SCHOLAR Study Guide National 5 Mathematics Course ...€¦ · National 5 Mathematics Course...

SCHOLAR Study Guide

National 5 Mathematics

Course MaterialsTopic 5: Arcs and sectors

Authored by:Margaret Ferguson

Reviewed by:Jillian Hornby

Previously authored by:Eddie Mullan

Heriot-Watt University

Edinburgh EH14 4AS, United Kingdom.

First published 2014 by Heriot-Watt University.

This edition published in 2016 by Heriot-Watt University SCHOLAR.

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Distributed by the SCHOLAR Forum.

SCHOLAR Study Guide Course Materials Topic 5: National 5 Mathematics

1. National 5 Mathematics Course Code: C747 75

AcknowledgementsThanks are due to the members of Heriot-Watt University's SCHOLAR team who planned andcreated these materials, and to the many colleagues who reviewed the content.

We would like to acknowledge the assistance of the education authorities, colleges, teachersand students who contributed to the SCHOLAR programme and who evaluated these materials.

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The Scottish Qualifications Authority for permission to use Past Papers assessments.

The Scottish Government for financial support.

The content of this Study Guide is aligned to the Scottish Qualifications Authority (SQA)curriculum.

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1

Topic 1

Arcs and sectors

Contents

5.1 Calculating the length of an arc . . . . . . . . . . . . . . . . . . . . . . . . . . 3

5.2 Finding the radius, diameter or angle given the length of an arc . . . . . . . . 6

5.3 Calculating the area of a sector . . . . . . . . . . . . . . . . . . . . . . . . . . 9

5.4 Finding the radius, diameter or angle given the area of a sector . . . . . . . . 12

5.5 Learning Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

5.6 End of topic test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2 TOPIC 1. ARCS AND SECTORS

Learning objectives

By the end of this topic, you should be able to:

• calculate the length of an arc;

• calculate the radius, diameter or angle given the length of the arc;

• calculate the area of a sector;

• calculate the radius, diameter or angle given the area of the sector.

© HERIOT-WATT UNIVERSITY

TOPIC 1. ARCS AND SECTORS 3

1.1 Calculating the length of an arc

Finding the circumference of a circle

Key point

Remember :

Circumference = π × Diameter ⇒ C = πD

Diameter = 2 × Radius ⇒ D = 2rTherefore, the formula for circumference may also be written as:

Circumference = 2 × π × Radius ⇒ C = 2πr

Example

Problem:

A wheel has a diameter of 36 cm. What is its circumference?

Solution:

C = πD = × 36 = 113 cm (to the nearest whole number)

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Q1:

A tart has a diameter of 12 cm. What is its circumference?Round your answer to 1 decimal place.

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Finding the length of an arc of a circle

Key point

Remember:

Length of Arc = angle360◦ × π × D

Diameter = 2 r

By combining the calculations for a fraction of a quantity and the circumference of acircle, we can find the length of an arc of a circle.

Example

Problem:

A fan has a radius of 30 cm. When fullyopen it makes an angle of 160◦ at thecentre.

What is the length of the outer edge ofthe fan?

Solution:

To find the length of the outer edge of the fan we must find a fraction of thecircumference of the whole circle.The formula to find the length of an arc is angle/360◦ × πDThe circle has radius 30 cm so the diameter D = 2 × 30 = 60 cm.The angle at the centre of the fan is 160◦.The length of the outer edge of the fan = 160◦

/360◦ × π × 60 = 83 · 8 cm (to 1 d.p.).

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Example

Problem:

Look at the pie. What is the length of theremaining crust?

Solution:

To find the length of the crust we must find a fraction of the circumference of the wholecircle.The formula to find the length of an arc is angle/360◦ × πDThe circle has radius 3·5 cm so the diameter D = 3 · 5 × 2 = 7 cm.The angle at the centre is 360◦ − 120◦ = 240◦ because the angle given is for thepiece of pie which has been eaten.The length of crust = 240◦

/360◦ × π × 7 = 14 · 7 cm (to 1 d.p.).

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Calculating the length of an arc practice

Go online

Q2:

A piece of cake has a radius of 12 cm.

It makes an angle of 40◦ at the centre.

What is the length of the outer edge ofthe piece of cake? Give your answer to 1decimal place.

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Q3:

A stuffed crust pizza has a radius of 21cm.

The angle of the space where the slicewas removed is 128◦.

What is the length of the remainingstuffed crust? Give your answer to 1decimal place.

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Calculating the length of an arc exercise

Go online

Q4:

A pie has diameter of 10 cm.

What is its circumference (in cm?)

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Q5:

A piece of cake has a radius of 10 cm.

The remaining cake has an angle of 190◦

at the centre.

What is the length, in cm, of the curvededge of the piece of cake?

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Q6:

A stuffed crust pizza has a radius of 32cm.

The angle of the space where the slicewas removed is 88◦.

What is the length of the remainingstuffed crust?

Give your answer to 1 decimal place.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.2 Finding the radius, diameter or angle given the length ofan arc

Key point

Remember:

Length of arc = angle360◦ × π × D

Diameter = 2 × Radius ⇒ D = 2r

By re-arranging the formula for the length of an arc we can find the angle or the diameter.



Examples

1.

Problem:

The length of the remaining icing aroundthe cake is 45 cm and the angle at thecentre is 260◦.

Calculate the radius of the circular cake.

Solution:

The icing is an arc and is 45 cm long.The angle at the centre of the remaining cake is 260◦.Remember the formula the length of an arc is length of arc = angle/360◦ × πDIf we replace what we know we get

45 =260◦

360◦× π ×D

(260◦

360◦× π = 2 · 269

)

45 = 2 · 269 ×D (re− arrange the equation)

45

2 · 269 = D

D = 19 · 83252534 (don′t round yet

)radius = D ÷ 2

= 9 · 9 cm (to 1d.p.). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.

Problem:

When a pendulum swings its path is anarc.

When it swings from left to right the archas length 28 cm and the pendulum is 19cm long.

Calculate the angle made as thependulum moves through its path.

Solution:

Length of arc = angle/360◦ × πDThe radius is 19 cm so the diameter = 2 × 19 = 38 and the length of the arc = 28.If we replace what we know we get



28 =angle

360◦× π × 38 (π × 38 ÷ 360 = 0 · 3316)

28 = angle × 0 · 3316 (re− arrange the equation)

28

0 · 3316 = angle

angle = 84 · 4◦ (to 1 d.p.). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Finding the radius, diameter or angle practice

Go online

Q7:

If the length of the arc is 24 cm, calculatethe radius.

Give your answer to the nearest wholenumber.

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Q8:

If the length of the arc is 251 m with aradius of 200 m, calculate the angle x ◦.


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Finding the radius, diameter or angle exercise

Go online

Q9: Finding the diameter.

If the length of the arc is 10·28 cm,calculate the diameter of the whole pizza.

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Q10: Finding the radius.

If the length of the arc is 109 m, calculatethe radius.

Give your answer to 1 decimal place.

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Q11: Finding the angle.

Calculate the angle at the x given thatthe length of the arc is 221 m and theradius is 39 m.


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1.3 Calculating the area of a sector

Area of a circle

Key point

Remember :

Area = π r 2

Example

Problem:

A coin has a radius of 1·2 cm. What is the area of one side?



Solution:

A = πr2 = π × 1 · 22 = 4 · 5 cm2 (to 1 decimal place)

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Q12:

A whole cheese has a radius of 16 cm. What is the area of one side?

Give your answer to the nearest whole number.

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Area of a sector of a circle

Key point

Remember:

Area of Sector = angle360◦ × π × r2

We can calculate the area of a sector of a circle by taking a fraction of the area of thewhole circle.

Example

Problem:

What area of the top of the pizza ismissing?



Solution:

To find the area of the sector we must find a fraction of the area of the whole pizzausing the formulaarea of sector = angle/360◦ × π × r2

Since we know the angle and the radius we can put the values into the formula giving,area of sector = 120◦

/360◦ × π × 62 = 37 · 7 cm2 (to 1 d.p.)

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Calculating the area of a sector practice

Go online

Q13:

What area of the top of this 7 cm pizza ismissing?


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Q14:

Calculate the area of the pizza remaining.


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Calculating the area of a sector exercise

Go online

Q15: Area of a sector.

Calculate the area of the shaded sectorof the circle.


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Calculate the area of the sector of pizzawhich has been eaten.


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What is the area of the pizza remaining(in cm2)?


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1.4 Finding the radius, diameter or angle given the area of asector

Key point

Remember:

Area of Sector = angle360◦ × π × r2

By re-arranging the formula for the area of a sector we can find the angle or the radius.



Examples

1.

Problem:

The area of the top of the cake remainingis 230 cm2 and the angle at the centre is260◦.

Calculate the radius of the circular caketo 1 decimal place.

Solution:

The area of the sector is 230.The angle at the centre of the remaining cake is 260◦.Remember the formula the area of a sector isarea of sector = angle/360◦ × π × r2

If we replace what we know we get,

230 =260◦

360◦× π × r2

(260◦

360◦× π = 2 · 269

)

230 = 2 · 269 × r2 (re− arrange the equation)

230

2 · 269 = r2

r2 = 101 · 3662406 (square root the answer)

r =√101 · 3662406

r = 10 · 1 cm (to 1 d.p.). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.

Problem:

This earring is in the shape of a slice ofcake.

The area of the top of the earring is 325mm2 and the radius is 25 mm.

Calculate the angle at the point of theearring to 1 decimal place.

Solution:

area of sector = angle/360◦ × π × r2

If we replace the area of the sector and the radius we get,



325 =angle

360◦× π × 252

(π × 252 ÷ 360 = 5 · 454)

325 = x × 5 · 454 (re− arrange the equation)

325

5 · 454 = x

x = 59 · 6◦ (to 1 d.p.). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Finding the radius, diameter or angle given the area of a sector practice

Go online

Q18:

If the area of the cake is 4.3 inches2,calculate the radius of this sector.

Calculate your answer to 1 decimal place.

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Q19:

If the area of the sector is 495 mm2,calculate the angle y◦.

Give your answer to the nearest degree.

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Finding the radius, diameter or angle given the area of a sector exercise

Go online

Q20: Finding the angle of a sector.

A wedge of cheese has an area of 440·5cm2 and a radius of 29·5 cm. Calculatethe angle of the wedge of cheese.

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Q21: Finding the radius of a sector.

A sector has an angle of 318◦. If the areaof the sector is 4 cm2 calculate theradius.


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Q22: Finding the diameter of a sector.

The area of the top of the cheese is 1309cm2. The angle of the remaining cheeseis 310◦.

Calculate the diameter of the cheese.

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1.5 Learning Points• Circumference = πD and Diameter = 2rtherefore C = 2πr

• Length of an Arc = angle360◦ × π × Diameter

• Area of a Sector = angle360◦ × π × radius2

• To find an angle, radius or diameter when the length of an arc or the area of thesector is known:

◦ put the values you know into the formula;

◦ re-arrange or change the subject of the formula.



1.6 End of topic test

End of topic 5 test

Go online

Q23: Fraction of a circumference

A piece of cake has a radius of 100 cm. Itmakes an angle of 120◦ at the centre.

What is the length, in cm, of the outeredge of the piece of cake?

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Q24: Fraction of the area of a circle

A pizza has radius 8·1 cm and is missinga sector with an angle of 230◦.

What is the area of the pizza remaining(in cm2)?

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Q25: Finding the radius of an arc

If the length of the arc is 81 cm, calculatethe length of the radius.


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Q26: Finding an angle in a sector

If the area of the sector is 924 cm2,calculate the size of the angle at thecentre of the pizza.


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Q27: Finding the diameter in a sector

The Pie Chart shows the occupationalstructure in 1831.

The area of the sector labelled labourers& servants is 40.2 cm2 with an angle200◦.

Calculate the diameter of the pie chart.Give your answer to 1 decimal place.

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ANSWERS: TOPIC 5 19

Answers to questions and activities

5 Arcs and sectors

Answers from page 3.

Q1: 37.7 cm

Calculating the length of an arc practice (page 5)

Q2: 8.4 cm

Q3: 85.0 cm

Calculating the length of an arc exercise (page 6)

Q4: 31·4 cm

Q5: 33 cm

Q6:

Diameter of the pizza = 2 × 32 = 64

Angle of the remaining pizza = 360-88 = 272

151·9 cm

Finding the radius, diameter or angle practice (page 8)

Q7: 13 cm

Q8: 72◦

Finding the radius, diameter or angle exercise (page 8)

Q9: 31 cm

Q10: 78·1 m

Q11: 324·7◦

Answers from page 10.

Q12: 804

Calculating the area of a sector practice (page 11)

Q13: 68.4 cm2


20 ANSWERS: TOPIC 5

Q14: 354.5 cm2

Calculating the area of a sector exercise (page 11)

Q15: 1165·7 cm2

Q16: 230·5 cm2

Q17: 241·3 cm2

Finding the radius, diameter or angle given the area of a sector practice (page 14)

Q18: 3.5 inches

Q19: 28◦

Finding the radius, diameter or angle given the area of a sector exercise (page14)

Q20: 58◦

Q21: 1·2 cm

Q22: 44 cm

End of topic 5 test (page 17)

Q23: 209 cm

Q24: 74·4 cm2

Q25: 27 cm

Q26: 63◦

Q27: 9·6 cm


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