Law of Sines Calculator
Find missing sides or angles of any triangle using the Law of Sines. Supports AAS (two angles + non-included side) and SSA (two sides + non-included angle) modes, including the ambiguous case.
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How to use this calculator
The Law of Sines relates each side of a triangle to the sine of its opposite angle. All three ratios are equal and can be used to find unknown sides or angles.
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Choose AAS if you know two angles and any one side; choose SSA if you know two sides and the angle opposite one of them.
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Enter angle A, angle B (AAS) or side b (SSA), and side a.
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The calculator solves for all missing parts and reports the area. In SSA mode it may show two valid solutions (ambiguous case).
Frequently asked questions
What is the ambiguous case in the Law of Sines?
The SSA configuration (two sides and a non-included angle) can produce zero, one, or two valid triangles. When the side opposite the given angle is shorter than the other given side but long enough to reach the base line, two different triangles satisfy the given data. This calculator reports all valid solutions.
When should I use AAS vs ASA?
AAS gives two angles and a side not between them; ASA gives two angles and the side between them. Both uniquely determine a triangle. In AAS mode this calculator takes angle A, angle B, and side a (opposite A). If you have ASA, compute the third angle first (C = 180 − A − B) then treat it as AAS.
Can the Law of Sines find an obtuse angle?
The inverse sine function always returns angles between 0° and 90°. An obtuse angle B is found as 180° − arcsin(sin B). This calculator automatically checks both possibilities and discards solutions where angles would sum beyond 180°.
How is the area formula derived from the Law of Sines?
Area = ½ · a · b · sin C. This follows because the altitude of a triangle equals one side times the sine of an adjacent angle. Combined with the Law of Sines, the area formula works whenever two sides and any angle are known.
Law of Sines — AAS, ASA, and the ambiguous SSA case
When the Law of Sines applies
The Law of Sines is the tool of choice when at least one side and its opposite angle are both known. AAS (two angles + any side) uniquely determines a triangle because the third angle is found by subtraction and all sides follow from the sine ratio. ASA reduces to AAS immediately. The trickier SSA case — two sides and a non-included angle — requires checking whether zero, one, or two triangles are consistent with the data.
The ambiguous case explained
In SSA mode with angle A and sides a and b: if a ≥ b, exactly one triangle exists. If a < b, construct a perpendicular of height h = b·sin A. If a < h, no triangle reaches the base; if a = h, exactly one right triangle; if h < a < b, two triangles fit the data. This situation is called the "ambiguous case" and is a classic source of errors on exams — always verify which solutions are geometrically valid.
Applications across science and engineering
The Law of Sines is fundamental in astronomy (calculating distances to stars via parallax), navigation (triangulating a ship's position from two known landmarks), and structural engineering (analyzing force components in trusses). Any time you can measure angles more easily than distances, the Law of Sines converts those angles into the distances you need.
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