# Arc Length & Area of a Sector

## Arc Length & Area of a Sector Revision

**Arc Length & Area of a Sector**

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

Make sure you’re up to snuff on your radians!

**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.

**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.

**Example 1: Arc Length**

The following circle has radius \textcolor{red}{r} = \textcolor{red}{5}\text{ cm} and angle \textcolor{blue}{\theta} = \textcolor{blue}{105}\degree.

By converting the angle into radians, find the length of \textcolor{purple}{l} to 2 decimal places.

**[3 marks]**

To convert degrees to radians, we multiply \textcolor{blue}{\theta} by \dfrac{\pi}{180}\\

\textcolor{blue}{105}\degree\times\dfrac{\pi}{180}=\textcolor{blue}{\dfrac{7\pi}{12}}\\

Now we can use this formula to calculate the arc length:

\textcolor{purple}{l} = \textcolor{red}{r}\textcolor{blue}{\theta}\\ \textcolor{purple}{l} = \textcolor{red}{5}\times\textcolor{blue}{\dfrac{7\pi}{12}}\\ \textcolor{purple}{l} = 9.16\text{ cm}\\

**Example 2: Area of a Sector**

The following circle has radius \textcolor{red}{r} = \textcolor{red}{4}\text{ cm} and angle \textcolor{blue}{\theta} = \textcolor{blue}{\dfrac{5\pi}{6}}\text{ radians}.

Find the area of the shaded sector to 2 decimal places.

**[2 marks]**

As the angle is in radians, we can use this formula to calculate the area of the sector:

Area = \dfrac{1}{2}\textcolor{red}{r}^2\textcolor{blue}{\theta}\\

Area = \dfrac{1}{2}\times\textcolor{red}{4}^2\times\textcolor{blue}{\dfrac{5\pi}{6}}\\

Area = 20.94\text{ cm}^2\\

## Arc Length & Area of a Sector Example Questions

**Question 1:** What is the perimeter of a section of a circle with angle \theta = 45° and radius r = 9\text{ mm}? Give your answer in the form 9(a + b\pi).

**[3 marks]**

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}

**Question 2:** A spinner of radius 4\text{ cm} has 6 identical sections. Using the equation for sector area, find the shaded area (in \text{cm}^2).

**[2 marks]**

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.

**Question 3:** Here is a plot for a garden, including a pool and a patio (grey). Find the area of the garden covered by grass, in the form a + b\pi.

**[4 marks]**

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