Updated · Built and checked by the PlinzoTools team, formulas tested against published sources
Solve a triangle's remaining side and angle from one side-angle pair and a second angle.
The Law of Sines relates a triangle's sides to the sines of their opposite angles -- useful for solving non-right triangles when you know one full side-angle pair plus one more angle.
Enter side a, its opposite angle A, and a second angle B. The remaining side and angle are calculated instantly.
a / sin(A) = b / sin(B) = c / sin(C)
The tool takes three inputs: side a, the angle A opposite it, and a second angle B. It returns three outputs: side b (opposite angle B), angle C, and side c (opposite angle C). Angle C is found first as 180 minus A minus B. Then the ratio a / sin(A) is computed once, and both remaining sides are that ratio multiplied by the sine of their opposite angle. Angles are entered in degrees, not radians.
Use a hypothetical triangle with a = 10, A = 40 degrees and B = 60 degrees.
The tool shows sides to four decimals and angle C to two. A quick reasonableness check: the largest angle (80 degrees) faces the longest side (c), and the smallest angle (40 degrees) faces the shortest side (a). If your result breaks that pattern, an input was likely mistyped.
The tool only produces an answer when side a is positive, angle A is between 0 and 180, angle B is positive, and the two angles add to less than 180 degrees. Otherwise it shows a prompt asking for a side and two angles summing below 180. That rule is not arbitrary: the third angle must be positive for the shape to exist.
A well-known trap with the Law of Sines is the ambiguous case, where you know two sides and an angle not between them, and there may be zero, one or two valid triangles. This calculator sidesteps it by asking for two angles instead, which pins down the triangle's shape uniquely. If your problem gives two sides and a non-included angle, you cannot enter it here directly; you would need to work through the ambiguity by hand first.
The Law of Sines is common in surveying, navigation and any task where you can measure angles from two points to a distant object. For real surveying or engineering work, verify results with professional tools and a qualified surveyor or engineer.
Use the Law of Sines when you know an angle-side pair plus one more angle or side; use the Law of Cosines Calculator when you know two sides and the included angle, or all three sides.
A triangle's three angles always sum to exactly 180° -- any two given angles must leave a positive remainder for the third.
Degrees. The tool converts internally, so convert any radian value to degrees before entering it.
No. Angles alone give the shape but not the size, so you need at least one side length.
No. It requires two angles and one side, which avoids the ambiguity. An SSA problem needs to be resolved by hand first.
The largest angle should face the longest side and the smallest angle the shortest side. Also, the three angles should add to 180 degrees.
Reference: NIST Weights and Measures