# Arc Length & Area of a Sector

A LevelAQAEdexcelOCRAQA 2022OCR 2022

## Arc Length & Area of a Sector

We’ll now look at how to calculate arc lengths and sector areas.

A Level

## Arc Length

Let’s say you’ve got a section of a circle, and you want to find the length of the curved edge.

For a circle with radius $\textcolor{red}{r}$ and angle $\textcolor{blue}{\theta}$, we have the arc length $\textcolor{purple}{l} = \textcolor{red}{r}\textcolor{blue}{\theta}$.

As mentioned, it’s important that you’re using radians for your value of $\textcolor{blue}{\theta}$.

We can actually use this formula to derive the circumference of a circle.

Set $\textcolor{blue}{\theta} = 2\pi$. Then we have $\textcolor{purple}{l} = 2\textcolor{red}{r}\pi = d\pi$, where $d$ is the diameter.

A Level

## Area of a Sector

Say we have the same section of a circle, but we now wish to calculate the area of the sector.

For a circle with radius $\textcolor{red}{r}$ and angle $\textcolor{blue}{\theta}$, we have the sector area $\textcolor{orange}{A} = \dfrac{1}{2}\textcolor{red}{r}^2\textcolor{blue}{\theta}$.

Again, using $\textcolor{blue}{\theta = 2\pi}$ gives us the equation for the area of a circle:

$\textcolor{orange}{A} = \dfrac{1}{2}\textcolor{red}{r}^2 \textcolor{blue}{2\pi} = \pi \textcolor{red}{r}^2$.

A Level

## Example Questions

Firstly, we have

$\theta = 45° = \dfrac{\pi}{4}$

Then the length of the arc is

$l = r\theta = \dfrac{9\pi}{4}$

Giving the total perimeter

$\dfrac{9\pi}{4} + 9 + 9 = 9(2 + \dfrac{\pi}{4})\text{ mm}$

First of all, the angle $\theta = 60° = \dfrac{\pi}{3}$.

The area of one sector is

$\dfrac{1}{2}r^2 \theta = \dfrac{1}{2} \times 4^2 \times \dfrac{\pi}{3} = \dfrac{8\pi}{3}\text{ cm}^2$

so the area of all three identical sectors is $8\pi\text{ cm}^2$.

Total area of garden:

$5 \times 3.5 = 17.5$

Area of patio:

$(1 \times 3.5) + (\dfrac{1}{2} \times 1^2 \times \pi) = 3.5 + \dfrac{\pi}{2}$

Area of pool:

$\dfrac{1}{2} \times 1.25^2 \times \dfrac{\pi}{2} = \dfrac{25\pi}{64}$

Then, the total area covered by grass is

$17.5 - (3.5 + \dfrac{\pi}{2} + \dfrac{25\pi}{64}) = 14 - \dfrac{57}{64}\pi$

A Level

A Level

## You May Also Like...

### A Level Maths Revision Cards

The best A level maths revision cards for AQA, Edexcel, OCR, MEI and WJEC. Maths Made Easy is here to help you prepare effectively for your A Level maths exams.

£14.99

### A Level Maths – Cards & Paper Bundle

A level maths revision cards and exam papers for Edexcel. Includes 2022 predicted papers based on the advance information released in February 2022! MME is here to help you study from home with our revision cards and practise papers.

From: £22.99

### Transition Maths Cards

The transition maths cards are a perfect way to cover the higher level topics from GCSE whilst being introduced to new A level maths topics to help you prepare for year 12. Your ideal guide to getting started with A level maths!

£8.99