Law of Sines
 
In any triangle, the ratio of a side length to the sine of its opposite angle
is the same for all three sides.   As a formula:
Law of sines
Try this Drag any vertex of the triangle. Note that the ratio of a side to the sine of its opposite angle is the same for all three sides.

See also Law of Cosines

The Law of Sines says that in any given triangle, the ratio of any side length to the sine of its opposite angle is the same for all three sides of the triangle. This is true for any triangle, not just right triangles.

Press 'reset' in the diagram above. Note that side 'a' has a length of 25.1, and its opposite angle A is 67°. The sine of 67° is 0.921, so the ratio of 25.1 to 0.921 is 27.27. If you repeat this for the other three sides, you will find they have the same ratio, designated here by the letter s.

As you drag the above triangle around, you will see that although this ratio varies, it is always the same for all three sides of the triangle.

Written as a formula

The Law of Sines is written formally as

Law of sines formula Calculator
where A is the angle opposite side a, B is the angle opposite side b, and C is the angle opposite side c.

What is it used for?

A triangle has three sides and three angles. The Law of Sines is one of the tools that allows us to solve the triangle. That is, given some of these six measures we can find the rest. Depending on what you are given to start, you may need to use this tool in combination with others to completely solve the triangle.

When do I use it?

You can use the Law of Sines if you already know

  1. One side and its opposite angle, and
  2. One or more other sides or angles
The first allows us to calculate the "Law of Sines" ratio s. Then we can use this ratio to find other sides and angles using the other givens.

Example

In the figure on the right, we are given side b and angle B, which opposite each other, so we can use them to calculate the 'Law of Sines' ratio (s) for this particular triangle: Law of Sines example 1 Equation.  s equals b over sine b, which equals 14 over sine 40, which equals 27.78 We are also given the length of side c. Because we know the Law of Sines ratio, we can find the opposite angle C: c over sine C equals s We now know both angles B and C, so using the fact that the interior angles of a triangle add up to 180°, we can find the third angle A: Using the same principle as above we know that so we solve this for a, the last unknown side:
Calculator
We have now solved the triangle, since we now know all three sides and all three angles.

The circumcircle connection

It turns out that the "Law of Sines" ratio is also the diameter of the triangle's circumcircle, which is the circle that passes through all three vertices of the triangle. This is sometimes formally written as where r is the circumradius - the radius of the triangle's circumcircle.

Summary

So if we are given one side and its opposite angle we can find the "law of Sines" ratio for the triangle. Then, using that ratio and the other given elements, we can solve the triangle.

Proof

See Proof of the Law of Sines.

Things to try

In the figure above
  1. click "hide details' then reshape the triangle by dragging its vertices.
  2. Solve the triangle using the Law of Sines.
When done, click on 'show details' to verify your answer.

Related triangle topics

General

Perimeter / Area

Triangle types

Triangle centers

Congruence and Similarity

Solving triangles

Triangle quizzes and exercises

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