Triangle Solver

Three measurements determine a triangle, with one exception

Given three sides, two sides and the angle between them, or two angles and any side, the triangle is fixed and every remaining measurement follows. Three angles are not enough, because they fix the shape and leave the size free: infinitely many similar triangles share them. The one genuinely awkward case is two sides and an angle not between them, which can produce two different valid triangles, one, or none at all.

The ambiguous case is a real ambiguity

With two sides and a non included angle, swinging the second side from the end of the first can meet the base line in two places, one, or neither, depending on whether it is long enough to reach and short enough to cross twice. A solver that silently returns one answer here is discarding a genuine solution. Both are reported below when both exist, because the input does not contain the information needed to choose between them.

Which rule to use, and why it matters

The cosine rule is numerically stable everywhere and is the right choice for finding an angle from three sides. The sine rule is simpler and has a trap: the inverse sine always returns an acute angle, so an obtuse angle computed that way comes out wrong without a separate check. Deriving the largest angle with the cosine rule first, then using the sine rule for the rest, avoids the problem entirely, which is what this solver does.

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