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# 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?## Step-by-Step:

# 1

Start with the formula:

Length of arc =

^{θ}⁄_{360°}× 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.## 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 =

^{θ}⁄_{360°}× 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.**Help Us To Improve Mathematics Monster**

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