Law of Sines Calculator
Find a missing side, angle, or area of any triangle using the Law of Sines. Use it when you know one side, its opposite angle, and one additional angle.
About this calculator
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). To find an unknown side, rearrange to b = a × sin(B) / sin(A), where a is the known side and A is its opposite angle. The third angle is found using C = 180° − A − B (since angles in a triangle sum to 180°). Once all sides and angles are known, the area can be calculated as Area = 0.5 × a × b × sin(C). This calculator handles both degrees and radians, making it useful for geometry homework, surveying, and engineering problems where two angles and one side (AAS or ASA configuration) are given.
How to use
Suppose a triangle has a known side a = 10 units, its opposite angle A = 40°, and a second angle B = 65°. First, the third angle is C = 180° − 40° − 65° = 75°. To find side b: b = 10 × sin(65°) / sin(40°) = 10 × 0.9063 / 0.6428 ≈ 14.1 units. To find the area: Area = 0.5 × 10 × 14.1 × sin(75°) = 0.5 × 10 × 14.1 × 0.9659 ≈ 68.1 square units. Enter these values, select degrees, and choose what to find.
Frequently asked questions
When should I use the Law of Sines instead of the Law of Cosines?
Use the Law of Sines when you have an Angle-Angle-Side (AAS) or Angle-Side-Angle (ASA) configuration — that is, two angles and one side are known. The Law of Cosines is better suited when you have two sides and the included angle (SAS) or all three sides (SSS). Mixing up the two laws can lead to incorrect or ambiguous results, so identifying your known values first is essential.
What is the ambiguous case in the Law of Sines and how does it affect results?
The ambiguous case (SSA) arises when you know two sides and a non-included angle, which can produce zero, one, or two valid triangles. This calculator assumes an AAS or ASA scenario to avoid ambiguity. If you are working with SSA data, you should manually check whether a second valid triangle exists by testing if sin(B) ≤ 1 for both possible configurations. Awareness of this case is critical in surveying and navigation.
How do I convert between degrees and radians when using the Law of Sines?
To convert degrees to radians, multiply by π/180 (e.g., 45° × π/180 ≈ 0.7854 rad). To convert radians back to degrees, multiply by 180/π. This calculator accepts both units automatically — simply select your preferred angle unit before entering values. Internally, all trigonometric functions operate in radians, so the conversion is handled for you.