CHAPTER
11
Preparation
Useful skills for this chapter:
1 Copy the shapes and label them with as many words that apply.
a
b
e
f
Rectangle
4 equal sides
Quadrilateral
G
Square
3 equal sides
c
PA
Triangle
2 equal sides
g
d
h
FI
N
AL
K
K I CF F
O
ES
• some understanding of triangles, rectangles, squares and circles.
Show what you know
1
Draw each shape.
a A rectangle with one side equal to 6 cm
b A 3 cm square
c A triangle with one side equal to 4 cm
d A rhombus
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11
CHAPTER
ES
Measurement and Geometry
G
Two-dimensional shapes
N
AL
PA
In this chapter we look at two-dimensional shapes, which are also known as
polygons.
A polygon is a two-dimension shape enclosed by
three or more line segments called sides. Exactly
two sides meet at each vertex, and the sides
do not cross.
FI
Polygons are named according to the number of sides
that they have, or their angles.
Polygons have no thickness, but there are solid objects that are like twodimensional shapes with thickness. Can you find some in your classroom?
CHAPTER 11
TWO-DIMENSIONAL SHAPES
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253
11a
Triangles
What is a triangle? Think of words that start with ‘tri’. a triathlon is a three-event race
and a tripod is a three-legged stand for keeping a camera or telescope steady. The
prefix ‘tri’ means ‘three’. so a triangle has three angles. It also has three straight sides.
Triangles can be sorted according to the lengths of their sides or according to the sizes
of their interior angles.
ES
Equilateral triangles
PA
G
a triangle with all of its sides the same length is called equilateral. ‘Equilateral’
comes from two Latin words meaning ‘equal’ and ‘sides’. Here are some pictures of
equilateral triangles.
If all three angles in a triangle are the same, we call it equiangular, from two Latin
words meaning ‘equal’ and ‘angles’.
Every equiangular triangle is also equilateral. This is a special property of triangles.
N
AL
This shape has equal sides but different angles.
4 cm
4 cm
4 cm
4 cm
FI
This shape has equal angles but different sides.
Isosceles triangles
a triangle with at least two sides the same length is called isosceles, from two Greek
words meaning ‘equal’ and ‘legs’. Every equilateral triangle is isosceles, but there are
isosceles triangles that are not equilateral. Here are some pictures of isosceles triangles.
Which one is equilateral and which ones are isosceles but not equilateral?
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If a triangle has exactly two angles the same, then it has to be isosceles, but need not
be equilateral. You can see in the pictures above that the triangle in the middle and
the one on the right have exactly two angles equal.
Scalene triangles
G
ES
The only other thing that can happen is that all of the sides of the triangle have
different lengths. We call these triangles scalene, from a Latin word meaning ‘to mix
things up’. Here are some pictures of scalene triangles.
PA
If all three angles in a triangle are different then the triangle has to be scalene. Draw a
few to convince yourself this is true.
Right-angled triangles
FI
N
AL
When one of the angles in a triangle is 90°, we call it a right-angled triangle. Here
are some right-angled triangles. Which ones are isosceles and which ones are scalene?
Can a right-angled triangle be equilateral? Either draw one or explain why there
aren’t any.
Can you see why there cannot be two right angles in a triangle?
Draw some diagrams to help.
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255
Obtuse-angled triangles
When the biggest angle is more than 90°, we call the triangle an obtuse-angled
triangle. Here are some obtuse-angled triangles. Which one is isosceles and which
one is scalene?
ES
Can an obtuse-angled triangle be equilateral? Either draw one or explain why there
aren’t any.
Acute-angled triangles
PA
G
If all of the angles are less than 90°, we call the triangle an acute-angled triangle. Here
are some acute-angled triangles. Which ones are isosceles and which one is scalene?
Can an acute-angled triangle be equilateral? Draw one.
N
AL
Example 1
a Draw an acute-angled triangle with one side 6 cm in length.
b Draw an isosceles triangle with one angle a right angle.
Solution
FI
a Acute-angled triangles have all angles
less than 90°. Here is one with one side
6 cm in length. Yours may look different.
6 cm
b Isosceles triangles have 2 sides equal.
The only way to draw this is with the
right angle between the 2 equal sides.
Here is one possibility.
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5 cm
5 cm
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11a
Draw and then cut out as many different types of triangles as you can from what
you have learnt so far. Label each triangle and make a poster to display your work.
11a
1
ConnECT, appLY anD BUILD
Individual
ES
1
Whole class
Use a ruler and a pencil to draw:
a an isosceles triangle with two sides 4 cm in length
G
b an obtuse-angled, scalene triangle with one side equal to 5 cm
c a right-angled triangle that is not scalene
2
PA
d a triangle with two angles equal to 60°.
We sometimes use the word ‘base’ to name the side of the triangle that it ‘sits’ on.
Use a protractor and a ruler to construct triangles using the base and the angles
shown. measure the third angle in your triangle and label the size of it in your
diagram.
a
30°
60°
b
60°
5 cm
N
AL
8 cm
c
50°
d
50°
90°
60°
4 cm
6 cm
3
60°
a Draw a square. now draw in its diagonals.
FI
b measure the four angles around the centre point where the diagonals cross.
What do you notice?
c Draw a rectangle that is not a square. now draw in its diagonals.
d measure the angles around the centre point where the diagonals cross.
e What do you notice?
4
Copy each shape and draw a line inside each to form two right-angled triangles.
a
b
c
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11B
Quadrilaterals
Quadruplets are four children born to the same mother at the same time.
a ‘quad’ vehicle has four wheels.
Can you think of other words that start with ‘quad’?
What is a quadrilateral?
ES
In Latin, ‘latus’ means ‘side’. so a quadrilateral is a shape with four sides. It has four
vertices also.
There are many different kinds of quadrilaterals; some have special names. We know
two kinds of quadrilaterals already. rectangles and squares have four sides.
G
Rectangle
a rectangle is a quadrilateral in which all the angles are right angles.
PA
The opposite sides of a rectangle have the same length. These
sides are parallel to each other.
Properties of a rectangle
all angles are right angles.
2
opposite sides are parallel.
3
opposite sides have the same length.
N
AL
1
Square
FI
a square is a very special kind of rectangle.
all of its sides have the same length.
Parallelogram
a parallelogram is a quadrilateral with opposite sides
parallel. It looks like a ‘pushed over’ rectangle.
rectangles and squares are special kinds of
parallelograms. They have four right angles as well
as opposite sides parallel.
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Trapezium
a trapezium has two sides that are parallel.
You might have seen a table at school
with this shape.
Rhombus
a square is a special kind of rhombus. If you have
a rhombus with four right angles, it is a square.
ES
a rhombus is a parallelogram with four equal sides.
Think of a rhombus as a square pushed sideways.
Look at a pack of cards and find a ‘diamond’ card. Can you see that
the diamond is a rhombus?
Kite
PA
G
a diamond is a rhombus drawn vertically.
a kite has two pairs of adjacent sides equal.
N
AL
so a rhombus and a square are special kinds of kite.
Example 2
FI
a Draw a parallelogram that is also a kite. What other name could you give to
this shape?
b Draw a rectangle with equal sides. What other name could you give to this
shape?
Solution
a a parallelogram that is also a kite
is a rhombus because all four
sides are equal.
b a rectangle with equal sides is a
square.
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259
11B
Whole class
ConnECT, appLY anD BUILD
1
Draw and then cut out as many different types of quadrilaterals as you can make
from what you have learnt. Label each quadrilateral and make a poster to display
your work.
2
Use the Venn diagram on the right to put
extra labels on the quadrilaterals you have
made above. For example, if you made a
square, the Venn diagram tells you that
a square is also a rectangle, a rhombus,
a parallelogram, a quadrilateral and a
trapezium. so you can put 6 labels on your
square. Isn’t that amazing?
quadrilateral
rectangle
parallelogram
ES
square
rhombus
1
PA
11B
G
trapezium
Individual
Draw:
N
AL
a a square with 5 cm sides
b a trapezium with a base of 6 cm and the side opposite its base equal to 4 cm
c a parallelogram with at least one angle equal to 130°
2
Construct quadrilaterals using the sides and angles shown. measure the missing
sides and missing angles and mark each on your drawing.
FI
3
d a rectangle with one pair of sides equal to 1 cm and the other pair longer than
your left thumb.
Draw a rhombus with at least one right angle. What do you notice?
a
b
12 cm
90°
4 cm
5 cm
90°
100°
90°
12 cm
2 cm
100°
10 cm
c What is the sum of the angles in each?
d What do you notice about the missing side in a?
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4
Draw a shape that has four sides of 4 cm and the angle
at M is as follows. (You may need to use trial and error
to get the sides to meet.) The first one has been done for
you:
a M = 60°
4 cm
4 cm
b M = 90°
4 cm
M = 60°
4 cm
c M = 150°
Other polygons
ES
11C
G
In this section we look at how shapes with more than four sides are named. as before,
the name of each shape tells us something about its properties.
PA
Pentagons
The Greek prefix ‘penta’ means ‘five’ and ‘gon’ means
‘angle’. so a pentagon has five angles. It also has five
vertices and five sides. Here are two pentagons.
Regular pentagons
N
AL
108°
108°
regular pentagons have five equal angles and
five equal sides. Each angle is 108°.
The marks on the sides in the diagram indicate that the side
lengths are all the same.
108°
108°
108°
FI
Hexagons
The Greek prefix ‘hexa’ means ‘six’ and ‘gon’ means ‘angle’. so a hexagon has six
angles. It also has six vertices and six sides. Here are two hexagons. The first one is a
regular hexagon. The second one is a non-convex irregular hexagon.
120°
120°
120°
120°
120°
120°
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261
Regular hexagons
Regular hexagons have six equal angles and six equal sides. Each angle is 120°.
Two-dimensional shapes are named according to the number of sides. We could start
the list below by calling a one-sided shape a monogon and a two-sided shape a digon.
But what would they look like? Try for yourself. Do you agree that one-sided shapes
and two-sided shapes do not make any sense?
We have already discussed a three-sided shape – which we call a triangle – but it could
also be called a trigon. A four-sided shape is known as a quadrilateral, but it could be
called a tetragon.
Greek or
Roman
prefix
Name
Penta
Pentagon
6
Hexa
Hexagon
Hepta
Heptagon
Octa
Octagon
FI
8
N
AL
7
Regular
polygon
PA
5
Irregular
example
G
Number
of sides
ES
A regular polygon has all sides equal and all angles equal.
262
9
Ennea
Nonagon
or enneagon
10
Deca
Decagon
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11C
1
Whole class
ConnECT, appLY anD BUILD
a Draw a sketch of a regular pentagon. now draw lines to show how you could
cut the pentagon into 5 isosceles triangles.
b Draw a regular pentagon. now draw lines to show how you could cut the
pentagon into 3 triangles.
Can the pentagon be cut into 3 triangles in another way?
11C
G
Individual
measure the sides and angles of these shapes.
a
A
C
N
AL
B
PA
1
ES
c are any of your triangles special, like equilateral, isosceles or scalene?
D
G
E
F
FI
b
c
H
M
I
L
J
K
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263
2 I am a shape. What shape am I?
a I have 6 equal sides and 6 equal angles.
b I have 12 sides.
c I have the same number of sides as an octopus has legs.
d I have 5 sides.
e I have 10 sides.
fMy prefix means 5 and the rest of my name is the same as 10–sided.
ES
3 Draw or trace these shapes to complete the questions below.
G
a Draw a rectangle. Draw a line to show how you could cut the rectangle into
2 right-angled triangles. In how many ways can you do this?
b Draw a square. Now draw a line to show how you could cut the square into
2 rectangles. How can you make them equal rectangles?
PA
c Draw a square. Now draw lines to show how you could cut the square into
3 equal rectangles.
d Draw a rhombus. Now draw a line to show how you could cut the rhombus
into 2 equal triangles. In how many ways can you do this?
e Draw or trace a regular hexagon. Now draw lines to show how you could cut
the hexagon into 6 equilateral triangles.
N
AL
f Draw a square. Now draw a line to show how you could cut the square into
one triangle and one irregular pentagon.
g Draw or trace a regular hexagon. Now draw a line to show how you could cut
the hexagon into one isosceles triangle and one irregular pentagon.
FI
1
3
Trace a third triangle that has an edge in common with the
second, and a vertex in common with the first and the third. Now
you should have a trapezium.
2
2
4 a
Take an equilateral triangle. (You might have one in a set of plastic shapes.)
Trace it. Now trace a second triangle that has one edge in common with the
first triangle.
1
b Keep tracing triangles, going around the vertex at the centre all triangles have
in common, until you have a hexagon.
c How many triangles did you draw?
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11D
Symmetry of twodimensional shapes
In mathematics when the pieces of a two-dimensional shape match up exactly across
a straight line, we say the shape is symmetrical about the line.
ES
For example, this triangle is symmetrical about the red dotted line:
N
AL
PA
G
In nature, we see symmetry in animals and in plants.
The line is called a line of symmetry.
FI
When we say that something is symmetrical, we mean that it is identical on both sides
of the line of symmetry. an example of symmetry is the drawing of the tree on the left.
Symmetric
Asymmetric
The opposite of symmetrical is asymmetrical, as shown in the picture of the tree on
the right.
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265
a shape can have more than one line
of symmetry.
The shape on the right has two lines
of symmetry.
ES
The shapes below have four lines of symmetry.
N
AL
PA
a circle has infinitely many lines of symmetry!
It would not be possible to draw them all.
G
Imagine folding the shape over along a line of symmetry. The two halves then match
each other exactly. The image is reflected in the line. We call the line the axis of
reflection or the axis of symmetry.
ConnECT, appLY anD BUILD
FI
11D Whole class
1
Create a picture using your class set of pattern blocks or use triangle-grid paper
to draw one that includes hexagons, trapezia, triangles and rhombuses. ask your
partner to make its reflection. Here is one example.
picture
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YEar 5
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2
Draw half of a picture and ask your partner to complete it so that the object you
have drawn is symmetrical about a line.
3
Use triangle grid paper (see BLM 15 in the Interactive Textbook) and create a
picture that has:
a one line of symmetry
b two lines of symmetry
1
ES
11D Individual
c three lines of symmetry
Copy these shapes and draw in their lines of symmetry.
b
c
2
PA
G
a
a Draw 5 regular polygons of different sizes.
b mark in the lines of symmetry with a dotted line.
Copy each diagram, then complete the missing parts of each shape. The dotted
lines are lines of symmetry.
N
AL
3
FI
a
c
b
d
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267
11E
Transformations and
tessellation
We see patterns all around us. many patterns are made by shapes fitting together.
rotation, reflection and translation are some of the different ways we can transform a
two-dimensional shape.
Rotation
ES
We rotate a shape about a point when we turn it through an angle about the point.
Point
PA
G
This shape has been rotated clockwise through 90° about the point marked with a red
dot. The word ‘image’ has been used to label the shape after rotation in the diagram
below.
Point
N
AL
Image
FI
We can rotate anticlockwise about a point.
This arrow has been rotated anticlockwise through 90°.
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Example 3
How has this shape been moved?
Before
ES
Image
After
G
Solution
PA
The shape has been rotated 90° in a clockwise direction.
Reflection
N
AL
A reflection is a transformation that flips a figure about a line. This line is called the
axis of reflection. A good way to understand this is to suppose that you have a book
with clear plastic pages and a triangle drawn, as in the first diagram below. If the page
is turned, the triangle is flipped over. We say it has been reflected; in this case the axis
of reflection is the binding of the book.
olleH
FI
Hello
This shape has been reflected in the vertical line.
Image
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269
Translation
When we translate a shape, we slide it. We can slide it left or right, up or down.
Translations move the shape without rotating it.
This shape has been
translated horizontally.
This shape has been
translated vertically.
Image
ES
Image
Tessellation
G
A tessellation is a tiling pattern made by fitting together two-dimensional shapes
with no gaps or overlaps. The tessellation can continue in all directions.
PA
For example, we could start with an equilateral triangle.
FI
N
AL
We can rotate it 180° and shift it so the triangles fit together perfectly. The tiling can
continue horizontally and vertically. We say that the equilateral triangle tessellates.
Circles do not tessellate because we cannot rotate and shift them to fill up the whole
space without gaps or overlaps.
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It is possible to tessellate two or more shapes.
Whole class
ConnECT, appLY anD BUILD
G
11E
ES
The tessellation below uses regular hexagons and equilateral triangles.
Use your class set of shapes or cut out some of your own. Take turns giving
instructions to your partner to translate a shape in different ways.
2
Look around the school for tessellating patterns. Take digital photographs of them
and describe the shapes used. Draw in the lines of symmetry.
N
AL
PA
1
11E
Use your class set of pattern blocks or use triangle-grid paper to draw and colour
a tessellating pattern that fills a 10 cm × 10 cm space on the page and uses:
FI
1
Individual
2
a only triangles
b only hexagons
c only trapezia
d hexagons and triangles
a Draw a tessellation pattern on triangle-grid paper using rhombuses and
trapezia.
b Draw a tessellation pattern on triangle-grid paper using hexagons, trapezia,
triangles and rhombuses.
3
a Draw the possible shapes that can be made from three identical regular
hexagons. rotations and reflections are considered the same.
b Create a shape made from 3 hexagons that will tessellate. Use triangle grid
paper to demonstrate your tessellation.
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271
11F
Enlargement
transformations
a scale drawing can be used when an object is too large to be shown at full size on a
page, for example a road map of a suburb or a plan of a building.
scale drawings are also helpful when we want to see a
very small object in a larger size so we can see more of its
detail, for example an enlargement of a diagram of a tiny
insect or a detailed picture of a leaf.
G
a scale drawing or enlargement has exactly the same
shape as the original object, but a different size.
ES
The human head louse is 1 to 2 millimetres in length. Here
is an enlarged picture of a head louse.
In this section we look at enlargements of some two-dimensional shapes.
PA
Below on the left is a square of side length 2 centimetres drawn on centimetre grid
paper. The perimeter of this square is 8 centimetres and the area is 4 cm2.
FI
N
AL
We can enlarge the square by doubling its side length as shown on the right.
The side length is now 4 cm, the perimeter of the larger square is 16 centimetres
and the area is 16 cm2. The side lengths were multiplied by 2, which doubled the
perimeter. The area is now four times the size.
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Example 4
ES
a Enlarge the following shapes drawn on centimetre grid paper (BLM 14) by
multiplying each side length by 3.
i
ii
G
b What happens to the perimeter and area of each shape when you enlarge
each side length by multiplying it by 3?
Solution
ii
FI
N
AL
PA
a i
b If each side length is multiplied by 3, the
perimeter is multiplied by 3 and the area
is multiplied by 9.
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273
ConnECT, appLY anD BUILD
Cut out a picture from a magazine. Trace it on to the 1-centimetre grid paper
in BLM 14 (available to download from the Interactive Textbook). Enlarge the
picture by tracing it onto 2-centimetre grid paper.
N
AL
1
Whole class
PA
11F
G
ES
Enlargements can be created using grid paper of two different sizes. on the left is a
picture of a cat traced onto 1-centimetre grid paper. on the right, using 2-centimetre
grid paper, is the enlargement. Focusing on one square at a time, we copy the drawing
into the corresponding square on the larger grid. It is helpful to number the squares so
that you know which you are copying from and to.
Individual
FI
11F
1
Enlarge each shape onto grid paper by multiplying each side length by the given
number.
a
274
×2
b ×4
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c
×3
d ×2
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These shapes have been drawn on centimetre grid paper. Calculate the area (A)
and perimeter (P) for each after it is enlarged by multiplying each side length by
the given number.
1
×6
b ×3
11G
c
×2
d ×5
ES
a
Review questions
Use a ruler and a protractor to draw:
G
2
PA
a a triangle with 1 side of length 5 cm
b a scalene triangle with 1 side equal to 5 cm
c a right-angled triangle that is not isosceles
d a triangle with 1 angle equal to 30°
e a quadrilateral with no right angles
N
AL
f a rectangle with 1 side equal to 6 cm
g a rhombus with 1 angle equal to 45°
2
Copy these shapes and draw in their lines of symmetry.
b
c
FI
a
3
You will need 2-centimetre grid paper. Enlarge the
cat’s face onto the larger grid paper by numbering
and copying each square.
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