Law of sines calculator

Choose the known-data pattern

Choose ASA/AAS when two angles A and B and one opposite side a are known. Whether the known side lies between the two angles changes the name, not this numeric procedure: find C = 180° − A − B, then scale sides with the sine rule.

Choose SSA when side a, its opposite angle A and another side b are known. SSA is ambiguous, so the calculator checks every candidate instead of assuming the acute inverse-sine angle is the only answer.

Formula and the ambiguous case

The law of sines is a/sin A = b/sin B = c/sin C. In the two-angle mode, b = a sin B/sin A and c = a sin C/sin A.

For SSA, sin B = b sin A/a. If this exceeds 1, no triangle exists. Otherwise B may be asin(value) or 180° − asin(value); each candidate is kept only when A + B < 180°. Equal right-angle candidates are deduplicated.

Input limits and page boundary

Angles must be strictly between 0° and 180°, sides must be positive finite decimals and all side lengths must share one unit. Two known angles must sum to less than 180°. Zero, negative, formulas and typed units are rejected.

This page handles ASA, AAS and SSA. Use the law of cosines page when three sides are known (SSS) or when two sides enclose a known angle (SAS). Ordinary display rounds to two decimals while calculations keep internal precision.

Two angles and one side

With a = 10, A = 30° and B = 60°, C = 90°, b ≈ 17.32 and c = 20. The same computation covers an ASA or AAS diagram when the labels correspond.

SSA with two solutions

With a = 3, A = 10° and b = 9, B can be about 31.4° or 148.6°. Both leave a positive C, so two distinct triangles are displayed.