the right triangle on the left, sin A = h / b, so h = b · sin A. On the right, sin B = h / a, so h = a · sin B. Equating the two expressions for h gives a · sin B = b · sin A, i.e. a / sin A = b / sin B. Repeat with a different altitude to include c / sin C.
Law of sines vs. law of cosines — when to use which
| You know… | Use | Why |
|---|---|---|
| Two angles + included side (ASA) | Law of sines | You already have a matched side–angle pair after finding the third angle. |
| Two angles + non-included side (AAS) | Law of sines | The known side is opposite one of the known angles. |
| Two sides + non-included angle (SSA) | Law of sines (ambiguous) | Handled here; the angle is opposite one of the sides. |
| Two sides + included angle (SAS) | Law of cosines | No matched side–angle pair to anchor the ratio. |
| Three sides (SSS) | Law of cosines | You need cosines to recover the angles first. |
| One angle = 90° | Right-triangle trig | Faster with the Right Triangle Calculator. |
What this tool does for you
- Solves ASA, AAS and SSA using the law of sines — every side, every angle, area, perimeter, semiperimeter, inradius and circumradius returned in one go.
- Handles the SSA ambiguous case explicitly: classifies no-triangle, one-triangle and two-triangle inputs, then shows both valid triangles side-by-side with their own diagrams.
- Draws each solution to scale from the solved side lengths, so the geometry — not just the numbers — makes the ambiguity obvious.
- Accepts input in degrees or radians with a one-click unit toggle.
- Prints personalised step-by-step working with your own numbers substituted into a / sin A = b / sin B = c / sin C.
- Skips SAS and SSS on purpose — those need the law of cosines and are handled by the full Triangle Calculator we link to.
- Copy result / download PNG / download PDF / print, matching the shared result-actions used across the site.
Frequently asked questions
+What is the law of sines in simple terms?
In any triangle, each side divided by the sine of its opposite angle gives the same number: a/sin A = b/sin B = c/sin C. That shared ratio equals 2R, the diameter of the circumscribed circle. Because the ratio is fixed for a given triangle, knowing any matched side–angle pair lets you solve for the other sides once you know their opposite angles.
+When should I use the law of sines instead of the law of cosines?
Use the law of sines when the values you know pair sides with their opposite angles — that's the ASA, AAS and SSA cases. Reach for the law of cosines when you know two sides and the angle between them (SAS) or all three sides (SSS): those have no matched side–angle pair, so the sine ratio has no starting point.
+What is the ambiguous case?
The ambiguous case is SSA: you know two sides and an angle that is not between them. Depending on the numbers, the given angle can produce no triangle at all, exactly one triangle, or two different triangles that both fit. This calculator classifies which of the three you have and shows both triangles side by side when there are two.
+Why do some SSA inputs give two answers?
Because arcsin(x) can return two valid angles between 0° and 180° — an acute angle B and its supplement B' = 180° − B. Both produce a real triangle as long as A + B' is still less than 180°. If the supplement pushes the angle sum past 180°, the second triangle is discarded and only the acute solution survives.
+Can I use this calculator for right triangles?
Yes — the law of sines works for right triangles too, and this tool will solve them. If you already know the triangle is right, the dedicated Right Triangle Calculator is faster because it uses sine, cosine, tangent and the Pythagorean theorem directly instead of the full ratio.
+What if my inputs describe an SAS or SSS triangle?
The law of sines can't start those — there is no side–angle pair to anchor the ratio. Use the full Triangle Calculator instead: it applies the law of cosines to SAS and SSS and returns the same full set of derived values (sides, angles, area, perimeter, R and r).