Law of Sines Calculator

Solve triangles using the Law of Sines for AAS, ASA, and SSA cases. Automatically handles the ambiguous case.

Select Case & Enter Values

Given two angles and a non-included side.
Ambiguous Case: Two valid triangles exist for these inputs. Both solutions are shown below.

Result

Triangle Solution
--

Step-by-Step Solution

a/sin(A) = b/sin(B) = c/sin(C)

Understanding the Law of Sines

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 enables us to solve triangles when we know certain combinations of sides and angles.

a / sin(A) = b / sin(B) = c / sin(C) = 2R

where R is the circumradius (radius of the circumscribed circle).

Cases Where the Law of Sines Applies

AAS (Angle-Angle-Side)

Two angles and a non-included side are known. Find the third angle, then use the law to find the remaining sides.

Always one unique solution

ASA (Angle-Side-Angle)

Two angles and the included side are known. Find the third angle, then use the law to find the remaining sides.

Always one unique solution

SSA (Side-Side-Angle)

Two sides and a non-included angle. This is the ambiguous case that may have 0, 1, or 2 solutions.

0, 1, or 2 solutions possible

The Ambiguous Case (SSA)

The SSA case is called "ambiguous" because knowing two sides and a non-included angle does not always determine a unique triangle. The number of solutions depends on the relationship between the given side and the altitude of the potential triangle:

  • No solution: The side opposite the given angle is too short to reach the other side.
  • One solution: The side opposite the given angle exactly equals the altitude (creating a right triangle), or the given angle is obtuse/right and the opposite side is longer than the adjacent side.
  • Two solutions: The side opposite the given angle is longer than the altitude but shorter than the adjacent side, and the given angle is acute.

When to Use Law of Sines vs. Law of Cosines

  • Use Law of Sines for AAS, ASA, and SSA cases.
  • Use Law of Cosines for SSS and SAS cases.
  • For SSA, consider starting with Law of Sines but verify with the ambiguous case check.

Practical Applications

  • Triangulation in surveying and mapping to determine unknown distances.
  • Navigation: Determining ship or aircraft position using angle measurements.
  • Astronomy: Computing distances to nearby stars using parallax angles.
  • Architecture and construction: Calculating roof pitch angles and beam lengths.
  • Physics: Analyzing force diagrams and vector resolution.