# The length of an arc and the perimeter of a sector

## Home learning focus

Learn how to find the length of an arc and the perimeter of a sector.

This lesson includes:

- learning summary
- two activity sheets

# Learn

Students looking to achieve grade 4 in GCSE Maths must understand how to calculate the length of an arc and the perimeter of a sector.

Read page 5 of our 'Circles, sectors and arcs' Bitesize revision guide and page 3 from our 'Sector, segment and arc' Bitesize revision guide to understand:

## Arc length

An arc is a portion of the circumference of a circle.

A chord separates the circumference of a circle into two sections - the **major arc** and the **minor arc**. It also separates the area into two segments - the **major segment** and the **minor segment**.

**Worked example**

Calculate the arc length to 2 decimal places (dp).

First calculate what fraction of a full turn the angle is.

The angle is 90°, so it is one quarter of the whole circle (360°). The arc length is ¼ of the full circumference.

The **circumference of a circle** is πd (or π × diameter) and the **diameter** is = 2 × radius.

Remember that **π** is = 3.14 (to 2 dp)

The formula to calculate the arc length is: **Arc length** = angle ÷ 360° × π × d

The arc length is: ¼ × π × 8 = 2π

**2π** is 2 × 3.14

The arc length is 6.28 cm (to 2 dp).

## Perimeter of a sector

The **perimeter** is the distance all around the outside of a shape. We can find the perimeter of a **sector** using what we know about finding the length of an arc.

A sector is formed between two radii and an arc. To find the perimeter, we need to add these values together.

Perimeter = Arc length + 2r

**Worked example**

In this diagram, the arc length (27.5 cm) and the radius (45 cm) are shown.

From this the perimeter can be calculated:

Perimeter = 27.5 + (2 × 45)

27.5 + 90 = 117.5 cm

The perimeter is **117.5 cm.**

To learn more about arcs look at the Circles, sectors and arcs Bitesize guide here.

# Practise

## Activity 1

**Finding the length of an arc**

Complete the activity sheet from TES on finding the length of an arc to test your knowledge. You can print it out or write your answers on a piece of paper.

## Activity 2

**The length of an arc**

Complete the activity sheet from White Rose Maths on finding the length of an arc to test your knowledge. You can print it out or write your answers on a piece of paper.

Click here to see the correct answers.

# Choose your exam specification

BBC Bitesize has GCSE exam board-relevant content for students in England, Northern Ireland and Wales. Chose the exam specification that matches the one you study.

# There's more to learn

Have a look at these other resources around the BBC and the web.