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

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

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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 $8\pi\text{ cm}^2$ .

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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]