law of sines calculator

What is the Law of Sines?

The Law of Sines is a trigonometry relationship used to solve non-right triangles. It links each side of a triangle with the sine of its opposite angle:

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

This formula is especially useful when you know:

• Two angles and one side (ASA or AAS), or
• Two sides and an angle opposite one of those sides (SSA, the ambiguous case).

How to use this calculator

Mode 1: ASA/AAS

Enter two triangle angles and one side. The calculator finds the third angle first, then computes the missing sides with the Law of Sines. This mode gives a single valid triangle when inputs are valid.

Mode 2: SSA (Ambiguous Case)

Enter one known angle and its opposite side, then one additional side. Depending on your numbers, there may be:

• No triangle (impossible geometry),
• One triangle (unique solution), or
• Two triangles (ambiguous case).

Why the ambiguous case happens

In SSA inputs, the equation for an unknown angle often becomes sin(X) = k. For values of 0 < k < 1, two different angles share the same sine value: X and 180° - X. If both produce a valid total angle sum under 180°, you get two different triangles.

Practical applications

A Law of Sines triangle solver is useful in many real-world contexts:

• Surveying and land measurement
• Navigation and bearings
• Architecture and roof geometry
• Physics vector triangles
• Engineering layout calculations

Common mistakes to avoid

• Using degrees in one step and radians in another.
• Forgetting that side a must pair with angle A, and so on.
• Entering all three angles with a sum not equal to 180°.
• Missing the second possible triangle in SSA cases.
• Rounding too early and propagating error.

Quick example

Suppose you know A = 40°, B = 65°, and a = 10.

First compute C = 180° − 40° − 65° = 75°. Then apply:

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

The calculator performs these steps automatically and returns all values instantly.