Law of Sines Calculator – Triangle Calculator
Solve triangles using the Law of Sines. Handle SSA ambiguous cases with confidence.
Select Known Values
Enter Values
Triangle Visualization
Triangle Solution
Two possible triangles exist with these measurements. The second solution is also shown above.
What is the Law of Sines?
The Law of Sines is a foundational trigonometric theorem establishing the relationship between the sides and angles of any triangle.
The Core Formula
The law states that the ratio of a side length to the sine of its opposite angle is constant: a/sin(A) = b/sin(B) = c/sin(C).
ASA & AAS Cases
Use this law when you know two angles and one side (Angle-Side-Angle or Angle-Angle-Side). You can easily find the remaining sides.
The SSA Case
When you know two sides and a non-included angle (Side-Side-Angle), it's called the ambiguous case because it can form 0, 1, or 2 different triangles.
Circumcircle Connection
A beautiful property of this law is that the constant ratio (a/sin(A)) is exactly equal to the diameter of the triangle's circumscribed circle.
Works for Any Triangle
Unlike basic SOH CAH TOA, which only works for 90-degree right triangles, the Law of Sines works perfectly for acute and obtuse triangles.
Paired with Cosines
The Law of Sines is best friends with the Law of Cosines. If you don't have a matching "side and opposite angle" pair, you use Cosines first.
Real-World Applications
From ancient star mapping to modern GPS, the Law of Sines is a crucial geometric tool used across numerous industries.
Land Surveying
Surveyors use a technique called triangulation. By measuring a baseline and two angles to a distant point, they can calculate exact distances.
Aviation & Navigation
Pilots use the Law of Sines to calculate crosswind correction angles, ensuring the plane flies on the correct path despite wind pushing it sideways.
Astronomy
Astronomers calculate the distance to nearby stars using parallax—measuring the star's angle from Earth at two different points in its orbit.
Structural Engineering
Engineers use non-right triangle mathematics to calculate force distributions and required beam lengths for complex roof trusses and bridges.
Forestry
Foresters can calculate the exact height of a leaning tree on a sloped hill by taking angle measurements from a known distance away.
Radio & GPS Tracking
Cell towers use angular relationships and the Law of Sines to triangulate the exact location of a mobile phone signal.
Pro Tips & The Ambiguous Case
Avoid common pitfalls and master the tricky "Ambiguous Case" with these expert trigonometry tips.
The 180° Rule
Before doing complex math, remember that all angles in a triangle must add up to exactly 180°. Use this to find the third angle instantly.
Size Relationships
As a quick sanity check: The longest side must always be opposite the largest angle, and the shortest side opposite the smallest angle.
Calculator Mode
The most common mistake! Make sure your scientific calculator is set to 'Degrees' (not 'Radians') when entering angle measurements.
Spotting Ambiguity (SSA)
If you are given two sides and an angle *not* between them (SSA), you might have 0, 1, or 2 possible triangles. Always check for a second angle!
Finding the 2nd Angle
In the ambiguous case, if your calculator gives you angle B, the second possible angle is always 180° - B. Check if it fits!
Domain Errors
If your equation requires you to find the inverse sine of a number greater than 1 (e.g., sin⁻¹(1.2)), no such triangle can physically exist.
