Mathematics, Statistics & Geometry

Law of Sines Calculator

Solve for missing sides and angles of any triangle using the Law of Sines (AAS format). Includes full step-by-step ratio breakdowns.

Angle C
80
Side b15.035
Side c15.757
Calculation StepsInputs: ∠A = 30°, ∠B = 70°, side a = 8 Sum of angles in a triangle is 180°: ∠C = 180° - ∠A - ∠B = 180° - 30° - 70° = 80° Law of Sines: a/sin(A) = b/sin(B) = c/sin(C) Ratio = 8 / sin(30°) = 8 / 0.5000 = 16.0000 Find side b: b = Ratio * sin(B) = 16.0000 * sin(70°) = 16.0000 * 0.9397 = 15.035082 Find side c: c = Ratio * sin(C) = 16.0000 * sin(80°) = 16.0000 * 0.9848 = 15.756924

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The Rule of Proportions

The Law of Sines Calculator makes solving oblique triangles incredibly simple. By establishing the perfect ratio between side lengths and their opposing angles, this tool instantly generates the full dimensions of your triangle from just three known inputs.

asin⁡(A)=bsin⁡(B)=csin⁡(C)\begin{aligned} \frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)} \end{aligned}

Where:
a, b, c=
The lengths of the three sides of the triangle
A, B, C=
The internal angles directly opposite to their respective sides

The Constant Ratio

The beauty of the Law of Sines is its elegance. Imagine a triangle getting larger while keeping its angles exactly the same. The ratio asin⁡(A)\frac{a}{\sin(A)} acts as a universal scaling factor for that specific triangle's geometry. If you know that ratio, you know everything.

Real-World Applications

  • Forestry and Firefighting: Using triangulation to locate a wildfire. If two watchtowers know the distance between them (Side) and the compass angles to the fire (Angles), they can use the Law of Sines to calculate exactly how far away the fire is.
  • Astronomy: Calculating the distance to near stars using parallax. The Earth's orbit provides the known base side, and the shift in the star's position provides the angles.
  • Architecture: Designing complex, asymmetric roof trusses where support beams must meet at specific non-right angles.

Frequently Asked Questions

It is a trigonometric rule stating that the ratio between a side length and the sine of its opposite angle is constant for all three sides of any triangle.

You use it when you are given AAS (Angle-Angle-Side) or ASA (Angle-Side-Angle). You must know at least one complete 'pair' (a side and its opposite angle) to establish the ratio.

When you are given SSA (Side-Side-Angle), the math can sometimes result in two completely different but valid triangles (one acute, one obtuse), or no valid triangle at all. This calculator uses AAS to avoid this ambiguity.

Yes, but it's usually overkill. If angle C is 90°, sin(90°) = 1, so the ratio becomes simply 'c/1'. You are better off using standard SOH CAH TOA.

Because all internal angles of a flat triangle must sum to exactly 180 degrees, the third angle is always just 180° minus the other two known angles.

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