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Law of Sines Calculator - Solve Triangles
Trigonometry Tool

Law of Sines Calculator

Solve triangles quickly using the Law of Sines. Find missing sides and angles from known side-angle relationships with accurate results.

Solve Your Triangle

Enter two side-angle pairs
Side & Angle Pair 1 a ↔ A
Enter the length of side a.
Enter the angle opposite side a.
Side & Angle Pair 2 b ↔ B
Enter the length of side b.
Enter the angle opposite side b.
Clear Calculator

What Is the Law of Sines?

The Law of Sines is a trigonometric relationship used to solve triangles when certain sides and angles are known. It connects each side of a triangle with the sine of its opposite angle.

The formula is written as a / sin(A) = b / sin(B) = c / sin(C). This relationship allows you to calculate unknown sides or angles when enough information about the triangle is available.

The Law of Sines is especially useful for oblique triangles, including triangles that are not right triangles. It is commonly used in geometry, trigonometry, surveying, engineering and physics.

How to Use This Calculator

Enter two known side-angle pairs. Each side must be matched with its opposite angle. The calculator then uses the Law of Sines to determine the remaining measurements.

1

Enter Side a

Enter the length of side a and its opposite angle A.

2

Enter Side b

Enter the length of side b and its opposite angle B.

3

Solve the Triangle

Click the solve button to calculate the missing side and angle values.

Law of Sines Example

Suppose a triangle has side a = 10 and angle A = 30°, while another angle is B = 45°.

Using the Law of Sines, the corresponding side b can be calculated with:

a / sin(A) = b / sin(B)

10 / sin(30°) = b / sin(45°)

b = 10 × sin(45°) / sin(30°)

The same relationship can be used to determine other missing sides and angles when enough triangle information is available.

Frequently Asked Questions

What is the Law of Sines?

The Law of Sines is a trigonometric formula that relates the sides of a triangle to the sines of their opposite angles.

What is the Law of Sines formula?

The formula is a/sin(A) = b/sin(B) = c/sin(C). Each side corresponds to the angle directly opposite it.

When should I use the Law of Sines?

It is useful when you know enough side and angle information to establish a side-angle relationship, especially in oblique triangles.

Can the Law of Sines solve an SSA triangle?

Yes. The SSA case can sometimes produce two possible triangles, one triangle, or no valid triangle. This is known as the ambiguous case.

Do angles need to be entered in degrees?

Yes. This calculator uses degrees for all angle measurements.

What does side a correspond to?

Side a is always opposite angle A. Similarly, side b is opposite angle B and side c is opposite angle C.

Can this calculator find the missing side?

Yes. When the entered measurements provide enough information for a valid triangle, the calculator determines the missing side.

What if my values do not make a triangle?

The calculator checks the entered measurements and displays an error when they cannot form a valid triangle using the Law of Sines.