📐 Law of Sines Calculator
Calculate triangle sides and angles
How to Use This Calculator
Select What to Find
Choose whether you want to find a missing side (b) or a missing angle (B).
Enter Known Values
For finding a side: enter side a, angle A, and angle B. For finding an angle: enter side a, angle A, and side b.
Calculate
Click "Calculate" to get the missing side or angle using the Law of Sines.
Review Results
Check the calculated value and review the formula that was used.
Formula
a / sin(A) = b / sin(B) = c / sin(C)
Law of Sines
To Find a Side:
b = a·sin(B) / sin(A)
Given: side a, angle A, and angle B
To Find an Angle:
sin(B) = b·sin(A) / a
Given: side a, angle A, and side b
When to Use:
- AAS (Angle-Angle-Side) cases
- ASA (Angle-Side-Angle) cases
- SSA cases (but beware of ambiguous case!)
- When Law of Cosines is not applicable
⚠️ Ambiguous Case (SSA):
When given two sides and a non-included angle, there may be zero, one, or two possible triangles. Always verify your result makes sense!
About Law of Sines Calculator
The Law of Sines Calculator finds missing sides or angles in any triangle using the Law of Sines. This law states that the ratio of any side to the sine of its opposite angle is constant for all three sides and angles in a triangle. It's particularly useful for solving triangles when you know two angles and one side (AAS or ASA).
What is the Law of Sines?
The Law of Sines states that for any triangle: a/sin(A) = b/sin(B) = c/sin(C), where a, b, c are the sides and A, B, C are the angles opposite those sides respectively. This relationship holds for all triangles, not just right triangles.
When to Use This Calculator
- AAS (Angle-Angle-Side): When you know two angles and a non-included side
- ASA (Angle-Side-Angle): When you know two angles and the included side
- Navigation: Calculate distances using angle and distance measurements
- Surveying: Determine unknown measurements in triangular plots
- Engineering: Solve problems involving triangular structures
Why Use Our Calculator?
- ✅ Flexible: Can find either missing sides or angles
- ✅ Formula Display: Shows the exact formula used with your values
- ✅ Accurate: Precise calculations using standard Law of Sines
- ✅ 100% Free: No registration required
- ✅ Educational: Helps understand the Law of Sines
Important Note: Ambiguous Case
When given SSA (two sides and a non-included angle), there may be zero, one, or two possible triangles. This is called the "ambiguous case." Our calculator returns one solution, but always verify that your triangle is valid and that the solution makes geometric sense.
Frequently Asked Questions
When should I use Law of Sines vs Law of Cosines?
Use Law of Sines when you have AAS (two angles and a side) or ASA (two angles and included side). Use Law of Cosines when you have SAS (two sides and included angle) or SSS (all three sides).
What is the ambiguous case?
The ambiguous case occurs with SSA (Side-Side-Angle) data. Depending on the values, there may be 0, 1, or 2 possible triangles. Always verify your solution geometrically.
Can I use Law of Sines for right triangles?
Yes, Law of Sines works for all triangles including right triangles. However, for right triangles, using basic trig ratios (sin, cos, tan) is often simpler.
Why does the Law of Sines work?
The Law of Sines comes from the relationship between the sides of a triangle and the angles opposite them. It's derived from the fact that all three ratios (side/sin(opposite angle)) are equal to twice the circumradius of the triangle.
What if I get an angle greater than 180°?
This shouldn't happen with valid triangle inputs. If you get an angle ≥ 180°, check your inputs - either the triangle doesn't exist or there's an error in your data.