Triangle Solver
SSS, SAS, ASA/AAS, and SSA-ambiguous cases
When One Dataset Splits Reality 🖖
The triangle does not merely get solved. It reveals whether your facts are enough to define reality. In the SSA case, the same measurements can describe two different triangles because sine cannot distinguish an angle from its mirror: {0}. So one dataset can produce two valid worlds.
Why you always need a length 🖖
Three known values usually pin down a triangle, but at least one must be a side. Three angles alone (AAA) fix the shape yet not the size: a tiny triangle and a giant one can share identical angles. The tool leans on two rules: the law of cosines when you know all three sides (SSS) or two sides and the angle between them (SAS), and the law of sines for the rest.
A hidden circle sets the ratio 🖖
In the law of sines, a/sin(A) = b/sin(B) = c/sin(C) is not just "some constant." That shared value equals 2R, the diameter of the unique circle passing through all three corners (the circumscribed circle). So every triangle secretly carries a circle, and the sine law is really measuring its size. Double the circumradius and every side-to-sine ratio doubles with it.
Example problems
- SSS - SSS case: 3-4-5 triangle gives a right angle.
- SAS - SAS case solved via law of cosines, then law of sines.
- ASA - ASA case solves the third angle first, then sides.
- SSA ambiguous - SSA ambiguous case can produce two valid triangles.