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
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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