This law of sines calculator solves a triangle from two angles and a side (AAS/ASA) — and checks the notorious ambiguous case when you give two sides and a non-included angle (SSA).
a/sin A = b/sin B = c/sin C = 2R — One Ratio Rules the Triangle
The ratio’s identity as 2R (twice the circumradius) is the elegant footnote most courses skip. Individual trig values live on our Trigonometry Calculator. Part of our Math Calculators collection.
Worked Example (Verified)
- A = 30°, B = 45°, a = 10 (defaults): C = 105°, ratio = 10/sin 30° = 20 → b = 14.142, c = 19.319
- SSA illustration: A = 30°, a = 10, b = 14 → sin B = 0.7 → B = 44.43° (then C = 105.57°, c = 19.28) or B = 135.58° (then C = 14.42°, c = 4.98) — two genuinely different triangles from identical data
- No-solution edge: same angles with b = 21 → sin B = 1.05 > 1 — the side cannot reach; the data describes nothing
- Right-triangle reunification: with A = 90°, the ratio gives a = 2R — the hypotenuse as diameter, Thales waving from across the millennia
Choosing Sines vs Cosines
- Known: an angle and its opposite side plus anything else → sines.
- Known: two sides + included angle, or all three sides → cosines (next tool).
- SSA is the only ambiguous configuration in triangle solving — treat its answers as candidates until matched against the figure.
- Round late: sines of rounded angles compound errors; carry extra digits until the final side.
Related Tools
- Law of Cosines Calculator — SAS and SSS triangles
- Trigonometry Calculator — the six functions and inverses
- Math Calculators — the full collection
Frequently Asked Questions
What is the law of sines?
In any triangle, side ÷ sine of its opposite angle is constant: a/sin A = b/sin B = c/sin C, and the constant equals the circumcircle’s diameter 2R. Given one complete side-angle pair plus another angle or side, every remaining part follows.
How do I solve an AAS triangle?
Get the third angle from 180° − A − B, form the ratio a/sin A, and scale it by the other angles’ sines. Defaults: A = 30°, B = 45°, a = 10 → C = 105°, b = 14.142, c = 19.319.
What is the ambiguous case (SSA)?
Two sides and a non-included angle can describe zero, one or two triangles, because sin B = x has an acute and an obtuse solution. This calculator tests both candidates against the 180° budget and reports every valid triangle — the honest treatment most quick tools skip.
When should I use the law of cosines instead?
When no complete side-angle pair exists: two sides with the included angle (SAS) or all three sides (SSS). The law of sines needs an opposite pair to anchor its ratio; the law of cosines manufactures one.
Why does the ratio equal 2R?
By the inscribed-angle theorem, side a subtends angle A at the circumcircle, and chord length = 2R sin(central half-angle) = 2R sin A. So a = 2R sin A — the law of sines is circle geometry in disguise.
Is this law of sines calculator free?
Yes — free, instant, no sign-up.