Now you have drawn a regular octagon, measure an
o
internal angle using your protractor. It should be 135 .
Label your drawing, ‘My regular octagon – DATE’
Exercise 3.7: A Regular Nonagon in a Circle
A nonagon is a nine-sided figure. Do the following:
1. Draw two circles of radius 6 cm separated by 4
cm on a sheet of A4 paper.
2. In the first circle, draw a vertical diameter.
3. Place the edge of a set-square along this vertical
diameter with the right-angle touching the
centre.
45
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. Extend the length of the horizontal diameter
beyond the east point by 3 cm.
7. Place compass point at the north point and draw
an arc from the centre until it intersects with the
circumference on the right of the circle.
8. Place the compass point on the south point and
extend the arm until the pencil touches the
intersect on the right side of the circumference
– you must do this as accurately as possible.
9. Draw an arc from the intercept point until it
passes through the extended line from the east
point.
46
10. Place the compass point on this intersect
and draw an arc which passes through the north
point and the south point.
11. Label the intercept of this arc with the
horizontal diameter with the letter A. The
distance from the west point to A is the side
length of the nonagon.
12. Place the compass point on the west point
of the first circle and adjust the arms until the
pencil touches point A – you must do this as
accurately as possible.
13. On the second circle, place the compasses
point close to the north point. Moving in a
clockwise direction, draw a circumference
intercept arc. Move the compass point to this
47
intercept point and draw another circumference
intercept arc.
14. Continue doing this until you arrive back to
where you started.
15. Using a colour pen, draw lines to connect
each intercept point. Once you have drawn all
nine lines you have your nonagon.
Label your drawing, ‘My regular nonagon – DATE’
48
Now you have drawn a regular nonagon, measure an
o
internal angle using your protractor. It should be 140 .
Since you have already created a pentagon, you are
able to draw a decagon (10 sided figure) where the
side length will be half that of the pentagon.
Exercise 3.8: Connecting the Vertices of
Polygons
Now you have created several polygons, you may
consider drawing a fresh set of polygons and draw
lines between their vertices, then add some colour.
You will find some interesting results.
The pentagon: you will see in
at the centre another
pentagon surrounded by five
isosceles triangles. The figure
also looks like a star.
The hexagon: you will see
another hexagon formed in
the centre surrounded by six
equilateral triangles.
49
The heptagon: although there
is another heptagon in the
centre which is surrounded by
equilateral triangles, there are
also irregular quadrilaterals
which make up another
heptagon
The octagon: like the
heptagon, has two inner
octagons, in the centre made
up of triangles the second
made up of equilaterals.
The nonagon: reducing the
number of connecting lines,
you can see the number of
triangles and equilaterals with
an irregular nonagon in the
centre. If you look carefully
you can see three large
triangles connecting the
vertices.
50
Once you start looking at the connecting lines in a
decagon it gets very complicated as you will discover
if you attempt to do it. Connecting lines on a polygon
has applications in communications networks. For
example, if there are 7 nodes (users) the colour lines
are used to prioritise the communication and it also
shows how some direct links are excluded. From the
diagram there is no direct link between A and D or
between B and F. But you will notice there is a direct
link between G and all the other nodes. So if E wanted
to communicate with C, they would have to go
through one of the other nodes.
51
4 Irregular Polygons
You may not realise it but you see irregular polygons
every day – for example the
letter T is an octagon as you can
see from the diagram on the
left. When we say irregular, we
mean the sides are not the all
the same length as you can see
in the letter T. The letter E is dodecagon – how many
sides has it got? When looking at irregular polygons
we are interested in the number of sides, their length
and the internal angles.
4.1 Triangle
Triangles fall into one of four categories as shown
below.
Equilateral Isosceles Right Irregular
angled
52
You will notice the setsquares in your Helix box are
right-angled triangles, one of them being an isosceles
right-angled triangle. Whatever shape triangle you
o
have the internal angles will always add up to 180 .
The equilateral triangle has three internal angles of
o
60 and the isosceles triangles have two internal
angles which are the same. The right-angled triangle
o
as the name suggests has one internal 90 angle.
Whatever shape triangle you have it can be converted
into two right angled triangles by dropping a
perpendicular line as illustrated in the diagram below.
Irregular triangle Two right angled
triangles
Exercise 4.1
Measuring the angles in an irregular triangle. Do the
following.
53
1. Draw a line of length 12 cm.
2. Draw another line of 10 cm from the end of the
first line with a random angle between them.
3. Draw a third line connecting the ends of the lines
you have already drawn.
4. Use your protractor to measure the three
internal angles. Add the three angle values
together.
o
5. _____ + _____ + _____ = 180
Have you noticed the triangular cross section of the
famous Toblerone chocolate package – it’s an
equilateral triangle.
54
4.2 Quadrilateral
There are several shapes of quadrilaterals, all of them
o
have four sides and the internal angles add up to 360 .
Square Parallelogram Isosceles
Trapezium
Rectangle Trapezium Concave
Rhombus Kite Irregular
55
As you can see, quadrilaterals take on many forms;
they are distinguished by having:
• Two or more parallel sides
• Two or more right-angles
• Two or more equal size angles
From the diagrams on the previous page can you
identify which quadrilaterals have these features? The
rhombus is a diamond shape and the rectangle is also
known as an oblong. A concave quadrilateral has an
o
internal angle larger than 180 and looks like an arrow
head.
The rhombus on the other
hand is an interesting shape –
it’s all a matter of perspective.
Looking at the chequered
pattern, the pattern closest to
the base of the diagram is made up of squares.
However as you look further away from the base, each
black and green square transforms into a rhombus.
56
To indicate parallel lines on a
quadrilateral certain arrow
marks are used as indicated
in the figure on the left. The
lines with single arrows are parallel and the sides with
double arrows are parallel.
Exercise 4.2
Drawing a concave quadrilateral.
1. On a sheet of A4 paper, using your ruler mark
two horizontal dots 10 cm apart.
2. Set your compasses with a separation of 5 cm,
place the point on the left dot and draw an arc.
3. Place the compasses point on the right-hand dot
and draw a touching arc.
4. Place the compasses point where the two arcs
touch and draw an arc vertically above this
point.
5. Increase the separation of your compasses to
6.5 cm.
6. Place the compasses point on right dot and draw
a horizontal arc.
57
7. Place the compasses point on the left dot and
draw an arc which intercepts the previous arc in
two places.
8. Draw vertical line which passes through these
intercepts until it intercepts the furthermost arc.
9. With one of your colour pens draw lines
between the left and right dots and the
furthermost intercept.
58
10. Draw lines between the left and right dots
and the midway intercept – you have your
concave quadrilateral.
11. Using your protractor, measure all four
internal angles.
12. Add all four angles, the result should be
o
360 .
4.3 Pentagon
The sum of the internal angles of a
o
pentagon is 540 . The following
exercise involves drawing an irregular
pentagon and then measuring all the
angles and adding up the result.
59
Exercise 4.3
1. Using your ruler, draw a horizontal line of length
5 cm on an A4 sheet of paper in landscape.
2. Move your ruler to the right end of the line and
draw another line of 4 cm forming an obtuse
angle.
3. Move you ruler to the end of the pervious line
and this time draw a line of 4.5 cm forming
another obtuse angle.
4. Again move you ruler to the end of the pervious
line and this time draw a line of 5.3. cm forming
another obtuse angle.
5. Draw a line connecting the end of this line to the
left end of the first line.
60
6. Label the five internal angles A to E.
7. Make a table along the right-hand side of the A4
sheet of paper similar to the one shown here.
8. Using your protractor, measure
each angle in turn and enter the
measurements in the table.
9. Add up all the angles, they
o
should sum to 540 .
4.4 Hexagon
The sum of the internal angles of a
o
hexagon is 720 . In the following
exercise your are you are expected
to draw an irregular hexagon.
61
1. Draw a horizontal line of length 6 cm.
2. Draw a second line from the right end of length
5 cm with an obtuse angle.
3. Draw a third line of 7 cm from the end of the
second line with another obtuse angle.
4. Draw a fourth line of length 6 cm with an acute
angle from the end of the third line.
5. Draw a fifth line of length 5 cm with a reflex
angle from the end of the fourth line.
6. Draw line connecting the end of the fifth line to
the left of the first line.
7. Label the internal angles A to F an make a table
similar to the one shown on the next page.
8. Measure each angle and enter its size into the
table.
62
9. To measure the reflex angle,
measure the external angle and
o
subtract 360 , enter this value in
the table.
10. Add up all the angles and
o
they should come to 720 .
4.5 Heptagon
The sum of the internal angles of
o
a heptagon is 900 . In this
exercise you are going to draw an
irregular heptagon and measure
all seven of its internal angles.
1. Draw a horizontal line 5 cm long.
2. From the right of this line draw a second line 4
cm long with an obtuse angle.
3. From the end of the second line draw a third line
of length 4 cm with an obtuse angle.
4. From the end of the third line, draw a fourth line
of length 2 cm with an acute angle.
63
5. From the end of the fourth line draw a fifth line
of length 4 cm with a reflex angle.
6. From the end of the fifth line, draw a sixth line
of length 7 cm and an acute angle.
7. Finally draw a seventh line connecting the sixth
line to the left of the first line.
8. Label the angles A to G.
9. Measure each of the angles with your protractor
and enter your measurements into a
table similar to the one shown here
on the left.
10. Add all the angles and they
o
should come to 900 .
64
5 More on the Circle
In this chapter we shall be
looking at more features
relating to the circle. First a
few more definitions. In the
diagram on the left you can
see a circle with several
lines.
1. A chord is a line that
touches the circumference in two places, but
does not pass through the centre.
2. A sector is an area within the circle as shown in
purple.
3. A segment is an area of a circle that lies
between a chord and the circumference as
shown in yellow.
4. A tangent is an external line
that touches the circumference at
one place.
Consider a circle with two chords
connected to two radii as shown
in the diagram on the left. A is the
65
angle subtended by the chords and B is the angle
subtended by the radii. A = 2B. You are now to show
this is true.
Exercise 5.1
1. Draw a circle of radius 5 cm and mark the centre
with a dot.
2. Draw two radii as shown in the circle on the
previous page.
3. Draw two chords from the ends of the radii to a
point on the circumference. Mark the point they
meet with a dot.
4. Mark your two angles A and B.
5. Using your protractor on one of the radii with
the dot under the cross-hairs and then measure
angle A.
6. Place your protractor alone one of the chords
with the circumference dot under the cross hairs
and measure the angle B.
7. Confirm that A = 2B.
Whenever a triangle is formed from two radii, it is
worth noting it will always be an isosceles triangle –
66
two of the angles are equal as
shown in the diagram on the left.
Tangents are external lines which
touch the circumference of a circle
at one place. Very often you will see
converging tangents
(intersecting to a point)
as shown in the
diagram on the left. You
are now going to create
a circle with two
tangents and measure the angles between them.
Exercise 5.2
1. Draw a circle of radius 5 cm to the left an A4
page.
2. Use a set square to draw a radius from the
centre to the south point.
3. Draw a horizontal tangent line of length 14 cm
which passes through the south point. There
should be a right-angle between this line and the
67
radius you have drawn. Confirm using your
protractor.
4. Draw another tangent line from the right end of
the horizontal line to the circumference.
5. Draw a radius from the centre to where the
second line touches the circumference.
6. Using your protractor, measure the angle
between the radii - mark this as A.
68
7. Measure the angle between the two converging
tangents on the right, mark this as B.
8. Confirm the angles A = 2B
Exercise 5.3
In this exercise you are going
to draw two triangles within a
circle similar to the one
shown on the left. The
purpose is to measure the
internal angles and see how
they are related.
1. Draw a circle of radius 6 cm.
2. Draw a chord on the left hand side of the circle.
69
3. Draw a second chord on the right-hand side of
the circle which is not parallel with the first
chord.
4. Draw connecting lines from the ends chords
which cross over within the circle.
5. There are four internal angles, label these A to
D. Measure each of these angles.
70
6. Enter your measurements into a table similar to
the one shown below.
7. What are your conclusions?
The results of your measurements
will show you which angles are
equal.
And finally here is an optical
illusion. When you look at the
image on the left you perceive a
triangle surrounded by three
circles. But there is no triangle,
your brain fills in the lines
between the circles to give the
illusion of a triangle that doesn’t exist in the drawing.
71
Here are some more interesting ebooks on maths.
72
Why not visit
www.CambridgePaperbacks.com
73
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