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

What Is the Length of an Arc of a Circle When the Angle Is in Radians?

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



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

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

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



Step 1
Multiply the angle θ by the radius r.
θ × r = θr.

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 When the Angle Is in Radians

Question: What is the length of the arc with an angle of 1 radian and a radius of 5cm, as shown below?



Step 1
Multiply the angle, 1 radian, by the radius, 5cm.
1 × 5 = 5cm.

The length of the arc with an angle of 1 radian and a radius of 5cm is 5cm.

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

The slider below shows another real example of how to find the length of an arc of a circle when the angle is in radians:

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

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

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



Read more about how to find the length of an arc (in degrees)
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Note
WHAT IS AN ARC?

An arc is a portion of the circumference.



WHAT ARE RADIANS?

Radians are a way of measuring angles.

1 radian is the angle found when the radius is wrapped around the circle.



There are 2π radians in a full angle.

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 radians, the full angle is 2π.

Hence for a general angle θ, the formula is the fraction of the angle θ over the full angle 2π multiplied by the circumference of the circle:



The 2π's cancel, leaving the simpler formula: