Sine Rule Calculator 2026
Instantly solve triangles using the Law of Sines. Find missing sides and angles, handle the ambiguous SSA case, and calculate the exact area of any triangle.
Triangle Parameters
Enter any 3 known values (at least one must be a side)
Length of side opposite to Angle A
Length of side opposite to Angle B
Length of side opposite to Angle C
Angle opposite to side a
Angle opposite to side b
Angle opposite to side c
Solved Triangle
Missing sides, angles, and area
Enter any 3 known values (including at least one side) and click Solve Triangle to reveal the missing measurements.
Sine Rule & Triangle Properties
Core formulas and properties used to solve triangles in Euclidean geometry.
| Concept | Formula / Rule | Description |
|---|---|---|
| Sine Rule | a/sin(A) = b/sin(B) = c/sin(C) | Relates the length of a side to the sine of its opposite angle. |
| Triangle Area | 0.5 × a × b × sin(C) | Calculates area using two sides and the included angle (SAS). |
| Ambiguous Case | SSA Configuration | Two sides and a non-included angle can yield 0, 1, or 2 valid triangles. |
| Angle Sum | A + B + C = 180° | The interior angles of any Euclidean triangle always sum to 180 degrees. |
| Cosine Rule | c² = a² + b² – 2ab·cos(C) | Used to find missing parts when given SSS or SAS configurations. |
Sine Rule & Trigonometry FAQ
Learn more about the Law of Sines, triangle solving strategies, and the ambiguous case.
The Sine Rule (or Law of Sines) states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant for all three sides. It is expressed as: a/sin(A) = b/sin(B) = c/sin(C). It is primarily used to find missing sides or angles when you know certain combinations of the triangle’s parts.
You should use the Sine Rule when you know either: 1) Two angles and any one side (AAS or ASA), or 2) Two sides and an angle opposite to one of them (SSA). If you know three sides (SSS) or two sides and the included angle (SAS), you must use the Cosine Rule first to find the missing parts.
The ambiguous case occurs when you are given two sides and a non-included angle (SSA). Because the sine function is positive in both the first and second quadrants (sin(θ) = sin(180° – θ)), there can be zero, one, or two valid triangles that satisfy the given measurements. This calculator automatically detects and displays both solutions if the ambiguous case applies.
Yes, the Sine Rule applies to all triangles, including right-angled ones. However, for right-angled triangles, it simplifies to the basic trigonometric ratios (SOH CAH TOA), which are generally faster and more direct to use than the full Sine Rule formula.
If you know two sides and the included angle (SAS), you can calculate the area directly using the formula: Area = 0.5 × a × b × sin(C). This is a direct derivation from the Sine Rule and is highly efficient for finding the area without needing to calculate the triangle’s height first.