# Length of an Arc(KS3, Year 7)

## The Lesson

The length of an arc of a circle is given by the formula: 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: ## 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? # 1

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

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. ## 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. 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:
Length of arc = θ360° × 2πr

## 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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