Law of Sines Calculator
The Law of Sines relates the sides of any triangle to the sines of their opposite angles. Enter two angles and a side (AAS or ASA) or two sides and an angle (SSA) to solve the triangle.
The Law of Sines
For a triangle with sides $a$, $b$, $c$ opposite to angles $A$, $B$, $C$ respectively:
$$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$This common ratio equals the diameter of the triangle's circumscribed circle.
Choosing the Right Case
| Case | Known | Notes |
|---|---|---|
| AAS | A, B, a | Always one unique solution |
| ASA | A, B, c (side between) | Always one unique solution |
| SSA | a, b, A | Ambiguous — may have 0, 1, or 2 solutions |
The Ambiguous SSA Case
When you know side $a$ (opposite to angle $A$), side $b$, and angle $A$, the number of solutions depends on the height $h = b \sin A$:
- If $A \ge 90°$: one solution if $a > b$, no solution if $a \le b$.
- If $A < 90°$ and $a < h$: no solution (side too short to reach the base).
- If $A < 90°$ and $a = h$: exactly one solution (right triangle).
- If $A < 90°$ and $h < a < b$: two solutions.
- If $A < 90°$ and $a \ge b$: one solution.
For triangles where you know three sides or two sides and the included angle, use the Law of Cosines calculator instead. For right triangles specifically, see the right triangle calculator.
Frequently Asked Questions
Can I use the Law of Sines on a right triangle?
Yes. The Law of Sines works on all triangles. For a right triangle with $C = 90°$, $\sin C = 1$, so the formula simplifies to $a/\sin A = c$, which is just $c = a/\sin A$ or $a = c\cdot\sin A$ — the same as basic SOH-CAH-TOA trig.
What does the ambiguous case mean in practice?
In surveying or navigation, if two solutions exist you need additional information (like whether the unknown angle is acute or obtuse) to decide which triangle is the real one. When both solutions are shown, check against the context of the problem.
