Area of a sector

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Area Of A Sector

Here we will learn how to calculate the area of a sector including how to identify a sector of a circle, form and use the formula for the area of a sector of a circle and calculate the area of a sector in various scenarios.

There are also area of a sector circle worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

What is the area of a sector?

The area of a sector is the space inside the section of the circle created by two radii and an arc. It is a fraction of the area of the entire circle.

Major sector: a major sector has a central angle which is more than $180°$ .
Minor sector: a minor sector has a central angle which is less than $180°$ .

Area of a sector $=\frac{\theta}{360} \times \pi r^{2}$

θ $=$ angle of the sector

$r$ $=$ radius of the circle

What is the area of a sector?

How to find the area of a sector

In order to find the area of a sector you need to be able to find the area of a circle. This is because the area of a sector is part of the total area of the circle.

‘How much’ of the circle is decided by the angle created by the two radii.

The sum of the angle around a point is equal to $360°$ .

Therefore, the area of a sector is the fraction of the full circle’s area.

The angle is out of $360$ or $\frac{\theta}{360}$ , where θ represents the angle, so we can multiply this by the area of the circle to calculate the area of a sector.

NOTE: At GCSE, all angles are measured in degrees. Make sure that your calculator has a small ‘d’ for degrees at the top of the screen rather than an ‘r’ for radians – these are not used until A Level.

Area of a sector formula:

Area of a sector = $\frac{\theta}{360} \times \pi r^{2}$

θ = angle of the sector

$r$ = radius of the circle

In order to solve problems involving the area of a sector you should follow the below steps:

Find the length of the radius $\pmb{r}$ .
Find the size of the angle creating the sector.
Substitute the value of the radius and the angle into the formula for the area of a sector.
Clearly state your answer.

How to find the area of a sector

Area of a sector worksheet

Get your free area of a sector worksheet of 20+ questions and answers. Includes reasoning and applied questions.

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Area of a sector worksheet

Get your free area of a sector worksheet of 20+ questions and answers. Includes reasoning and applied questions.

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Area of a sector is part of our series of lessons to support revision on sector of a circle and circles, sectors and arcs. You may find it helpful to start with the main circles, sectors and arcs lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:

Area of a sector examples

Example 1: calculate the area of a sector (quadrant)

Calculate the area of the segment shown below. Give your answer to 3 decimal places.

Find the length of the radius.

Radius = $6cm$ .

2Find the size of the angle creating the sector.

Angle = $90°$ (shown by the symbol of the right angle).

3Substitute the value of the radius and the angle into the formula for the area of a sector.

Area of a sector = $\frac{\theta}{360} \times \pi r^{2}$

= \frac{90}{360} \times \pi\times 6^{2}

= 9\pi

4Clearly state your answer

The question asked you to round your answer to 3 decimal places

Area of a sector = ${9\pi}cm^2$

= $28.274333\dotscm^2$

= $28.274cm^2$ (3 d.p.)

Example 2: calculate the area of a sector (semicircle)

Calculate the area of the semicircle shown below. Give your answer in terms of $\pi.$

Find the length of the radius.

Diameter = $24cm$

Radius = $12cm$ – as the radius is half the diameter

Find the size of the angle creating the sector

Angle = $180°$ . This is because the shape shown is a semicircle. Therefore, the angle of the straight line is $180$ degrees.

Substitute the value of the radius, the angle into the formula for the area of a sector.

Area of a sector= $\frac{\theta}{360} \times \pi r^{2}$

$= \frac{180}{360} \times \pi \times 12^{2}$

$= 72\pi$

Clearly state your answer

The question asked you to round your answer in terms of $\pi$ :

Area of a sector = $72\pi cm^{2}$

Example 3: calculate the area of a sector (with an angle given)

Calculate the area of the sector shown below. Give your answer to 3 significant figures.

Find the length of the radius.

Radius = $8cm$

Find the size of the angle creating the sector.

Angle = $115°$

Substitute the value of the radius, the angle into the formula for the area of a sector.

Area of a sector = $\frac{\theta}{360} \times \pi r^{2}$

$= \frac{180}{360} \times \pi\times 8^{2}$

$= \frac{184}{9}\pi$

Clearly state your answer.

The question asked you to round your answer to 3 significant figures.

Area of a sector = $\frac{184}{9}\pi cm^{2}$

= $62.228…cm^{2}$

= $62.2cm^{2}$ (3 s.f.)

Example 4: calculate the area of a sector (the angle is not given)

Calculate the area of the sector $AOB$ below.

The length of the radius $(OB)$ is $9cm$

The length of a chord $(AB)$ is $10cm$ .

Give your answer to 2 decimal places

Find the length of the radius

Radius = $9cm$

Find the size of the angle creating the sector

In this example you are not given the angle of the sector, you need to calculate it first.

Here you can use the triangle created by the two radii and the chord to find the angle (see below)

In this example you need to apply the cosine rule to find the angle.

$a^2=b^2+c^2-2bcCos(A)$

$A$ is the angle you are trying to find. You can therefore use the rearranged cosine rule to find the angle.

\[\begin{aligned} \operatorname{Cos} A&=\frac{b^{2}+c^{2}-a^{2}}{2 b c} \\ \operatorname{Cos} A&=\frac{9^{2}+9^{2}-10^{2}}{2 \times 9 \times 9} \\ \operatorname{Cos} A&=\frac{31}{81} \\ A&=\operatorname{Cos}^{-1}\left(\frac{31}{81}\right) \\ A&=67.497977…^{\circ} \end{aligned}\]

The size of the angle creating the sector (made by the two radii) is $67.498°$ .

Substitute the value of the radius, the angle into the formula for the area of a sector.

Area of a sector $=\frac{\theta}{360} \times \pi r^{2}$

$= \frac{67.498}{360} \times \pi\times 9^{2}$

$= \frac{184}{9}\pi$

$= 47.71152\dots$

Notice in this example how your calculator does not automatically give your answer in terms of pi.

Clearly state your answer.

The question asked you to round your answer to 2 decimal places

Area of a sector $= 47.71152… cm^{2}$

$= 47.71 cm^{2}$ (2 d.p.)

How to solve problems involving an area of a sector

Sometimes you may be given the area of the sector and asked to find a property of the circle such as the radius. In this case you need to ‘reverse’ the process.

Clearly state which of the properties you know and do not know (e.g. radius, angle of sector, area of sector).
Using the formula for the area of a sector create an equation and solve for the unknown property you have been asked to find.
Clearly state your answer (check how the questions asks for the answers to be rounded).

Solving problems involving an area of a sector examples

Example 5: finding the radius given the area of the sector

The sector below has an area of $20cm^2$ . Calculate the length of $x$ . Give you answer to $2$ decimal places

Clearly state which of the properties you know and do not know (e.g. radius, angle of sector, area of sector).

Radius = $x$

Angle of sector = $115°$

Area of sector = $20cm^2$

Using the formula for the area of a sector create an equation and solve for the unknown property you have been asked to find.

\[\begin{aligned} \text { Area of a sector }&=\frac{\theta}{360} \times \boldsymbol{\pi} r^{2} \\ 20&=\frac{115}{360} \times \boldsymbol{\pi} \times x^{2} \quad \quad \quad \text {simply the fraction (if possible) }\\ 20&=\frac{23}{72} \times \boldsymbol{\pi} \times x^{2} \quad \quad \quad \text { divide each side of the equation by the fraction }\\ \frac{1440}{23}&=\boldsymbol{\pi} \times x^{2} \quad \quad \quad \text { divide each side of the equation by pi } \\ 19.928967&=x^{2} \quad \quad \quad \text { square root each side of the equation } \\ \text { 4.464187… }&=x \quad \quad \quad \text { only the positive answer is given as x represents a length } \end{aligned}\]

Clearly state your answer (check how the questions asks for the answers to be rounded).

The question asks you to give your answer to 2 decimal places

\begin{aligned} &x=4.464187\\ &=4.46\,(2\,d.p.) \end{aligned}

Example 6: finding the angle of a sector given the area of the sector

The sector below has an area of $62cm^2$ . Calculate the length of $x$ . Give you answer to $2$ decimal places

Clearly state which of the properties you know and do not know (e.g. radius, angle of sector, area of sector).

Radius = $8.5$

Angle of sector = $x°$

Area of sector = $62cm^2$

Using the formula for the area of a sector create an equation and solve for the unknown property you have been asked to find.

\[\begin{aligned} \text { Area of a sector }&=\frac{\theta}{360} \times \boldsymbol{\pi} r^{2} \\ 62&=\frac{x}{360} \times \boldsymbol{\pi} \times 8.5^{2} \\ 62&=\frac{x}{360} \times \boldsymbol{\pi} \times 72.25 \quad \quad \quad \text { divide both slides by 72.25 }\\ 0.85813&=\frac{x}{360} \times \boldsymbol{\pi} \quad \quad \quad \text { divide both slides by pi }\\ 0.27315&=\frac{x}{360} \quad \quad \quad \text { multiply both slides by 360 } \\ 98.3344&=x \end{aligned}\]

Clearly state your answer (check how the questions asks for the answers to be rounded).

The question asks you to give your answer to 2 decimal places:

\begin{aligned} &x=98.3344\\ &=98.33\, (2\, d.p.) \end{aligned}

Common misconceptions

Confusing the segment/sector

A segment is made from a chord whilst a sector will have lines (radii) coming from the origin.

Sectors are a portion of a circle – it can be helpful to think of a sector as a pizza slice.

Finding the area of the whole circle not the area of the sector

Remember to find the fraction of the circle that makes the sector not just the area of the whole circle.

Use

$\frac{\theta}{360} \times \pi r^{2}$

Not

\pi r^2

To find the area of a sector

Incorrect use of the cosine rule

Many mistakes are made when applying other rules within a sector question, such as the cosine rule. Take your time with these parts and regularly ask if your answer makes sense within the context of the question. If you need to review the cosine rule do so before continuing on further.

Practice area of a sector questions

90^{\circ}

180^{\circ}

270^{\circ}

360^{\circ}

A quadrant is a quarter of a circle. Angles around a point are equal to $360^{\circ}$ so a quadrant will be a quarter of $360^{\circ}$ .

$360\div4 =90$

90^{\circ}

180^{\circ}

270^{\circ}

360^{\circ}

A semi circle is half a circle. Angles around a point are equal to $360^{\circ}$ so a half will be half of $360^{\circ}$ .

$360\div2 =180$

9\pi cm^{2}

100\pi cm^{2}

10\pi cm^{2}

25\pi cm^{2}

Area of a sector = $\frac{\theta}{360} \times \pi r^{2}$

$\frac{90}{360} \times \pi\times 10^{2} =25\pi$

157.08 cm^{2}

100\pi cm^{2}

25\pi cm^{2}

50\pi cm^{2}

Area of a sector = $\frac{\theta}{360} \times \pi r^{2}$

$\frac{180}{360} \times \pi\times 10^{2 }= 50\pi$

150.796 cm^{2}

48\pi cm^{2}

64\pi cm^{2}

8\pi cm^{2}

Area of a sector = $\frac{\theta}{360} \times \pi r^{2}$

$\frac{270}{360} \times \pi\times 8^{2} = 48\pi$

175.93 cm^{2}

56\pi cm^{2}

64\pi cm^{2}

8\pi cm^{2}

Area of a sector = $\frac{\theta}{360} \times \pi r^{2}$

$\frac{315}{360} \times \pi\times 8^{2} = 56\pi$

Area of a sector GCSE questions

1. The diagram shows a sector of a circle with centre $O$ .

The radius of the circle is $13cm$ .
The angle of the sector is $150^{\circ}.$

Calculate the area of the sector, give your answer correct to 3 significant figures.

(3 marks)

Show answer

$\frac{150}{360}\times \pi \times 13^2$

(1)

$221.220\dots$

(1)

$221 cm^{2}$

(1)

2. The diagram shows a sector of a circle with centre $O$ .

The radius of the circle is $6.5cm$ .
The acute angle $AOB$ is $70^{\circ}.$

Calculate the area of the sector, give your answer correct to 3 significant figures.

(3 marks)

Show answer

$\frac{290}{360}\times \pi \times 6.5^2$

(1)

$106.923\dots$

(1)

$107 cm^{2}$

(1)

3. The diagram shows a sector of a circle with centre $O$ .

The radius of the circle is $4.9cm$ .
The acute angle $AOB$ is $60^{\circ}.$

Calculate the area of the sector, give your answer correct to 2 decimal places.

(3 marks)

Show answer

$\frac{60}{360}\times \pi \times 4.9^2$

(1)

$12.5716\dots$

(1)

$12.57 cm^{2}$

(1)

4. Below is a sector of a circle with an area of $5\pi cm^{2}$ and a radius of $10cm$ . Find the size of the angle labelled $x$ .

(3 marks)

Show answer

$5\pi=\frac{x}{360} \times \pi \times 10^2$

Correct attempt to form an equation for the area of the sector

(1)

Attempt to solve equation for $x$ , with two steps completed correctly

(1)

$18$

(1)

5. The sector below has an area of $30 cm^{2}$ .

Calculate the length of $x$
Give you answer to $2$ decimal places.

(4 marks)

Show answer

$30=\frac{135}{360} \times \pi \times x^2$

Correct attempt to form an equation for the area of the sector

(1)

$\frac{80}{\pi}$ or $25.46479\dots seen$

(1)

$5.04626$ negative solution cannot be the solution

(1)

$5.05$

(1)

Learning checklist

You have now learned how to:

Identify and apply circle definitions and properties, including: centre, radius, chord, diameter, circumference

Identify and apply circle definitions and properties, including tangent, arc, sector and segment

Calculate areas of a sector

Calculate a property of a circle given the area of its sector

The next lessons are

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