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# Length of an Arc (Radians)

(KS3, Year 7)

## The Lesson

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

## 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?## 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.## 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.## 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.**Help Us To Improve Mathematics Monster**

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