The law of sines lets you solve any triangle — not just right triangles — as long as you know an angle-side pair plus one more value. This calculator handles the two cases where that formula applies directly (ASA/AAS and SSA), and it specifically flags the one case, SSA, where the same inputs can describe two entirely different valid triangles.

How the Law of Sines Calculator works

Every triangle obeys a/sinA = b/sinB = c/sinC — the ratio of each side to the sine of its opposite angle is constant across the whole triangle. In "Two Angles + Side" mode, the third angle falls out immediately from the 180° angle sum, and that constant ratio then gives you the two remaining sides directly. This case is never ambiguous: two angles and a side pin down the triangle's exact size and shape with no room for a second answer.

In "Two Sides + Angle" mode (SSA), the calculator solves sin(B) = b·sinA/a for the second angle. Because sine is positive in both the first and second quadrants, there are generally two mathematically valid angles that satisfy that equation — B itself, and its supplement 180° − B. The calculator checks both: a supplement is only a valid second solution if adding it to angle A still leaves the total under 180°. That's the mechanism behind zero, one, or two solutions.

Inputs and what they mean

Angle A and Angle B are entered in degrees. Side a is always opposite Angle A, and side b is always opposite Angle B — the calculator uses that convention throughout, whether you're in ASA/AAS mode (angle A, angle B, side a) or SSA mode (side a, side b, angle A). Getting the opposite-side pairing right matters: swapping which side is "opposite" which angle changes the entire solve.

In SSA mode specifically, the relationship between side a, side b, and the height h = b·sinA determines the outcome: if a < h, no triangle exists (side a is too short to ever reach side b); if a = h exactly, there's exactly one triangle and it happens to have a right angle at B; if h < a < b, there are two valid triangles; and if a ≥ b, there's exactly one triangle, because the supplementary angle would push the angle total past 180°.

Limits and edge cases

This calculator solves plane (Euclidean) triangles with exact input values — it doesn't model measurement uncertainty or spherical geometry (used, for example, in some navigation and astronomy problems). If you know two sides and the angle between them (SAS) or all three sides (SSS), the law of sines can't solve the triangle directly — those cases need the law of cosines first to find one more angle or side before the law of sines can take over.

The SSA ambiguous case is a real feature of the mathematics, not a bug: with only side-side-angle information, two different triangles genuinely satisfy the same measurements. Real-world context — an additional angle range, a diagram, or the physical situation being modeled — is what tells you which of the two solutions is the correct one.