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.

Question

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

Step-by-Step:

1

Start with the formula:
Length of arc = rθ

2

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

Answer:

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.

arc

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

Beware

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.
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See Also

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