Key Stage 3 Geometry

with your Helix Oxford

Mathematical

Instruments

Allen Brown

Cambridge

Paperbacks

Cambridge Paperbacks

www.CambridgePaperbacks.com

First published by Cambridge Paperbacks 2019

© Allen Brown 2019

All rights reserved. No part of this publication may be reproduced or

transmitted in any form or by any means, electronic or mechanical, including

photocopy, recording, or any information storage and retrieval system without

permission in writing from the author.

Disclaimer

Although the author and publisher have made every effort to ensure that the

information in this book was correct during preparation and printing, the

author and publisher hereby disclaim any liability to any party for any errors

or omissions.

Read this First

There have been many generations of school students

who have benefited from the Helix Oxford set of

Mathematical Instruments. If you learn how to use

them properly, you also will gain a good knowledge of

geometry which is vital for your gain a high grade in

your GCSE Maths.

Geometry is all about shapes and in this ebook you

will be shown how to use your Mathematical

Instruments to create many different 2D shapes.

However it must be stressed it does take practice to

get them right.

You are recommended to use A4 size

plain printer paper. You will find a ream

of A4 paper in most supermarkets these

days. Typically the weight of each page is

80 grams. Using lined paper is not a good

idea as the lines cause

distraction from the drawings

you are creating. Once you have

completed a drawing, us a two-

hole paper punch and file in an

A4 ring binder; there are many different colours

available.

In this ebook some basic calculations are required and

these can be performed on a 4 function calculator

which you probably already possess.

As you progress through this ebook you will acquire a

number of useful skills and education is about

learning new knowledge and skills and of course

gaining confidence. This can only be achieved through

practice – working through the exercises yourself.

If you are home-schooled you will benefit greatly from

the skills you will be taught. And may you have a

happy adventure learning about practical geometry.

Allen Brown

Cambridgeshire

Contents

1 Contents of your Helix Oxford set of Mathematical

Instruments .......................................................................... 3

2 Angles and the Protractor ................................................. 6

2.1 The Protractor ........................................................... 11

Exercise 2-1 ..................................................................... 12

Exercise 2-2 ..................................................................... 13

Exercise 2-3 ..................................................................... 14

2.2 The Compasses .......................................................... 16

o

Exercise 2-4: 60 Angle .................................................... 18

o

Exercise 2-5: 30 Angle .................................................... 19

Exercise 2-6 ..................................................................... 21

Exercise 2-7 ..................................................................... 22

o

Exercise 2-8: 90 Angle .................................................... 24

o

Exercise 2-9: 45 Angle .................................................... 25

3 The Circle ......................................................................... 27

3.1 Inscribing Polygons in a Circle ................................... 28

Exercise 3.1: A Regular Triangle in a Circle...................... 29

Exercise 3.2: A Regular Quadrilateral in a Circle ............. 30

Exercise 3.3: A Regular Pentagon in a Circle ................... 33

Exercise 3.4: A Regular Hexagon in a Circle .................... 38

1

Exercise 3.5: A Regular Heptagon in a Circle ................... 39

Exercise 3.6: A Regular Octagon in a Circle ..................... 41

Exercise 3.7: A Regular Nonagon in a Circle .................... 45

Exercise 3.8: Connecting the Vertices of Polygons .......... 49

4 Irregular Polygons ........................................................... 52

4.1 Triangle ...................................................................... 52

Exercise 4.1 ..................................................................... 53

4.2 Quadrilateral ............................................................. 55

Exercise 4.2 ..................................................................... 57

4.3 Pentagon ................................................................... 59

Exercise 4.3 ..................................................................... 60

4.4 Hexagon .................................................................... 61

4.5 Heptagon ................................................................... 63

5 More on the Circle ........................................................... 65

Exercise 5.1 ..................................................................... 66

Exercise 5.2 ..................................................................... 67

Exercise 5.3 ..................................................................... 69

2

1 Contents of your Helix Oxford set of

Mathematical Instruments

Your Helix Oxford set of mathematical instruments

has a long history stretching back many decades and

has helped many a school student to gain an

understanding of geometry.

Over the years there have been slight changes in the

contents of the Helix box. Today’s Helix box contains

lots of really useful items to help you gain an

understanding of geometry. When you open your

Helix box this is what you will see inside.

There will be a pair of

compasses, two set

squares, a protractor, a

pencil for the compasses,

a rubber and a 6 inch

ruler.

3

The Protractor

This has two roles, firstly to

create an angle of required size,

and secondly, to measure an

angle between two lines.

The Compasses

An instrument for drawing

circles and arcs. The pencil

found in the tin slides into

the opening and is fixed into position by rotating the

screw against the pencil.

o

A 45 Set Square

Although it’s called a set

square, it is not square,

although it does have

one right angle. You will

notice it also has a metric

scale in cm and also the imperial scale in inches. Very

convenient for drawing a right angle or an angle of

o

45 .

4

o

The 60 Set Square

Very similar to the other

set square except the

o

internal angles are 60

o

and 30 . Again a very

convenient instrument for drawing either of these

angles. You will also see measuring scales in both

centimetres and inches.

There is also a letter and number stencil which you

can use for labelling you exercise diagrams.

In the next chapter there is a fuller description on the

nature of angles and how they are made and

measured. There are also several exercises for you to

work through using these instruments.

5

2 Angles and the Protractor

Whenever two lines cross,

angles subtend between them

as shown in the diagram on the

left. Angles are measured in

degrees which is indicated by a

o , each angle is subdivided into 60 minutes, indicated

by a ’ and each minute is subdivided into 60 seconds

(or arcseconds) indicated by a ”. In the above diagram

o

all the angles add up to 360 , what you would find in

a full circle. Therefore,

+ + + = 360

or

2 + 2 = 2( + ) = 360

Dividing both sides by 2

+ = 180

Angles fall into four categories, as shown below.

Acute angle Right angle

6

Obtuse angle Reflex angle

o

As you can see the acute angle is less than 90 and an

o

obtuse angle is greater than 90 and a reflex angle is

o

greater than 180 .

o

The very special angle is the right angle, this is 90 and

you will come across this angle many times in

geometry and trigonometry. Right angles are

indicated by a small square in the corner. It indicates

o

that one line is perpendicular to the other (90

between them).

o

o

Angles occur in the range 0 to 360 if you are moving

o

o

in a clockwise direction or 0 to -360 if moving in an

anticlockwise direction. Very often you will see the

Greek letter theta θ used to label an angle. Therefore,

0 ≤ ≤ 360

The symbol ≤ means small than or equal to. This

expression is a way of say that the angle θ lies in the

o

range of 0 to 360 .

7

Let’s have a closer look at some angles starting with

o

o

o

30 and increasing in steps of 30 until you reach 360

in a clockwise direction.

o

30 o 60 o 90

o

120 o 150 o 180

o

210 o 240 o 270

o

300 o 330 o 360

8

Can you identify which ones are acute, obtuse and

reflex angles?

o

Once you reach 360 you will have completed a whole

circle. When you combine all 12 sectors you get,

Each sector has an angle of

o

30 . When you look closely at

your protractor you may think

o

an angle of 1 is small and as

mentioned each of these is

divided into 60 minutes and

each minute is divided in 60

arcseconds – which appears to be a pretty small angle.

However, when performing measurements in

astronomy, the arcsecond is quite

common. When you look up into

the night sky to observe the

planet Jupiter the angle formed

between its equatorial axis and

your eye is about 33 arcseconds.

When you are looking at the

moon which is much closer, the

angle subtended by its equatorial

9

axis and your eye is about ½ degree which is roughly

the same as the sun.

Below are some rotated images to allow you to get a

better feel for the size of angles.

0 o 1 o 2 o

3 o 5 o 10 o

20 o 30 o 50 o

10

o

Looking at the rotation of 1 , it’s not very noticeable.

o

Even for 2 rotation it’s still only slightly different. By

o

the time you reach 5 the rotation is quite evident and

o

by the time you have a rotation of 50 you will notice

a big difference.

Having gained an insight into the size of angles, we are

going to look closely at your protractor for drawing

angles and also for measuring angles.

2.1 The Protractor

A very useful instrument in your Helix box,

In this diagram you can see the details of the angles.

o

Starting on the left-hand side at 0 as you move

11

o

clockwise the angles increase until you reach 180 on

the right-hand side. But you will notice there are two

o

scales, from 0 on the right-hand side - working in an

o

anticlockwise direction to 180 on the left-hand side.

An important point on the protractor is the cross-hairs

at the bottom centre. This is the point from where you

measure an angle or draw an angle.

Exercise 2-1

o

Using your protractor, draw an angle of 50 .

Work through the following.

1. Using the ruler, draw a line of length 12 cm and

mark a dot in the middle of the line. Then place

your protractor on the line with the cross hairs

over the dot as shown below.

12

o

2. Place a mark at the 50 angle by the protractor.

3. Remove the protractor and draw a line from the

centre of the line to the mark. There you have

o

drawn a 50 angle.

Exercise 2-2

Measuring an angle with your protractor. Work

through the following.

o

1. With your 60 set square, lay it flat on a sheet of

o

A4 paper and draw an out-line around the 60

angle (see diagram on the next page).

2. Remove the setsquare and mark the corner of

the angle with a dot.

3. Place your protractor with the base line on the

line your have drawn and the cross-hairs on the

dot.

13

4. You will observe on the protractor, the angle you

o

have drawn is 60 .

Exercise 2-3

Using your protractor and ruler to draw a regular

pentagon (five sides). The internal angle of a pentagon

o

is 108 . Do the following.

1. Draw a horizontal line of 5 cm long and mark the

left end with a dot.

2. Place the protractor alone the line with the dot

under the cross-hairs and mark out an angle of

o

108 .

3. Place the ruler so that its edge touches both the

o

end of the line and 108 mark. Move the ruler

alone until the end of the line is on the 10 cm

mark.

14

4. Draw a line of 5 cm long and mark the end of the

new line with a dot. Repeat stage 4 three more

times - you have your pentagon.

The final line connecting the last point and the first

point should also be 5 cm in length.

15

2.2 The Compasses

Although the compasses

are used for drawing

circles, they are also used

to draw arcs which are

also useful for dividing up distances. For example,

given a line, the mid-point can be determined by using

the compasses. This is how you do it.

1. Draw a horizontal line of 8 cm.

2. Open up your compasses so the separation is 5

cm.

3. Place the compasses point at one end of your

line and draw an arc which passes through the

line.

16

4. Place the compasses point at the other end of

your line and draw another arc which passes

through the line.

5. The arcs intersect at two places. Draw a line

between these two intersections.

6. The point where this line passes through your

line is the mid-way between the two ends.

7. Confirm this by measuring it, it will be 4 cm.

17

This technique will be used in the next chapter to

inscribe regular polygons within circles.

o

Exercise 2-4: 60 Angle

o

Your compasses can be used to create an angle of 60 .

This is how it’s achieved.

1. Draw a horizontal line 14 cm long.

2. Open your compasses with a separation of 6 cm.

3. Position the compasses point roughly mid-way

along the line and draw an arc roughly quarter

of a circle which intercepts the line.

4. Place the compasses point on the intersect of

the arc and the line and draw an intercept arc

with the previously drawn arc.

18

5. Draw a line from the first position of the

compasses to the new intercept.

6. Place your protractor on the line with the angle

corner on the cross-hairs.

o

7. Read off an angle of 60 on your protractor.

o

Exercise 2-5: 30 Angle

o

Your compasses can be used to create an angle of 30 .

19

1. Draw a horizontal length of 10 cm.

2. Separate your compasses with a distance of 5

cm.

3. Place the compasses point on the right end of

the line and draw an arc above the line.

4. Place the compasses point on the centre of the

line and draw another arc which intercepts the

previous arc.

5. Draw a line from the intercept to the left end of

the line.

20

6. Use your protractor to confirm the angle

o

subtended is 30 .

Exercise 2-6

In this exercise we are going to draw some parallel

lines and measure angles. Do the following.

1. Place your ruler on a sheet of A4 paper in a

horizontal position.

2. Draw two lines above and below the straight

edges of your ruler 12 cm long. You now have

two parallel lines.

3. Rotate the ruler so it crosses the parallel lines

and draw two more as before. You now have

two sets of parallel lines crossing each other at

an angle you now going to measure.

21

4. Label the angles as shown in the diagram below;

angles A to D.

5. Using your protractor, measure the angles A to

D.

You will discover that A is equal to C and B is equal to

D. This is a result of having parallel lines – in the

diagram you will notice a number of angles at each

intercept is the same.

Exercise 2-7

Drawing parallel lines using your compasses.

1. Draw a horizontal line 12 cm long.

2. Mark a point above the line where you want

your parallel to be.

22

3. Place the point of your compasses anywhere on

your line and open them so that the pencil point

is on your mark.

4. Draw an arc that passes through your mark and

the line.

5. Place the compasses point on your mark and

draw an arc to the left of your mark.

6. Place your compasses point where the first arc

passes through the line.

7. Draw an arc which intercepts the second arc.

23

8. With your ruler draw a line that passes through

your mark and the intersection of the two arcs.

You have your two parallel lines.

o

Exercise 2-8: 90 Angle

o

Drawing a perpendicular line using your ruler and 60

set square.

1. Draw a horizontal line 12 cm long.

o

2. Position your 60 setsquare along the line.

3. Draw another line along the edge of the

setsquare.

24

o

You now have perpendicular lines with a 90 between

them.

o

Exercise 2-9: 45 Angle

o

In this exercise you are going to draw an angle of 45

using you compasses and ruler. Do the following.

1. Draw a horizontal line of length 10 cm.

2. Mark the mid-point of you line using your ruler

(5 cm).

3. Using a setsquare, draw a vertical 6 cm

perpendicular to the mid-point.

4. With your compasses, place the point at the

mid-point of you line and extend until the

pencil touches the left end of your horizontal

line.

5. Draw and arc which passes through the

horizontal line and the vertical line.

25

6. Using your ruler, draw a line between the two

intercepts.

o

7. The angle between them is 45 , confirm this

result using your protractor.

In this chapter you have been able to draw angles of

o

o

o

o

60 , 30 , 90 and 45 by using your compasses and

ruler. You have also been able to confirm the sizes of

the angles using your protractor.

This is a very useful skill set. In the past, before the

advent of computers, engineering drawing skills

included the ability to draw various angles without the

use of a protractor. The

skilled craftsperson would

spend their time standing at

one of these tables drawing

plans for product designs.

26

3 The Circle

There are four features you need

to know about at this stage in the

description of a circle.

1. The circumference which

is the visible rim of the circle (also

known as the perimeter).

2. The centre O which is equidistant from every

part of the circumference.

3. The diameter is a line that stretches from one

side of the circumference to the other and

passes through the centre. You will often come

across vertical diameter (which goes from north

point to the south point) and horizontal

diameter (which goes from east point to west

point).

4. The radius which stretches from the centre to

the circumference.

If the diameter is D, the length of the circumference is

πD, where π is 3.141592… a never ending number

referred to as a transcendental number. In the

following exercises you will be expected to use your

27

compasses to draw circles. When using your

compasses, push the point firmly into the paper, hold

the grip at the top and just rotate them in an even

manner - this may take some practice to get an even

circle.

3.1 Inscribing Polygons in a Circle

In this section we are going to draw regular polygons

within a circle – regular means

all sides are of equal length. It

will be useful to have some

coloured gel pens (similar to the

ones shown on the left) to draw

each polygon. We shall start

with a three-sided figure (a

triangle) and work up to a nonagon (a nine-sided

figure). In the following exercises we shall refer to

specific points on the circle

circumference as indicated by

the image on the left. You

should know your east point

from your west point!

28

Exercise 3.1: A Regular Triangle in a Circle

A triangle with three equal sides is called an

equilateral triangle and the internal angles are the

o

o

same – each being 60 adding up to 180 .

Do the following:

1. Using your compasses, draw a circle of radius

5cm.

2. Draw a vertical diameter.

3. Position the compasses point on the south point

and draw an intersection arcs through the

circumference on the left and on the right.

4. Using one of your colour pens, connect the two

intersections points and the north point.

5. You have your equilateral triangle.

29

Now you have created an equilateral triangle; label

your drawing ‘My equilateral triangle - DATE’.

Exercise 3.2: A Regular Quadrilateral in a

Circle

A quadrilateral is the general name given to a four-

sided figure.

Do the following:

1. Using your compasses, draw a circle of radius 5

cm.

2. Draw a horizontal diameter.

3. Expand the compass arm separation to 7 cm.

30

4. Place the compass point at the east point and

draw an arc which intersects the circumference

at two places.

5. Place the compasses point at the west point and

draw another arc which again intersects the

circumference at two places.

31

6. Draw a line which passes though the

intersection points of the two arcs.

7. You will observe the two points where this line

intersects with the circumference are the north

32

point and south points of the circle. This is a

vertical diameter.

8. Using one of your colour pens draw lines

between the north point and the east point,

between the east point and the south point,

between the south point and the west point,

between the west point and the north point.

You now have a regular quadrilateral inscribed in a

circle; label your drawing ‘My regular quadrilateral

– DATE’.

Exercise 3.3: A Regular Pentagon in a Circle

A pentagon is a five-sided figure. Drawing this figure

in a circle is a little bit more involved and requires 18

33

stages. Nonetheless it is a worthwhile exercise and is

excellent practise in the use of your compasses. We

shall draw two circles, in the first circle we shall find

the separation for the points of the compasses. In the

second circle we shall mark out the length of the sides.

Do the following:

1. Using your compasses, draw two circles of

radius 6 cm separated by 2 cm.

2. In the first circle draw a vertical diameter.

3. Place the edge either of your set squares along

this diameter with the right angle at the centre.

4. Draw a horizontal radius from the centre to the

west point.

5. Draw a diameter from east point to the centre,

you now have a horizontal diameter.

6. Reduce the angle in your compasses by 1 .5 cm.

7. Place the compasses point on the east point and

draw an arc in the right-hand side of the circle.

8. Place the compasses point on the centre and

draw another arc in the right-hand side.

9. Draw a line between the points where the arcs

intersect.

34

10. Where this line crosses the diameter, label

this point A.

11. Expand the compasses so the point is on A

and the pencil is on the north point.

12. Draw an arc from the north point that

intersects the diameter on the left-hand side.

Label this this point B.

35

13. Draw a line from the north point to B. This

distance is the side length of the pentagon.

14. Adjust the compasses with the point on

the north point and the pencil point on B.

15. Place the compasses anywhere on the

circumference of the second circle and draw an

intersection arc on the circumference.

16. Place the compass point on this first

intersection arc and draw the second

intersection arc on the circumference.

17. Continue doing this until you arrive back to

where you started.

36

18. Using one of you colour pens, connect all

the intersection arcs on the circumference, you

have created your regular pentagon.

Now you have drawn a regular pentagon, measure an

o

internal angle using your protractor. It should be 108 .

Label your drawing, ‘My regular pentagon – DATE’

37

Exercise 3.4: A Regular Hexagon in a Circle

A hexagon is a six-sided figure. Do the following:

1. Draw a circle of radius 6 cm.

2. Place the compass point close to the south point

of the circle.

3. Draw an arc intersection with the circumference

on the left side of the circle.

4. Place the compass point on the intersection and

draw another arc intersection further along the

circumference of the circle.

5. Repeat step 4 until you arrive back to where you

started.

6. Use one of your colour pens and the ruler to

draw connecting lines between the intersecting

arcs. You have created a hexagon.

38

Now you have drawn a regular hexagon, measure an

o

internal angle using your protractor. It should be 120 .

Label your drawing, ‘My regular hexagon – DATE’

Exercise 3.5: A Regular Heptagon in a Circle

A heptagon is a seven-sided figure. Do the following:

1. Draw two circles of radius 6 cm separated by 2

cm.

2. In the first circle, draw vertical diameter.

3. Place the compass point at south point and draw

an arc which intersects the circumference at two

places. Label these A on the left side and B on

the right side.

39

4. Using your ruler, draw a horizontal line between

A and B. Label the intersection of this line with

the vertical diameter C.

5. The distance between A and C is the side length

of your heptagon. Adjust the compasses very

accurately with the point on A and the pencil on

C.

6. Place the compass point on the circumference

close to the south point on the second circle and

draw an intersection arc with the circumference

on the left side.

7. Place the compass point very accurately on this

intersection and draw another intersection

further along the circumference.

8. Repeat step 7 until you arrive back at the south

point.

40

9. Using a colour pen draw lines connecting the arc

intersection points along the circumference.

When you have drawn all seven lines, you have

created your heptagon.

Now you have drawn a regular heptagon, measure an

internal angle using your protractor. It should be

o

128.6 . Label your drawing, ‘My regular heptagon –

DATE’

Exercise 3.6: A Regular Octagon in a Circle

An octagon is an eight-sided figure. Do the following:

1. Draw a circle of radius 6 cm on a sheet of A4

paper.

2. Draw a vertical diameter.

41

3. Place the edge of a set-square along this vertical

diameter with the right-angle touching the

centre.

4. Draw a horizontal radius from the centre to the

east point.

5. Draw a diameter from the east point to the west

point.

6. Reduce the arms of your compass by 1 cm.

7. Place the compass point on the north point and

draw an arc which intercepts the circumference

in two places.

8. Place the compass point on the west point and

draw an arc which intersects the first arc in two

places.

42

9. Place the compass point on the east point and

draw an arc which intercepts the first arc in two

places.

10. Starting at the intercept point on the left

of the circle, draw a line from this point through

the centre to the circumference opposite.

43

11. Draw another line from the intercept point

on the right of the circle through the centre to

the circumference opposite. We now have the

eight vertices of your octagon.

12. Using a colour pen draw a line between the

north point and the north-west point.

13. Draw a line between the north-west point

to the west point.

14. Draw a line between the west point to the

south-west point.

15. Continue this until you arrive back at the

north point. You have now drawn your octagon.

44