Law of Sines Calculator
Enter a known side-angle pair and one more value to solve the triangle.
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Request a ToolHow to Use the Law of Sines Calculator
This calculator applies the law of sines to find unknown sides and angles in a triangle.
- Enter a known pair. Provide at least one complete side-angle pair, such as side a and angle A (the angle opposite side a).
- Add one more value. Enter a second side or angle. The calculator needs at least 3 values total (with one complete pair) to solve.
- Read the results. The calculator solves for all unknowns including the third side, third angle, and the common ratio. Results appear instantly.
This calculator works for AAS, ASA, and SSA configurations. For SSA (the ambiguous case), it finds one valid solution if it exists.
About the Law of Sines
The law of sines states that in any triangle, the ratio of a side to the sine of its opposite angle is constant: a/sin(A) = b/sin(B) = c/sin(C). This constant ratio equals the diameter of the triangle's circumscribed circle. The law is especially useful when you know a side and its opposite angle plus one additional piece of information. It complements the law of cosines, which handles cases where you know two sides and the included angle.
Frequently Asked Questions
What is the ambiguous case of the law of sines?
The ambiguous case occurs with SSA (two sides and a non-included angle). In this configuration, there may be zero, one, or two valid triangles. The ambiguity arises because the arcsin function can return two different angles for the same sine value. This calculator finds one valid solution or reports that no solution exists.
What does the common ratio represent?
The common ratio a/sin(A) = b/sin(B) = c/sin(C) equals the diameter of the circumscribed circle (the circle that passes through all three vertices of the triangle). This geometric interpretation connects the law of sines to circle geometry.
Can I use the law of sines for right triangles?
Yes, but it simplifies because sin(90) = 1, making the hypotenuse equal to the common ratio. For right triangles, the basic SOH-CAH-TOA ratios are usually simpler to use, but the law of sines works correctly for all triangles including right triangles.