Law of Sines Calculator

The Law of Sines

The Law of Sines (also called the Sine Rule) states that in any triangle, the ratio of a side length to the sine of its opposite angle is constant:

a / sin(A) = b / sin(B) = c / sin(C)

where a, b, c are side lengths and A, B, C are the angles opposite those sides. This ratio equals the diameter of the triangle’s circumscribed circle (2R).

When to Use the Law of Sines

Use the Law of Sines when you know:

• ASA — two angles and the side between them
• AAS — two angles and a side not between them
• SSA — two sides and an angle not between them (the ambiguous case)

For SSS or SAS, use the Law of Cosines instead.

The ASA and AAS Cases

These are straightforward. Since the three angles of a triangle always sum to 180°, knowing two angles gives you the third: C = 180° - A - B. Then apply the Law of Sines to find the unknown sides.

Example (AAS): A = 40°, B = 60°, a = 10. Then C = 80°. b = a × sin(B) / sin(A) = 10 × sin(60°) / sin(40°) ≈ 13.47. c = a × sin(C) / sin(A) ≈ 15.32.

The SSA Ambiguous Case

When you know two sides and the angle opposite one of them (SSA), there may be 0, 1, or 2 valid triangles. This is because the given side may not be long enough to reach the base (no solution), just touches it (one solution — right triangle), crosses it once (one solution), or crosses it twice (two solutions). The calculator checks all possibilities.

Applications

The Law of Sines is used in navigation (finding position from two known bearings), surveying (triangulation), astronomy (parallax distance measurement), and physics (force vector decomposition). Whenever you know angles and one side, the Sine Rule lets you fill in the rest of the triangle.

Area Formula

Once all sides and angles are known: Area = (1/2) × a × b × sin(C). This gives the area in square units matching the side units.