Length of an Arc (Radians)
(KS3, Year 7)

The Lesson

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

length of arc equals radius times angle in radians In this formula, r is the radius of the circle and θ is the angle (in radians) subtended by the arc. The image below shows what we mean by the length of an arc:

arc, angle and radius

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

Finding the length of an arc of a circle, when the angle is in radians, is easy.


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



Start with the formula:
Length of arc = rθ


Substitute the radius and the angle into the formula. In our example, r = 5 and θ = 1.
Length of arc = 5 × 1 Length of arc = 5 cm


The length of an arc of a circle with a radius of 5 cm, which is subtended by an angle of 1 radian, is 5 cm.

Lesson Slides

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. Open the slider in a new tab

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.

1 radian equals 1 radius wrapped around circumference

Why Does the Formula Work?

An angle in radians can be found using the formula:
Angle = Arc length ⁄ Radius
We can rearrange this formula to make the arc length the subject:
Arc length = Angle × Radius Arc length = θ × r


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.
Help Us To Improve Mathematics Monster

  • Do you disagree with something on this page?
  • Did you spot a typo?
Please tell us using this form

See Also

What is a circle? What is an angle? What are radians?