Law of Sines Calculator
Solve any triangle using the Law of Sines. Enter known sides and angles to find missing measurements.
Enter Known Values
Law of Sines Formula:
sin(A)/a = sin(B)/b = sin(C)/c
Frequently Asked Questions
What is the Law of Sines?
The Law of Sines states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant. The formula is: sin(A)/a = sin(B)/b = sin(C)/c, where a, b, c are the sides and A, B, C are the opposite angles.
When should I use the Law of Sines?
Use the Law of Sines when you know: (1) Two angles and one side (AAS or ASA cases), or (2) Two sides and a non-included angle (SSA case - the ambiguous case). For other cases like SSS or SAS, the Law of Cosines is more appropriate.
What is the ambiguous case (SSA)?
The SSA case (two sides and a non-included angle) can sometimes result in two different valid triangles, one valid triangle, or no valid triangle at all. This occurs when sin(B) could have two possible angle values between 0° and 180°.
How many values do I need to solve a triangle?
You need at least three values to solve a triangle, and at least one of them must be a side length. Common combinations include: two angles and one side (AAS/ASA), two sides and one angle (SSA/SAS), or three sides (SSS).
Can the Law of Sines be used for right triangles?
Yes, the Law of Sines works for all triangles including right triangles. However, for right triangles, basic trigonometric ratios (sin, cos, tan) are often simpler to use.
What if the sum of angles doesn't equal 180°?
In a valid triangle, the sum of all three angles must equal exactly 180°. If your calculated angles don't sum to 180°, there may be an error in the input values or the triangle cannot exist with the given measurements.
How is the area calculated?
Once all sides and angles are known, the area is calculated using the formula: Area = (1/2) × a × b × sin(C), where a and b are two sides and C is the included angle between them.
Why might I get "no solution" errors?
You may get no solution if: (1) The given values don't form a valid triangle (e.g., angles sum to more than 180°), (2) The sides don't satisfy the triangle inequality (sum of any two sides must be greater than the third), or (3) In SSA cases where sin(angle) > 1.