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Circles, sectors and arcs Area of a circle Circumference of a circle Arc length Solving equations Trigonometry Sine rule Cosine rule Rounding numbers Square numbers and square rootsThis topic is relevant for:

Here we will learn about sectors of a circle, including how to find the area of a sector, the perimeter of a sector and solve problems involving sectors of circles.

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

A** sector of a circle** is formed when a circle is divided using two radii.

Each sector has an angle between the two **radii**. The sector with an angle less than 180 degrees is called a** minor sector** and the sector with an angle greater than 180 degrees is called a** major sector**. If the central angle formed equals 180 degrees, the two sectors would be semicircles.

For example,

In GCSE mathematics you will need to know how to solve problems involving the area of a sector and the perimeter of a sector. You may be asked to give exact answers in terms of \pi or using decimals.

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.

The formula to find the area of a sector is,

\text{Area of a sector } = \frac{\theta}{360} \times \pi r^{2}. \theta = \text{ angle of the sector} r= \text{ radius of the circle}The** perimeter of a sector** is the distance around a sector.

We can calculate the perimeter of a sector by adding together the lengths of the two radii and the arc length of the sector.

To find the arc length of a sector we need to use the formula,

\text{Arc length } = \frac{\theta}{360} \times \pi\times d. \theta = \text{ Angle of the sector} d = \text{ Diameter of the circle}Or

\text{ Arc length } = \frac{\theta}{360} \times 2\times\pi\times r. \theta = \text{ Angle of the sector} r = \text{ Radius of the circle}The total perimeter of a sector formula would be,

\begin{aligned} \text{Perimeter of a sector } &= \text{ Arc length } + \text{ radius } + \text{ radius} \\\\ &= \frac{\theta}{360} \times \pi d+d \;\; \text{ or } \;\; \frac{\theta}{360} \times 2\pi r +2r \end{aligned}In order to calculate the area of a sector:

**Find the length of the radius.****Find the size of the angle creating the sector.****Substitute the values you know into the formula for the area of a sector.****Calculate the answer.**

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

COMING SOONGet your free sector of a circle worksheet of 20+ questions and answers. Includes reasoning and applied questions.

COMING SOONCalculate the area of the sector shown below.

Give your answer to 3 significant figures.

**Find the length of the radius.**

Radius = 8 \ cm

2**Find the size of the angle creating the sector.**

Angle = 115^o

3**Substitute the values you know into the formula for the area of a sector.**

4**Calculate the answer.**

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

\begin{aligned} \text{Area of a sector } &=\frac{184}{9}\pi \ cm^2 \\\\ &= 62.228...cm^2 \\\\ &=62.2 \ cm^2 \ (3 \ s.f) \end{aligned}Calculate the area of the sector shown below.

Give your answer to 3 significant figures.

**Find the length of the radius.**

Radius = 4m

**Find the size of the angle creating the sector.**

The angle creating the sector is the angle included between the two radii.

To find it in this case we need to use the angle fact, angles around a point add to 360^{o}.

360-240=120

The angle creating the sector is 120^{o}.

**Substitute the values you know into the formula for the area of a sector.**

\begin{aligned}
\text { Area of a sector }&=\frac{\theta}{360} \times \pi \times r^{2} \\\\
&=\frac{120}{360} \times \pi \times 4^{2}
\end{aligned}

**Calculate the answer.**

\begin{aligned}
&=16.75516083 \\\\
&=16.8 \ m^{2}
\end{aligned}

Below is a sector of a circle.

The diameter of the circle is 12 \ cm.

Calculate the area of the sector.

**Find the length of the radius.**

Diameter = 12 \ cm

The radius is half of the diameter, so the radius = 6 \ cm.

**Find the size of the angle creating the sector.**

The angle creating the sector is the angle included between the two radii.

To find it in this case we need to use the angle fact, angles around a point add to 360^{o}.

360-205=155

The angle creating the sector is 155^{o}.

**Substitute the values you know into the formula for the area of a sector.**

\begin{aligned}
\text { Area of a sector }&=\frac{\theta}{360} \times \pi \times r^{2} \\\\
&=\frac{155}{360} \times \pi \times 6^{2}
\end{aligned}

**Calculate the answer.**

\begin{aligned}
&=48.69468613 \\\\
&=48.7 \ cm^{2}
\end{aligned}

In order to calculate the perimeter of a sector:

**Find the length of the radius.****Find the size of the angle creating the sector.****Substitute the values you know into the formula for the arc length.****Add together the arc length and the two radii.****Calculate the answer.**

Calculate the perimeter of the sector shown below.

Give your answer to 3 significant figures.

**Find the length of the radius.**

Radius = 5.5 \ cm

**Find the size of the angle creating the sector.**

Angle = 117^o

**Find the arc length of the sector.**

\begin{aligned} \text{Arc length } &=\frac{\theta}{360} \times 2\times\pi \times r \\\\ &=\frac{117}{360} \times 2\times\pi \times 5.5 \\\\ &=\frac{143}{40} \pi \end{aligned}

**Add together the arc length and the two radii.**

Arc length: \frac{143}{40} \pi \ cm

Radius: 5.5 \ cm

\text{Total perimeter of sector } = \frac{143}{40} \pi + 5.5 +5.5 \ cm

\text{Total perimeter of sector } = \frac{143}{40} \pi + 11 \ cm

**Calculate the answer.**

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

\begin{aligned}
\text{Perimeter of a sector } &= \frac{143}{40} \pi + 11 \ cm \\\\
&=22.2311...cm \\\\
&=22.2 \ cm
\end{aligned}

Calculate the perimeter of the sector shown below.

Give your answer to 3 significant figures.

**Find the length of the radius.**

Radius = 4 \ cm

**Find the size of the angle creating the sector.**

The angle creating the sector is the angle included between the two radii.

To find it in this case we need to use the angle fact, angles around a point add to 360^{o}.

360-240=120

The angle creating the sector is 120^{o}.

**Substitute the values you know into the formula for the arc length.**

\begin{aligned}
\text{Arc length } &=\frac{\theta}{360} \times 2 \times \pi \times r \\\\
&=\frac{120}{360} \times 2 \times \pi \times 4 \\\\
&=8.37758041 \mathrm{~cm}
\end{aligned}

**Add together the arc length and the two radii.**

= 8.37758041+4+4

**Calculate the answer.**

\begin{aligned} & =16.37758041 \\\\
& =16.4 \ cm
\end{aligned}

Below is a sector of a circle.

The diameter of the circle is 12 \ cm.

Calculate the perimeter of the sector.

**Find the length of the radius.**

Diameter = 12 \ cm

The radius is half of the diameter, so, the radius = 6cm.

Although we can use the radius or diameter as long as we use the correct formula for each.

**Find the size of the angle creating the sector.**

The angle creating the sector is the angle included between the two radii.

To find it in this case we need to use the angle fact, angles around a point add to 360^{o}.

360-205=155

The angle creating the sector is 155^{o}.

**Substitute the values you know into the formula for the arc length.**

\begin{aligned}
\text{Arc length } &=\frac{\theta}{360} \times 2 \times \pi \times r \\\\
&=\frac{155}{360} \times 2 \times \pi \times 6 \\\\
&=16.23156204 \mathrm{~cm}
\end{aligned}

**Add together the arc length and the two radii.**

= 16.23156204+6+6

**Calculate the answer.**

\begin{aligned} &=28.23156204 \\\\
&=28.2 \ cm \end{aligned}

**Confusing the segment/sector**

A segment of a circle 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.

**Finding the length of the arc not the perimeter of the sector**

You must remember that the perimeter of the sector is the **combined **length of the arc and the two radii.

**Units**

Remember the perimeter is a length and therefore the units will not be squared or cubed. Square units are for area and cubed units are for volume.

**Incorrect angle unit set on calculator**

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.

1. Find the area of the sector in terms of \pi.

16\pi \ cm^2

8\pi \ cm^2

\frac{8}{3}\pi \ cm^2

8.4 \ cm^2

Substitute the values we know into the formula for the area of a sector.

\begin{aligned} \text{ Area of a sector }&=\frac{\theta}{360} \times \pi \times r^{2} \\\\ &=\frac{60}{360} \times \pi \times 4^{2} \\\\ &=\frac{8}{3} \pi \ cm^{2} \end{aligned}

2. A sector has an area of 18\pi \ cm^2 and a radius of 12 \ cm. What is the angle of the sector?

45^{\circ}

30^{\circ}

60^{\circ}

90^{\circ}

Substituting the values we know into the formula for the area of a sector,

\frac{\theta}{360} \times \pi \times 12^{2}=18 \pi.

Solving for angle \theta ,

\theta=45^{\circ}.

3. A sector has an area of 21\pi \ cm^2 and an angle of 210^{\circ}. Find the radius of the sector.

12 \ cm

6 \ cm

9 \ cm

36 \ cm

Substituting the values we know into the formula for the area of a sector,

\frac{210}{360} \times \pi \times r^{2}=21 \pi.

Solving for r ,

\begin{aligned} &r^{2}=36 \\\\ &r=\sqrt{36}=6 \mathrm{~cm}. \end{aligned}

4. Find the perimeter of the sector to 2 decimal places.

8.19 \ cm

16.38 \ cm

10.09 \ cm

12.19 \ cm

Substituting the values we know into the formula for the arc length,

\begin{aligned} \text{Arc length }&=\frac{\theta}{360} \times 2 \times \pi \times r \\\\ &=\frac{60}{360} \times 2 \times \pi \times 4 \\\\ &=4.188790205 \mathrm{~cm}. \end{aligned}

Adding together the arc length and the two radii,

\begin{aligned} &=4.188790205+4+4 \\\\ &=12.188790205 \\\\ &=12.19 \mathrm{~cm}. \end{aligned}

5. A sector has a radius 5 \ cm and a perimeter 2\pi +10 \ cm. What was the angle of the sector?

72^{\circ}

60^{\circ}

90^{\circ}

108^{\circ}

Substituting the values we know into the formula for the total perimeter,

\left(\frac{\theta}{360} \times 2 \times \pi \times 5\right)+5+5=2 \pi+10.

Solving for \theta ,

\theta=72^{\circ}.

6. A sector has an angle 24^{\circ} and a perimeter 29.03 \ cm. What was the radius of the sector to the nearest centimetre?

24 \ cm

12 \ cm

9 \ cm

10 \ cm

Substituting the values we know into the formula for the total perimeter,

\left(\frac{24}{360} \times 2 \times \pi \times r\right)+2 r=29.03.

Rearranging to change the subject of the formula to r,

\begin{aligned} &r\left(\frac{48 \pi}{360}+2\right)=29.03 \\\\ &r=29.03 \div\left(\frac{48 \pi}{360}+2\right) \\\\ &r=12.00142701 \\\\ &r=12 \mathrm{~cm}. \end{aligned}

1. The diagram shows a sector with radius 5 \ cm and area 16 \ cm^{2}.

Find the perimeter of the sector, giving your answer to three significant figures.

**(4 marks)**

Show answer

Use of 16=\frac{\theta}{360} \times \pi \times 5^{2}.

**(1)**

Finding angle \theta =73.3385…

**(1)**

Use of \frac{\text{their angle}}{360} \times \pi \times 10 for arc length.

**(1)**

**(1)**

2. The diagram shows triangle ABD and sector BDC.

Angle BDA = 50^{\circ}

Angle BAD = 62^{\circ}

Angle BDC = 30^{\circ}

AB = 7 \ cm.

Find the area of the sector BDC to three significant figures.

**(4 marks)**

Show answer

Use of sine rule \frac{BD}{\sin{62}}=\frac{7}{\sin{50}}.

**(1)**

**(1)**

Use of \frac{30}{360} \times \pi \times "8.068"^{2}.

**(1)**

Area = 17.0 \ cm^2

**(1)**

3. A sector has a perimeter 50 \ cm and angle 200^{\circ}.

Find the radius of the sector. Give your answer to three significant figures.

**(3 marks)**

Show answer

Use of 50=\frac{200}{360} \times \pi \times d + d (or equivalent with r ).

**(1)**

Rearranging the formula to d=\frac{50}{\frac{200}{360} \times \pi +1} (or equivalent with r).

**(1)**

Radius = 9.11 \ cm

**(1)**

You have now learned how to:

- Calculate areas of a sector
- Calculate a property of a circle given the area of its sector
- Calculate the perimeter of a sector
- Calculate a property of a circle given the perimeter of a sector

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