GCSE Maths Geometry and Measure 2D Shape

Circles, Sectors & Arcs

Circles, Sectors And Arcs

Here we will learn about circles, arcs and sectors, including how to find the area and circumference of a circle and how to find the area and arc length of a sector.

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

What are circles, arcs and sectors?

Circles are round plane figures whose boundaries consist of points equidistant from a fixed point.

Sector of a circle

Sectors are sections of a circle that are created by two radii and an arc.

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°.

Step-by-step guide: Sector of a circle

Arc of a circle

Arcs are portions of the circumference of the circle.

Major arc – a major arc is greater than half the circumference.

Minor arc – a minor arc is less than half the circumference.

Step-by-step guide: Arc of a circle

Parts of a circle

The parts of a circle have specific names and properties which you need to know for all circle related questions.

Step-by-step guide: Parts of a circle

Area and circumference of a circle

• Area of a circle (\pi r^{2})

The area of a circle can be found by multiplying the square of the radius by \pi .

For example,

The radius of a circle is 4cm.

The area of the circle is \pi \times 4^{2}=16 \pi \ \mathrm{cm}^{2}= 50.265… \mathrm{cm}^{2}

Step-by-step guide: Area and circumference of a circle

• Circumference of a circle (\pi d)

The circumference of a circle can be found by multiplying the diameter by \pi .

For example,

The diameter of a circle is 5 \ cm.

The circumference of the circle is \pi \times 5=5 \pi \ \mathrm{cm}=15.7079… cm

Step-by-step guide: Circumference of a circle

Area of a sector and arc length

• Area of a sector

The area of a sector is a part of the area of the full circle.

It can be found by using the formula \frac{\theta}{360} \times \pi r^{2}.

For example,

In this sector,

\theta=30^{o}, r=8 \mathrm{~cm}.

The area of the sector can be found using the formula.

\begin{aligned} \text{Area of sector }&=\frac{30}{360} \times \pi \times 8^{2}\\\\ &=\frac{16}{3} \pi \ \mathrm{cm}^{2}\\\\ &=16.755… \ \mathrm{cm}^{2} \end{aligned}

Step-by-step guide: Area of a sector

• Arc length

The arc of a circle is part of the circle’s circumference.

Its length can be found using the formula \frac{\theta}{360} \times \pi d.

For example,

In this sector,

\theta=30^{o} and the radius is 8 \ cm. This means the diameter is 16 \ cm.

The arc length can be found using the formula.

\begin{aligned} \text{Arc length} &= \frac{30}{360} \times \pi \times 16\\\\ &=\frac{4}{3} \pi \ \mathrm{cm}\\\\ &=4.188… \ \mathrm{cm} \end{aligned}

Step-by-step guide: Arc of a circle

• Perimeter of a sector

To find the perimeter of a sector you need to find the arc length and then add it to the two radii (i.e. find the length around the edge of the sector).

Step-by-step guide: Perimeter of a sector

• Segment of a circle

A segment of a circle is created when a chord is drawn within the circle.

You can find the area of a segment by finding the area of the sector and subtracting the area of the triangle, as shown in the diagram below.

Step-by-step guide: Segment of a circle

Equation of a circle

• Equation of a circle

The equation of a circle (at GCSE) is given in the form

x^{2}+y^{2}=r^{2}.

The circles that we study at GCSE have centre (0,0) and radius r.

Step-by-step guide: Equation of a circle

• Equation of tangent

A tangent to a circle is a straight line which touches the circle at one point only.

The tangent line is always perpendicular to the radius at that point. To find the equation of the tangent to a circle at a given point,

1. Find the gradient of the radius at that point.
2. Find the gradient of the tangent at that point.
3. Substitute the x and y coordinates of the given point to find the y -intercept of the tangent, and hence the equation of the tangent line.

Step-by-step guide: Equation of tangent

Circles, arcs and sectors examples

Example 1: find the radius of a circle from its diameter

Find the radius of the circle shown below.

The diagram shows the diameter of the circle as 12 \ cm.

We know the diameter is twice the length of the radius.

12 \ cm \div 2=6 \ cm

The radius of the circle is \bf{6 \ cm} .

Example 2: calculate the area of a circle

Find the area of the circle shown below. Give your answer to 1 decimal place.

Find the radius of the circle.

Use the formula \pi r^2 to calculate the area of the circle.

Give your answer clearly with the correct units.

Example 3: calculate the circumference of a circle

Find the circumference of the circle shown below. Give your answer in terms of \pi .

Find the radius or diameter of the circle

Use the relevant formula to calculate the circumference of the circle

Give your answer clearly with the correct units

Example 4: calculate the area of a sector

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

Find the length of the radius \textbf{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.

Example 5: calculate the arc length

Find the length of the arc. Give your answer in terms of pi.

Find the length of the radius/diameter.

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

Substitute the value of the radius/diameter and the angle into the formula for the arc length.

Example 6: calculate the perimeter of a segment

Find the perimeter of the segment shaded in the diagram below. Give your answer to 3 significant figures.

Identify what we need to calculate.

Calculate the arc length.

Use what you have calculated to work out the perimeter.

Example 7: equation of a circle

Construct the graph of x^{2}+y^{2}=16.

Work out the radius of the circle.

Construct the circle

Example 8: equation of a tangent

The circle x^{2}+y^{2}=68 passes through the point P(8, 2).

Find the equation of the tangent to the circle at the point P.

Find the gradient of the radius at that point.

Find the gradient of the tangent at that point.

Find the \textbf{y} -intercept and hence the equation of the tangent.

Common misconceptions

• Radius and diameters

You will notice some formulas use the radius whilst others refer to the diameter. Make sure you know which one the question gives you, this allows you to find the other if required.

• Not including the correct units

Remember the value of the radius, the diameter, the arc length and the circumference are measures of length.

The value of the area of a sector or segment are measures of area and therefore the units are squared.

• Not giving answer in terms of \textbf{π}

Sometimes the question may ask you to give the answer ‘in terms of \bf{\pi} ’. This means you do not give the numerical answer that is produced when you multiply it by \bf{\pi}

For example,

6 \times \pi = 6 \bf{\pi} (this is an answer in terms of pi)

6 \times \pi = 18.8495592… (this answer is not in terms of pi)

• Misuse of calculator

Ensure you know how to correctly use the \pi button on your calculator.

Practice circles, sectors and arcs questions

1. What is the radius of the circle?

10 \ cm

20 \ cm

5 \ cm

100 \ cm

The radius is half the diameter.

The diameter is given on the diagram as 10 \ cm.

10 \div 2=5

Radius = 5 \ cm

2. What is the area of the circle? Give your answer in terms of pi.

25 \pi \ \mathrm{cm}^{2}

10 \pi \ \mathrm{cm}^{2}

5 \pi \ \mathrm{cm}^{2}

100 \pi \ \mathrm{cm}^{2}

For this circle, the radius is 5 \ cm.

\begin{aligned} \text { Area } &=\pi r^{2} \\\\ &=\pi \times 5^{2} \\\\ &=25 \pi \end{aligned}

Units are cm^{2}.

3. What is the circumference of the circle? Give your answer to 1 decimal place.

15.7 \ \mathrm{cm}

78.5 \ \mathrm{cm}

314 \ \mathrm{cm}

31.4 \ \mathrm{cm}
\begin{aligned} \text {Circumference }&=\pi d \\\\ &=\pi \times 10 \\\\ &=10 \pi \\\\ \end{aligned}

Units are cm.

4. What is the area of the sector? Give your answer in terms of pi.

15 \pi \ \mathrm{cm}^{2}

25 \pi \ \mathrm{cm}^{2}

3 \pi \ \mathrm{cm}^{2}

6 \pi \ \mathrm{cm}^{2}

The radius is 5 \ cm and the angle of the sector is 216°.
\begin{aligned} \text{Area }&=\frac{\theta}{360} \times \pi r^{2} \\\\ &=\frac{216}{360} \times \pi \times 5^{2} \\\\ &=15 \pi \\\\ \end{aligned}

Units are for area so are cm^{2}.

5. What is the length of the major arc? Give your answer to 3 significant figures.

47.1 \ \mathrm{cm}

18.8 \ \mathrm{cm}

31.4 \ \mathrm{cm}

15.7 \ \mathrm{cm}
\begin{aligned} \text{Arc length }&=\frac{\theta}{360} \times \pi d \\\\ &=\frac{216}{360} \times \pi \times 10 \\\\ &=6 \pi \\\\ \end{aligned}

Units are for a length so are cm.

6. What is the perimeter of the sector? Give your answer correct to 1 decimal place.

18.8 \ \mathrm{cm}

28.8 \ \mathrm{cm}

10 \ \mathrm{cm}

23.8 \ \mathrm{cm}

Arc length = 6 \pi (see Q5)

The perimeter of a sector is the length of the two radii plus the arc length.

Therefore the perimeter is

6 \pi + 5+5=28.84955…

Perimeter is a measure of length so the units are cm.

Perimeter =28.8 \ cm

7. Calculate the area of the shaded segment. Give your answer to 3 significant figures.

44.7 \ \mathrm{cm}^2

12.7 \ \mathrm{cm}^2

13.2 \ \mathrm{cm}^2

34.4 \ \mathrm{cm}^2

First we need to calculate the area of the sector.

\begin{aligned} \text{Area of sector }&=\frac{80}{360} \times \pi \times 8^{2}\\\\ &=\frac{128}{9} \pi \end{aligned}

Next we need the area of the triangle. To calculate this we can use

\text{Area of triangle }=\frac{1}{2}ab \sin C.

We get,

\begin{aligned} \text{Area of triangle }&=\frac{1}{2} \times 8 \times 8 \times \sin (80) \\\\ &=31.5138… \end{aligned}

Finally we can subtract the area of the triangle from the area of the sector.

\begin{aligned} \text{Area of segment }&=\frac{128}{9} \pi -31.5138… \\\\ &=13.16658… \end{aligned}
The area of the segment is 13.2cm^2 \ (3sf).

8. A circle, centre (0, 0), has a radius of 9. What is the equation of the circle?

x^{2}+y^{2}=9

x^{2}+y^{2}=3

x^{2}+y^{2}=81

x^{2}+y^{2}=18

The general form for the equation of the circle is x^{2}+y^{2}=r^{2}.

Here r=9 and therefore r^{2} = 81 and x^{2}+y^{2}=81.

Circles, sectors and arcs GCSE questions

1. Look at the diagram below showing a circle.

(a) What is the radius of the circle?

(b) Calculate the area of the circle. Give your answer to 1 decimal place.

(c) Calculate the circumference of the circle. Give your answer to 1 decimal place.

(5 marks)

(a)

8 \ cm

(1)

(b)

\pi \times 8 \times 8 \ oe \ or \ 201.06…

(1)

201.1

(1)

(c)

\pi \times 16

(1)

50.3

(1)

2. OAB is a sector of a circle.

(a) Calculate the area of the sector.

Give your answer to 3 significant figures.

(b) Calculate the length of the arc AB.

Give your answer to 3 significant figures.

(c) Calculate the perimeter of the sector.

Give your answer to 3 significant figures.

(9 marks)

(a)

\frac{115}{360}

(1)

\frac{115}{360} \times \pi \times 9^{2}

(1)

81.2887…

(1)

81.3

(1)

(b)

\frac{115}{360} \times \pi \times 18

(1)

18.064…

(1)

18.1

(1)

(c)

“18.1” + 18 \ ft

(1)

36.1

(1)

3. The area of a circle is 36\pi.

Find the radius of the circle.

(2 marks)

36\pi=\pi r^{2}

(1)

r=6

(1)

4. A circle has the equation x^{2}+y^{2}=10.

(a) Show that the point P (-3, 1) lies on the circle.

(b) Find the equation of the tangent to the circle at the point (-3, 1).

(6 marks)

(a)

(-3)^{2}+(1)^{2}=10

(1)

9+1=10 therefore (-3, 1) lies on the circle.

(1)

(b)

(1)

Gradient of tangent = 3

(1)

y=3x+c

(1)

c=10 \ so \ y=3x+10

(1)

Learning checklist

You have now learned how to:

• Identify and apply circle definitions and properties, including centre, radius, chord, diameter, and circumference
• Calculate the area of a circle
• Calculate the circumference of a circle
• Calculate the area of a sector
• Calculate the arc length of a sector
• Give answers in terms of \textbf{π}

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