Follow Us on Twitter

How to Find the Length of an Arc of a Circle (Mathematics Lesson)

What Is the Length of an Arc of a Circle?

The length of an arc of a circle is given by the formula:



where θ is the angle (in degrees) subtended by the arc and r is the radius of the circle.

How to Find the Length of an Arc of a Circle

Question: What is the length of the arc with angle θ and radius r, as shown below?



Step 1
Multiply the radius r by π.
π × r = πr.

Step 2
Multiply by 2.
2 × πr = 2πr.

Step 3
Multiply by the angle θ.
θ × 2πr.

Step 4
Divide by 360°.
θ360° × 2πr.

This is the length of the arc with angle θ and radius r.

A Real Example of How to Find the Length of an Arc of a Circle

Question: What is the length of the arc with an angle of 60° and a radius of 5cm, as shown below?



Step 1
Multiply the radius, 5cm, by π.
π × 5 = 15.7cm.

Step 2
Multiply by 2.
2 × 15.7 = 31.4cm.

Step 3
Multiply by the angle 60°.
60° × 31.4 = 1,885cm.

Step 4
Divide by 360°.
1,885 ÷ 360° = 5.2cm.

The length of the arc with an angle of 60° and a radius of 5cm is 5.2cm.

Another Real Example of How to Find the Length of an Arc of a Circle

The slider below shows another real example of how to find the length of an arc of a circle:

How to Find the Length of an Arc of a Circle When the Angle Is in Radians

The formula used on this page works when the angle subtended by the arc is in degrees.

When the angle subtended by the arc is given in radians, the formula for the length of the arc becomes:



Read more about how to find the length of an arc (in radians)
Curriculum
Geometry Lessons
Interactive Test
  show
 
Note
WHAT IS AN ARC?

An arc is a portion of the circumference.



WHY DOES THE FORMULA WORK?

The length of an arc is just a fraction of the circumference of the circle of the same radius.

For example, an arc that is halfway round a circle is half the circumference of a circle.



An arc that is a quarter way round a circle is quarter the circumference of a circle.



In each case, the fraction is the angle of the arc divided by the full angle of the circle.



When measured in degrees, the full angle is 360°.

Hence for a general angle θ, the formula is the fraction of the angle θ over the full angle 360° multiplied by the circumferece of the circle:



IS THE ANGLE GIVEN IN DEGREES OR RADIANS?

The formula to find the length of an arc of a circle depends on whether the angle at the center of the arc is given in degrees or radians.

Make sure you check what units the angle is given in.