Free law of sines calculator solves oblique triangles instantly. Handles ASA, AAS, and SSA cases including ambiguous case. Shows step-by-step solutions with formulas.
Real-World Scenarios
Enter Angle A, Side c (included side), and Angle B
Angles
Sides
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The Law of Sines is a fundamental trigonometric relationship used to solve oblique (non-right) triangles. Enter any valid combination of angles and sides, and our calculator instantly finds all missing measurements with step-by-step solutions. Whether you're working with ASA, AAS, or the challenging SSA (ambiguous) case, this calculator handles them all with clear explanations.
The Law of Sines states that in any triangle, the ratio of a side length to the sine of its opposite angle is constant. This powerful relationship allows us to solve triangles when we know certain combinations of sides and angles. Unlike the Pythagorean theorem (which only works for right triangles), the Law of Sines works for ALL triangles—acute, right, or obtuse—making it essential for solving oblique triangles.
Law of Sines Formula
a/sin(A) = b/sin(B) = c/sin(C)Works for all non-right triangles where the Pythagorean theorem and basic trigonometry won't help.
Supports ASA (Angle-Side-Angle), AAS (Angle-Angle-Side), and SSA (Side-Side-Angle) configurations.
Automatically identifies when SSA produces zero, one, or two valid solutions and shows all possibilities.
See exactly how each calculation is performed with detailed formulas and mathematical steps.
Interactive visualization shows your solved triangle drawn to scale with all measurements labeled.
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Use the Law of Sines when you know: (1) two angles and any side (ASA or AAS), or (2) two sides and an angle opposite one of them (SSA). Use the Law of Cosines when you know: (1) all three sides (SSS), or (2) two sides and the included angle (SAS). The Law of Sines is generally easier to use when applicable, but the Law of Cosines is necessary for SSS and SAS cases.
The ambiguous case occurs with SSA (two sides and an angle opposite one side). Depending on the values, there may be zero solutions (impossible triangle), one solution, or two valid triangles that satisfy the given conditions. Our calculator automatically detects and shows all possible solutions when the ambiguous case applies.
The Law of Sines states: a/sin(A) = b/sin(B) = c/sin(C), where a, b, c are the side lengths and A, B, C are the opposite angles respectively. This ratio is equal to the diameter of the circumscribed circle (2R). The formula can be rearranged to solve for any unknown side or angle when enough information is provided.
Yes, the Law of Sines works for any triangle, including right triangles. However, for right triangles, basic trigonometric ratios (SOH-CAH-TOA) and the Pythagorean theorem are often simpler to use. The Law of Sines is most valuable for oblique (non-right) triangles where these basic methods don't apply.
An oblique triangle is any triangle that does NOT have a 90° angle. It can be acute (all angles less than 90°) or obtuse (one angle greater than 90°). The Law of Sines and Law of Cosines are the primary tools for solving oblique triangles, since basic right-triangle trigonometry (SOH-CAH-TOA) and the Pythagorean theorem don't apply.
In SSA cases, if the given side opposite the known angle is shorter than the height of the triangle (calculated as the other side × sin of the known angle), no triangle is possible. Our calculator automatically detects and reports this. For other cases (ASA, AAS), as long as the sum of two angles is less than 180°, a valid triangle exists.
These describe what information you know about a triangle. ASA = Angle-Side-Angle (two angles with the side between them). AAS = Angle-Angle-Side (two angles and a side not between them). SSA = Side-Side-Angle (two sides and an angle opposite one). Each requires different solving approaches, and SSA is special because it can produce 0, 1, or 2 solutions (the ambiguous case).