The Law of Sines - Math Open Reference (2024)

Try thisDrag 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 aswhere 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 below, 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:Notice here we are also given the length of side c. So, because we know the Law of Sines ratio for this triangle (s - 21.78), we can find the opposite angle C:

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 thatso we solve this for a, the last unknown side: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 thatratio 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.

Other triangle topics

General

• Triangle definition
• Hypotenuse
• Triangle interior angles
• Triangle exterior angles
• Triangle exterior angle theorem
• Pythagorean Theorem
• Proof of the Pythagorean Theorem
• Pythagorean triples
• Triangle circumcircle
• Triangle incircle
• Triangle medians
• Triangle altitudes
• Midsegment of a triangle
• Triangle inequality
• Side / angle relationship

Perimeter / Area

• Perimeter of a triangle
• Area of a triangle
• Heron's formula
• Area of an equilateral triangle
• Area by the "side angle side" method
• Area of a triangle with fixed perimeter

Triangle types

• Right triangle
• Isosceles triangle
• Scalene triangle
• Equilateral triangle
• Equiangular triangle
• Obtuse triangle
• Acute triangle
• 3-4-5 triangle
• 30-60-90 triangle
• 45-45-90 triangle

Triangle centers

• Incenter of a triangle
• Circumcenter of a triangle
• Centroid of a triangle
• Orthocenter of a triangle
• Euler line

Congruence and Similarity

• Congruent triangles

Solving triangles

• Solving the Triangle
• Law of sines
• Law of cosines

Triangle quizzes and exercises

• Triangle type quiz
• Ball Box problem
• How Many Triangles?
• Satellite Orbits

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