THE NUMORIX GUIDE
How to use the Law of Sines Calculator
Last reviewed September 14, 2026
What this calculator does
Convert the supplied angles to radians, use b = a*sin(B)/sin(A), find C = 180-A-B, then use c = a*sin(C)/sin(A) and area = 0.
Formula and method
Convert the supplied angles to radians, use b = a*sin(B)/sin(A), find C = 180-A-B, then use c = a*sin(C)/sin(A) and area = 0.5*a*b*sin(C).
Variables and inputs
Side a defaults to 10, angle A to 30 degrees, and angle B to 45 degrees. The UI assumes an ASA/AAS-style valid triangle with angles in degrees.
Worked example
For a=10, A=30 degrees, B=45 degrees: b=10*sin45/sin30=14.1421; C=180-30-45=105 degrees; c=10*sin105/sin30=19.3185; area=0.5*10*14.1421*sin105=68.3013 square units.
How to interpret the result
The law of sines relates each side to the sine of its opposite angle. It is especially useful when one side-angle pair is known.
Common mistakes to avoid
Match side a with angle A and side b with angle B. Check that A+B is less than 180 degrees and be alert to the ambiguous SSA case, which this fixed input layout does not model.
Assumptions and limitations
The UI has only one known side and two angles, so it does not expose the full ambiguous-case decision tree. Invalid angles can produce nonsensical results rather than a detailed validation error.
Practical use and checks
The Law of Sines Calculator solves a triangle from one known side and its opposite angle plus a second angle. Enter side a = 10, angle A = 30 degrees, and angle B = 45 degrees as a check. The remaining angle C should be 105 degrees, side b about 14.1421, side c about 19.3185, and area about 68.3013 square units. The side check follows a/sin(A) = b/sin(B) = c/sin(C), so compare ratios rather than only the rounded lengths. Use the calculator when the known side is correctly paired with its opposite angle, and verify that the two supplied angles sum to less than 180 degrees. Angles are entered in degrees; the engine converts them to radians for sine. The ambiguous SSA case is not fully represented by this fixed input arrangement, so it will not warn you about two possible triangles in every situation. Angles near 0 or 180 degrees make the sine denominator or remaining triangle extremely sensitive to rounding, and invalid angles can produce negative or nonphysical results. The model assumes a flat Euclidean triangle and consistent side units. For surveying or construction, keep measurement tolerances and the chosen angle convention with the output rather than treating four displayed decimals as field accuracy.