Law of Sines Calculator
Solve any triangle using the law of sines. Enter AAS, ASA, or SSA values to find all sides, angles, area, perimeter, circumradius, and more. Handles the ambiguous case with step-by-step solutions.
Given: Angle A = 40°, Angle B = 60°, Side a = 10
Step 1: Find Angle C = 180° - 40° - 60° = 80°
Step 2: Use Law of Sines: a/sin(A) = b/sin(B)
b = a × sin(B) / sin(A) = 10 × sin(60°) / sin(40°)
b = 13.472964
Step 3: c = a × sin(C) / sin(A) = 15.320889
Triangle Solution
Scalene AcuteAngles
A
40°
B
60°
C
80°
Sides
a
10
b
13.473
c
15.3209
Properties
Heights
Law of Sines Formula
a/sin(A) = b/sin(B) = c/sin(C)
Where a, b, c are sides opposite to angles A, B, C respectively.
How to Use the Law of Sines Calculator
Select Your Configuration
Choose AAS (two angles + non-included side), ASA (two angles + included side), or SSA (two sides + non-included angle). Each mode shows the appropriate input fields. Use preset examples to see how different triangle types work.
Enter Known Values
Type the values you know - angles in degrees and sides in any consistent unit. The calculator validates inputs in real-time and warns if the values cannot form a valid triangle (e.g., angles summing to more than 180°).
Review the Complete Solution
See all three sides, all three angles, area, perimeter, circumradius, inradius, and heights. For SSA (ambiguous case), both possible triangles are shown if two solutions exist. The step-by-step panel shows the exact calculation process.
Why Use Our Law of Sines Calculator
3 Input Configurations
Solve AAS, ASA, and SSA triangles. Each mode guides you to enter exactly the right values for a complete solution.
Complete Triangle Solution
Get all 3 sides, all 3 angles, area, perimeter, circumradius, inradius, and all 3 altitudes in one calculation.
Step-by-Step Solutions
See every calculation step from the given values to the final answer. Perfect for checking homework or learning the process.
Ambiguous Case Handling
SSA mode correctly detects 0, 1, or 2 solutions and shows all valid triangles. No more guessing about the ambiguous case.
6 Preset Examples
30-60-90, 45-45-90, equilateral, obtuse, ambiguous SSA, and navigation examples. Learn by exploring different triangle types.
High Precision
Results calculated to 6 decimal places with clean display. Angles in degrees, sides in your input units. All computation in-browser.
Complete Guide to the Law of Sines
The Law of Sines Formula and What It Means
The law of sines establishes a fundamental relationship in every triangle: the ratio of any side to the sine of its opposite angle is the same for all three side-angle pairs. Written mathematically: a/sin(A) = b/sin(B) = c/sin(C) = 2R, where R is the circumradius (the radius of the circle passing through all three vertices).
This means that longer sides always face larger angles, and the relationship is precisely proportional through the sine function. In a triangle where angle A is twice angle B, side a is not twice side b - it is sin(2B)/sin(B) = 2cos(B) times side b. This non-linear relationship is what makes the law of sines both powerful and sometimes counterintuitive.
Understanding the Ambiguous Case (SSA)
The ambiguous case is the trickiest application of the law of sines. When you know two sides (a and b) and the angle opposite one of them (A), you are solving sin(B) = b*sin(A)/a. Since sine produces the same value for supplementary angles (sin(30°) = sin(150°) = 0.5), there might be two valid angles for B.
Specifically: if b*sin(A)/a is greater than 1, no triangle exists. If it equals exactly 1, there is one right triangle. If it is less than 1, then B could be either arcsin(result) or 180° - arcsin(result). Both values are valid only if each produces a positive third angle C = 180° - A - B. When both work, you have two different triangles that satisfy the given measurements.
Solving Triangles: AAS vs ASA vs SSA
AAS (Angle-Angle-Side) gives you two angles and a side not between them. This is the simplest case: find the third angle (C = 180° - A - B), then use the law of sines twice to find the other two sides. There is always exactly one solution.
ASA (Angle-Side-Angle) gives you two angles and the included side. Again, find the third angle first, then use the law of sines. There is always exactly one solution. SSA (Side-Side-Angle) is the ambiguous case described above. It requires checking whether zero, one, or two triangles exist, making it the most complex scenario for the law of sines.
Real-World Applications of the Law of Sines
Navigation and surveying are classic applications. If you know the distance between two landmarks and can measure angles to a third point, the law of sines gives you the distances to that third point. This technique, called triangulation, has been used since ancient times and remains fundamental to modern GPS and mapping.
In engineering, the law of sines helps analyze forces in trusses and structural frames where force vectors form triangles. In astronomy, it is used to calculate distances to stars using parallax measurements. In computer graphics, it helps with mesh calculations and ray-triangle intersection tests. Even in everyday problem-solving, such as determining the height of a building from two different positions, the law of sines provides an elegant solution.
Law of Sines vs Law of Cosines: When to Use Which
The law of sines requires at least one known side-angle pair (a side and its opposite angle). It works for AAS, ASA, and SSA configurations. The law of cosines does not require opposite pairs - it works with SAS (two sides and the included angle) and SSS (three sides).
When both laws could technically work, the law of sines is usually preferred because the math is simpler - it involves basic ratios rather than the quadratic-like formula of the law of cosines. However, for SAS and SSS problems, only the law of cosines works directly. A common strategy is to use the law of cosines to find one angle from SSS or SAS, then switch to the law of sines for the remaining calculations since it is simpler.
Law of Sines Calculator FAQ
Common questions about the law of sines, triangle solving, and the ambiguous case.
Disclaimer: This Law of Sines Calculator provides solutions based on standard trigonometric formulas. Results are computed using floating-point arithmetic and may have minor rounding differences compared to exact symbolic solutions. Always verify critical calculations independently.