Solve a triangle using the law of sines — find missing sides or angles from a known side-angle pair and one additional measurement.
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How to use
Enter a known side a, its opposite angle A, and a second angle B to find side b and solve the full triangle.
Also in Triangles
Trigonometry
The law of sines calculator solves a triangle when you know one side and its opposite angle plus one additional measurement. It handles ASA, AAS, and the ambiguous SSA case, returning all missing sides, angles, and the area.
The law of sines states that the ratio of each side to the sine of its opposite angle is constant across the triangle. This constant equals the diameter of the circumscribed circle.
a / sin(A) = b / sin(B) = c / sin(C)
Each side divided by the sine of its opposite angle gives the same value.
When you know two sides and a non-included angle, the law of sines can produce zero, one, or two valid solutions. The ambiguous case arises because the sine function gives the same value for an angle and its supplement. A good calculator checks for all valid solutions and reports each one.
Use the law of sines when you have at least one side-angle pair and one other known measurement. Use the law of cosines for SSS or SAS configurations where no side-angle pair is directly available.
Frequently asked questions
It occurs in SSA (two sides and a non-included angle). The given information may produce two valid triangles, one triangle, or no triangle at all.
If you know one side-angle pair and one more angle or side, yes. The third angle is 180 minus the other two.
Related
These related calculators come from the same leaf category, nearby sibling categories, or the same top-level topic.
Solve a triangle using the law of cosines — find a missing side from two sides and the included angle, or find an angle from three sides.
Solve for the missing side of a right triangle using the Pythagorean theorem, with perimeter and area included.
Calculate sine, cosine, and tangent ratios for a right triangle from side lengths or find missing sides from an angle.
Calculate the angle of depression from height and horizontal distance, or find a missing side given the angle.
