Law of Sines Calculator

The law of sines states that in any triangle the ratio of each side to the sine of its opposite angle is the same: a / sin(A) = b / sin(B) = c / sin(C). Side a faces angle A, side b faces angle B, and side c faces angle C. You can use it to find a missing side when you know an angle and its opposite side, or to find a missing angle when you know a side and its opposite side.

It solves the three cases where the law of sines applies. AAS uses two angles and a side that is not between them. ASA uses two angles and the side between them. SSA uses two sides and an angle opposite the first side. For SAS and SSS triangles you need the law of cosines instead, since the law of sines alone cannot start those.

SSA is called the ambiguous case because two sides and a non-included angle can describe zero, one, or two different triangles. The calculator computes sin(B) = b * sin(A) / a. If that value is greater than 1, no triangle fits. If it equals 1, there is one right-angled triangle. If it is less than 1, angle B can be the acute value or its supplement (180 minus that value), and each is kept only when the angles still add up to less than 180 degrees.

Each side is named for the angle directly across from it, which is the convention the law of sines relies on. Side a sits opposite angle A, side b opposite angle B, and side c opposite angle C. When you enter SSA values, the angle you provide must be the one opposite the first side you enter, or the result will not match your triangle.

Once all three angles and sides are known, the area comes from the side-angle-side formula Area = (1/2) * a * b * sin(C), where C is the angle between sides a and b. The circumradius, the radius of the circle through all three corners, comes from the same law of sines relationship: 2R = a / sin(A). Both are shown for each valid triangle, so the two SSA solutions each get their own area and circumradius.

Yes, enter every angle in degrees, and the results report angles in degrees as well. The three interior angles of any triangle always sum to 180 degrees, so the calculator rejects any pair of angles that adds up to 180 or more before it tries to solve.