SSA Triangle Calculator

Solving an SSA triangle relies heavily on the Law of Sines, which states that the ratio of the sine of an angle to its opposite side is constant for all three angles in a triangle: (a / sin(alpha)) = (b / sin(beta)) = (c / sin(gamma)). When given two sides (such as 'a' and 'b') and one corresponding angle ('alpha'), the calculator determines the unknown angle ('beta') by rearranging the formula to: beta = arcsin((b * sin(alpha)) / a). Because the sine function is positive in both the first and second quadrants, this configuration can yield zero, one, or two valid triangles—a scenario commonly known as the ambiguous case. The algorithm evaluates side length relationships and height thresholds (h = b * sin(alpha)) to accurately isolate valid geometric configurations.

Let us solve a triangle where side a = 31, side b = 27, and angle alpha = 46 degrees. First, apply the Law of Sines to find angle beta: sin(beta) = (b * sin(alpha)) / a. Plugging in our values yields sin(beta) = (27 * sin(46°)) / 31 = (27 * 0.7193) / 31 = 0.6267. Taking the inverse sine, we find that beta is approximately 38.8 degrees. Next, find the third angle, gamma, by subtracting alpha and beta from 180 degrees: gamma = 180° - 46° - 38.8° = 95.2°. Finally, find the remaining side 'c' using the Law of Sines: c = (a * sin(gamma)) / sin(alpha) = (31 * sin(95.2°)) / sin(46°) = 42.9. This completes our full set of measurements for the triangle.

Always check for the ambiguous case before finalizing your answers. If the given side opposite the known angle is shorter than the adjacent side height, no valid triangle exists. Conversely, if it is longer than the adjacent side, only one triangle is formed. If it falls strictly between the height and the adjacent side length, you must calculate both acute and obtuse possibilities for the secondary angle.