Law of Sines Calculator — Solve Any Triangle
Side b
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Side c
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Angle C
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Area
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How the Law of Sines Works
The Law of Sines states that a/sin(A) = b/sin(B) = c/sin(C), where lowercase letters are sides and uppercase are opposite angles. This ratio is constant for any triangle and equals the diameter of the circumscribed circle.
This law is most useful when you know AAS (two angles and a non-included side) or ASA (two angles and the included side). Once you know two angles, the third is simply 180 minus their sum. Then use the ratio to find unknown sides.
The ambiguous case (SSA) can produce zero, one, or two valid triangles. This calculator handles the standard AAS case. For navigation, the Law of Sines helps determine positions using triangulation from known landmarks.
Frequently Asked Questions
What is the ambiguous case?
The ambiguous case occurs with SSA (two sides and a non-included angle). There may be 0, 1, or 2 possible triangles. It arises because the sine function gives the same value for supplementary angles.
Can the Law of Sines find angles?
Yes. Rearranging: sin(B) = b*sin(A)/a. Then B = arcsin(b*sin(A)/a). Be aware of the ambiguous case when using this approach.
What does the ratio a/sin(A) represent geometrically?
This ratio equals the diameter (2R) of the circumscribed circle (circumcircle) of the triangle. Every triangle can be inscribed in a unique circle.