Length of an Arc
(KS3, Year 7)

The Lesson

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

length of arc equals angle divided by 360 all times 2 times pi times radius In this formula, θ is the angle (in degrees) subtended by the arc and r is the radius of the circle. The image below shows what we mean by the length of an arc:

angle, arc and radius on a circle

How to Find the Length of an Arc of a Circle

Finding the length of an arc of a circle is easy.

Question

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

Step-by-Step:

1

Start with the formula:
Length of arc = θ360° × 2πr
Don't forget: π is pi (≈ 3.14) and / means ÷.

2

Substitute the angle and the radius into the formula. In our example, θ = 60° and r = 5.
Length of arc = 60°360° × 2 × π × 5 Length of arc = (60° ÷ 360°) × 2 × 5 × π Length of arc = 5.2 cm

Answer:

The length of an arc of a circle with a radius of 5 cm, which is subtended by an angle of 60°, is 5.3 cm.

Lesson Slides

The slider below shows another real example of how to find the length of an arc of a circle. Open the slider in a new tab

What Is an Arc?

An arc is a portion of the circumference.

arc

Why Does the Formula Work?

The length of an arc is just a fraction of the circumference of the circle of the same radius. The circumference is given by 2πr, where r is the radius. For example, an arc that is halfway round a circle is half the circumference of a circle.

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

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

angle 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:
Length of arc = θ360° × 2πr

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 is pi? What are degrees? Finding the circumference of a circle What is a full angle?