Law of Sines
Use the Law of Sines to solve triangles given two angles and one side. Free online ASA triangle solver with instant results for sides b, c, and angle C.
About This Calculator
The Law of Sines Calculator helps you solve triangles by applying the sine rule formula. Whether you are a student studying trigonometry, a teacher preparing lessons, or a professional working with triangular measurements, this tool computes missing angles and side lengths quickly and accurately.
The Law of Sines states that the ratio of a side length to the sine of its opposite angle is constant for all three sides of a triangle: a/sin(A) = b/sin(B) = c/sin(C) = 2R, where R is the circumradius. Given side a and angles A and B, angle C is found as 180 deg - A - B, then sides b and c are computed using the proportional relationship. This is the standard ASA (angle-side-angle) triangle solution method.
Practical applications include surveying and land measurement where distances are determined from angle observations, navigation for calculating distances between points using bearing angles, architecture and construction for roof truss and structural design, astronomy for measuring distances to celestial objects, and engineering for force vector analysis and mechanical design.
Frequently Asked Questions
What is the Law of Sines formula?
The Law of Sines states that a/sin(A) = b/sin(B) = c/sin(C), where a, b, c are side lengths and A, B, C are the opposite angles. This ratio is also equal to 2R, where R is the circumradius of the triangle.
When should I use the Law of Sines vs the Law of Cosines?
Use the Law of Sines when you know two angles and one side (AAS or ASA) or two sides and a non-included angle (SSA). Use the Law of Cosines when you know three sides (SSS) or two sides and the included angle (SAS).
What information do I need to use this calculator?
You need the length of side a and the measures of angles A and B (both in degrees). The calculator will compute the missing angle C and the lengths of sides b and c.
Can I use the Law of Sines for any triangle?
Yes, the Law of Sines works for any triangle -- acute, obtuse, or right. It is valid for all triangles in Euclidean geometry, not just right triangles.
What happens if my angles sum to more than 180 degrees?
If angles A and B sum to 180 degrees or more, no valid triangle exists. The calculator will not compute results because a triangle's interior angles must sum to exactly 180 degrees.
Does the Law of Sines work in radians?
Yes, the Law of Sines works in any angular unit as long as you are consistent. This calculator uses degrees, but the mathematical relationship holds identically in radians.
What real-world problems does the Law of Sines solve?
Surveyors use the Law of Sines to measure distances across inaccessible terrain by measuring angles from two known points. Navigators apply it to calculate distances between waypoints using bearing angles. Astronomers use it to determine distances to celestial objects from Earth-based angle observations.
What is the relationship between the Law of Sines and the circumradius?
The common ratio a/sin(A) = b/sin(B) = c/sin(C) equals 2R, where R is the circumradius of the triangle (the radius of the circle that passes through all three vertices). This means the Law of Sines also gives you the diameter of the circumscribed circle directly from any side and its opposite angle.