Law of Sines Calculator

Enter two angles and a side, or two sides and an angle, to solve a triangle using the law of sines.

How to Use the Law of Sines Calculator

In Two Angles & Side mode, enter angles A and B plus the side a opposite angle A. In Two Sides & Angle mode, enter sides a and b plus angle A (opposite side a) instead — this second mode can return zero, one, or two valid triangles.

The law of sines ties every side of a triangle to the sine of its opposite angle, and keeps that ratio constant across all three pairs:

a ÷ sin(A) = b ÷ sin(B) = c ÷ sin(C)

Given two angles and one side, the third angle is just 180° minus the other two, and the missing sides fall straight out of the ratio above. That case is always unambiguous — exactly one triangle fits.

The Ambiguous Case (SSA)

Two sides and a non-included angle is a different story. Solving for angle B via sin(B) = b × sin(A) ÷ a can land on a value above 1 (no triangle fits), exactly 1 (one right-angled triangle fits), or a valid sine that corresponds to two different angles — one acute, one obtuse — each completing a separate, equally valid triangle. This calculator checks all three outcomes and lists every triangle that actually closes (angle sum under 180°), rather than assuming there's only ever one answer.

It's the one case in triangle-solving where "enter three values, get one triangle" quietly breaks down — which is exactly why SSA doesn't get its own congruence rule in geometry class the way SAS and ASA do.