About This Tool
The SSA (Side-Side-Angle) case is known as the "ambiguous case" because it can have zero, one, or two valid triangles depending on the measurements. This happens when you know two sides and an angle that is NOT between them.
This calculator uses the Law of Sines to solve for the unknown angle, then checks for all possible solutions. It clearly shows whether your inputs produce no triangle, one unique triangle, or two different triangles (the ambiguous case).
Understanding SSA is crucial for surveying, navigation, and any field where indirect measurements might produce this configuration.
How to Use
1. Identify the angle and the two sides you know
2. Enter the known angle (the one that is NOT between your two sides)
3. Enter the two known sides โ one should be opposite the angle, one adjacent
4. The calculator checks all possibilities and shows the result(s)
5. If two triangles exist, both solutions are displayed
6. View the step-by-step analysis explaining why 0, 1, or 2 triangles exist
Formula
Law of Sines (to find the unknown angle):
If you know angle A, side a (opposite A), and side b:
sin(B) = b ยท sin(A) / a
Three possible outcomes:
1. sin(B) > 1: No triangle exists (impossible)
2. sin(B) = 1: One right triangle (B = 90ยฐ)
3. sin(B) < 1: One or two triangles
Checking for two triangles:
If sin(B) < 1, then B could be arcsin(sinB) OR 180ยฐ - arcsin(sinB)
The second solution is valid only if A + Bโ < 180ยฐ
Third side (Law of Cosines):
c = a ยท sin(C) / sin(A)