Sin Triangle Calculator — Law of Sines Solver
Enter two angles (A and B) and one known side to solve the triangle via the Law of Sines — getting all three sides, the third angle, perimeter and area.
°
°
Known side
a/sin A = b/sin B = c/sin C — the common constant
- 1
Third angle C = 180° − A − B
180 − 45 − 60 = 75 - 2
sin A — sine of the known side's opposite angle
sin(45°) = 0.707107 - 3
Law of Sines ratio = side ÷ sin(opposite angle)
10 ÷ 0.707107 = 14.1421a/sin A = b/sin B = c/sin C — all three pairs share this constant ratio.
How does this calculator work?
Law of Sines: a/sin A = b/sin B = c/sin C. Enter two angles and one side. The third angle is 180° − A − B; missing sides follow from the ratio. Area is then found by Heron's formula. The common ratio equals 2× the circumradius.
Formula
How this is calculated
The Law of Sines states that in any triangle, the ratio of a side to the sine of its opposite angle is constant: a/sin A = b/sin B = c/sin C. Given two angles and one side (AAS or ASA), the third angle is found as C = 180° − A − B, then the missing sides are computed from the ratios.
For example, with side a known: b = a × sin B / sin A and c = a × sin C / sin A. The common ratio a/sin A equals the diameter of the triangle's circumscribed circle (circumradius × 2). This calculator uses the AAS/ASA case, which always produces a unique triangle. The ambiguous SSA case (two sides and a non-included angle) can yield zero, one or two solutions and is not handled here.
Once all three sides are known, area is computed by Heron's formula: area = √(s(s−a)(s−b)(s−c)) where s = (a+b+c)/2. This is an exact cross-check independent of the sine rule. Units are whatever unit you use for the side length — the ratio is dimensionless.
Frequently asked questions
Use the Law of Sines when you know two angles and any one side (AAS or ASA), or two sides and an angle opposite one of them (SSA, with caution). Use the Law of Cosines when you know three sides (SSS) or two sides and the included angle (SAS) — those cases cannot be solved by the sine rule alone.
When two sides and a non-included angle are known (SSA), the Law of Sines may give two valid triangles, one, or none. This calculator avoids that ambiguity by requiring two angles and one side, which always yields a unique solution.
It equals 2R, where R is the radius of the circle that passes through all three vertices of the triangle (the circumradius). It is constant for all three side-angle pairs in the same triangle.
TG we-Calculate Editorial Team. (2026). Sin Triangle Calculator — Law of Sines Solver [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/sin-triangle-calculator
TG we-Calculate Editorial Team. "Sin Triangle Calculator — Law of Sines Solver." TG we-Calculate. 2026. https://we-calculate.com/calculator/sin-triangle-calculator.
TG we-Calculate Editorial Team, "Sin Triangle Calculator — Law of Sines Solver," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/sin-triangle-calculator
@misc{wecalculate_sin_triangle_calculator, title = {Sin Triangle Calculator — Law of Sines Solver}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/sin-triangle-calculator}}, year = {2026}, note = {TG we-Calculate} }
Did this calculator help you?