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!

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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 $textcolor{blue}{theta} = 2pi$ . Then we have $textcolor{purple}{l} = 2textcolor{red}{r}pi = dpi$ , 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 $textcolor{orange}{A} = dfrac{1}{2}textcolor{red}{r}^2textcolor{blue}{theta}$ .

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

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

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 $dfrac{pi}{180}$

$textcolor{blue}{105}degreetimesdfrac{pi}{180}=textcolor{blue}{dfrac{7pi}{12}}$

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

$textcolor{purple}{l} = textcolor{red}{r}textcolor{blue}{theta}$

$textcolor{purple}{l} = textcolor{red}{5}timestextcolor{blue}{dfrac{7pi}{12}}$

$textcolor{purple}{l} = 9.16text{ cm}$

Example 2: Area of a Sector

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

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

Area $= dfrac{1}{2}timestextcolor{red}{4}^2timestextcolor{blue}{dfrac{5pi}{6}}$

Area $= 20.94text{ cm}^2$

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 $r = 9text{ mm}$ ? Give your answer in the form $9(a + bpi)$ .

[3 marks]

Firstly, we have

$theta = 45° = dfrac{pi}{4}$

Then the length of the arc is

$l = rtheta = dfrac{9pi}{4}$

Giving the total perimeter

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

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Question 2: A spinner of radius $4text{ cm}$ 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

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

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

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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 $a + bpi$ .