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.

The Hidden Constant: 2R

The common ratio a/sin(A) = b/sin(B) = c/sin(C) is not just a number for solving triangles. It equals the diameter of the triangle’s circumscribed circle, 2R, where R is the circumradius. This is the Extended Law of Sines. It means that any triangle is determined, up to congruence, by knowing one side, its opposite angle, and the circle it can be inscribed in.

A practical consequence: if you measure a/sin(A) and find it equals, say, 12, you instantly know any triangle inscribed in the same circle with these side-and-angle pairs will give the same ratio. Astronomers used this property for centuries to calculate stellar parallax distances.

Avoiding the Ambiguous Case

When you have SSA configuration and don’t want to think through 0/1/2 triangles, here’s a trick: use the Law of Cosines instead. Treating one of the sides as unknown turns SSA into a quadratic equation in that side. The quadratic gives 0, 1, or 2 positive roots, and each root corresponds to a valid triangle. The math handles all the ambiguity cleanly without separate case analysis. This calculator does the SSA check the traditional Law-of-Sines way (with explicit case detection) so you can see exactly when two triangles are possible.