Circle Sectors and Arcs | Circle Segements | Maths Made Easy

# Circles Sectors and Arcs Worksheets, Questions and Revision

Level 6 Level 7

## Circle Sectors and Arcs

It can be useful to calculate the area and perimeter of segments of circles.

Make sure you are happy with the following topics before continuing:

Paying special attention to calculating the Area of a circle and circumference of a circle

## Area of a Sector

Area of a sector = $\dfrac{\textcolor{blue}{x}}{360\degree} \times \textcolor{red}{\text{Area of full circle}}$

Where $\textcolor{blue}{x}$ is the angle of the sector.

The $\textcolor{red}{\text{area of the circle}}$ will be equal to $\textcolor{red}{\pi r^2}$

Calculating the area of the circle segment shown involves using the angle of the sector $\textcolor{blue}{82\degree}$ and the radius $\textcolor{red}{12}$.

We apply these to the equation above:

$\dfrac{\textcolor{blue}{82\degree}}{360\degree} \times \textcolor{red}{\pi \times 12^2} = 103.04 \text{ cm}^2$

## Arc Length

The arc length is the length of the part of the circumference which is part of the circle segment. The equation for this is:

Length of arc = $\dfrac{\textcolor{blue}{x}}{360\degree} \times \textcolor{red}{\text{circumference of full circle}}$

Where $\textcolor{blue}{x}$ is the angle of the sector.

The $\textcolor{red}{\text{circumference of the full circle}}$ will be equal to $\textcolor{red}{\pi D}$

We can calculate the arc length of the circle segment shown by using the angle of the sector $\textcolor{blue}{118\degree}$ and the diameter $2 \times\textcolor{red}{9} = \textcolor{red}{18}$.

Now we apply these to the equation above:

$\dfrac{\textcolor{blue}{118\degree}}{360\degree} \times \textcolor{red}{\pi \times 18} = 18.54$ cm

## Example 1: Area and Circumference

The circle shown has centre $C$.

Find its area and circumference to $1$ decimal place.

[2 marks]

Because the line passes through the centre, we know it is a diameter of the circle. So,

Circumference $= \pi d = \pi \times 42 = 131.9 \text{cm (1d.p.)}$

The diameter is twice the length of the radius, so the $\text{radius } = 42 \div 2 = 21$cm. So:

Area of circle $= \pi r^2=\pi\times (21^2) = 1385.4\text{cm}^2\text{ (1d.p.)}$

Note: some questions will ask you to “leave your answer in terms of $\pi$”, this means that the answer should be in the form of “something $\times \pi$”.

## Example 2: Circle Sectors & Arc Length

The sector of a circle has centre $C$ as shown.

Find the area of the sector and the arc length to $3$ significant figures.

[2 marks]

The angle is $102\degree$, which means that this sector is $\frac{102}{360}$ as a fraction of the whole circle.

So, we get:

Sector area = $\dfrac{102}{360} \times \pi \times 8^2$

$= 57.0\text{cm}^2$

Arc length = $\dfrac{102}{360} \times \pi \times 2 \times 8$

$= 14.2\text{cm}$

### Example Questions

The formula for the area of a circle is $\pi r^2$. In this question we are given the diameter rather than the radius. Given that the diameter is twice the length of the radius:

$\text{radius} = 5.24 \div 2 = 2.62 \text{cm}$

Therefore,

$\text{area } = \pi r^2 = \pi \times (2.62)^2 = 21.6 \text{cm}^2$

The formula for the area of a circle is $\pi r^2$. The area of the sector is,

$\text{area} =\dfrac{50\degree}{360\degree} \times \pi (5)^2 = \dfrac{50\degree}{360\degree} \times 25 \pi = 10.9\text{m}^2$

The formula for the area of a circle is $\pi r^2$. The area of this sector, $26.15 \text{m}^2$, must be equal to $\frac{x}{360}$ of the total area of the circle. So, as an equation, this looks like:

$26.15 = \dfrac{x\degree}{360\degree} \times \pi \times 9^2 = \dfrac{x\degree}{360\degree} \times 81\pi$

Dividing the left-hand side and right-hand side by $81 \pi$ gives

$\dfrac{26.15}{81\pi} = \dfrac{x\degree}{360\degree}$

Then, multiply both sides by 360 to get,

$x=\dfrac{26.15}{81\pi}\times 360\degree$

$x = 37.0\degree$ (to 1d.p.)

The question asks for the total perimeter of the shape. We know that one side is 14 mm but the other two are missing. Immediately we can identify that the other straight side is also 14 mm. Then, all that remains is to calculate the arc length.

The angle in this sector is 165 degrees, meaning that the arc length will be equal to $\frac{165}{360}$ of the total circumference. The formula for the circumference is $\pi d$, or alternatively (and more helpfully in this case), $2\pi r$. So, we get:

Arc length $= \dfrac{165}{360} \times 2 \pi \times 14 = 40.3 \text{mm}$

Therefore,

$\text{total perimeter} = 14 + 14 + 40.3 = 68.3 \text{mm}$

The formula for the area of a circle is $\pi r^2$. The area of this sector, $160 \text{m}^2$, must be equal to $\frac{x}{360}$ of the total area of the circle. So, as an equation, this looks like:

$160 = \dfrac{x\degree}{360\degree} \times \pi \times 9^2 = \dfrac{x\degree}{360\degree} \times 81\pi$

Dividing the left-hand side and right-hand side by $81 \pi$ gives

$\dfrac{160}{81\pi} = \dfrac{x\degree}{360\degree}$

Then, multiply both sides by 360 to get,

$x=\dfrac{160}{81\pi}\times 360\degree$

$x = 226\degree$ (to 3s.f.)

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