Math Smart - 7

nevision
© Oscar pays $13.56 for 4 reams of white paper.

O Mrs Lee made a jug of orange juice by mixing a) Find the cost of 8 reams of paper.
1 glass of orange squash and 3 glasses of b) Find the cost of 2 reams of paper.
water. How many glasses of orange squash and
A ream of paper Is a packet containing 500
glasses of water are needed to make 6 jugs of
identical sheets of paper.
orange juice?
Q Yim mixes 2 bowls of fish stock and 5 bowls
o These are the ingredients that are needed for
of water to make some soup. Fah wants to making enough shortcrust pastry for 12 pies.
make 3 times the amount of soup that Yim
shortcrwst pastry
makes. How many bowls of stock and bowls
A/lakes 12 pies:
of water does she need?
150 g flour
Q A machine produces yellow gloves and pink A pinch of salt
gloves. For every 2 yellow gloves it produces, 75 Q butter
5 pink gloves are produced. If the machine 45 mi water
produced 400 yellow gloves, how many pink
gloves did it produce? a) What Is the ratio of the amount of
flour needed to the amount of butter
O Write these ratios in their simplest form.
needed?
a) 8:48 b) How many grams of butter is needed to
b) 8; 14 make 24 pies?
c) 12:10 c) Write down the quantities of the

d) 21 :28 ingredients needed to make 6 pies.
e) 125:30 d) Write down the quantities of the
ingredients needed to make 30 pies.
f) 32:72

Simplify these ratios. (E) Iris is 8 years old. The ratio of Iris's age to
Lim's age is4 : 7.
a) 6 weeks to 3 days
a) HowoldisLim?
b) 400 ml to 6/
b) What will the ratio of Iris' age to Lim's
c) 150 g to 2.4 kg
age be next year?
o Divide the following quantities in the given
Challenge!
ratios.
In the figure on the right,
a) 280 biscuits in the ratio 1 : 3
the shaded square, C, is
b) 125 sweets in the ratio 4 : 1
the part where square A
c) $315 in the ratio 5 : 2 and square B overlap.
d) $1320 in the ratio 4 : 7 The unshaded part of
of
square A is |
unshaded
the
There are 27 610 people at a football match.
part of square B. The ratio of
The ratio of the number of males to the
the shaded area to the total unshaded area
number of females is 8 : 3.
is 1 : 50. Given that the shaded area is 4 cm^
How many males are there at the match? find the area of square B.

Ratio, rate and proportion

O The ratio of one quantity to another quantity tells us how much the first quantity Is as
compared to the second quantity.
No units are used when we write ratio.

Example
The ratio of 4 kg to 1 kg is written as 4 : 1, without any units.
O A ratio can be expressed In Its simplest form, like fractions.
O Equivalent ratios are found by multiplying or dividing each part of a ratio by the same number.
O Quantities can be expressed as a proportion of a whole. Proportion is a comparison of the
quantity of a part to the quantity of a whole.

Example
1 in every 4 pens is a red pen.
The proportion of red pens is T of the total number of pens.

If
O two quantities are in the same ratio, they are said to be in direct proportion.
The two quantities increase or decrease at the same rate.

10 machines ►90tovcars~^ [?^ ► 45
10 —^ 90
? machines -►45 toy cars [I] = i^lO = 5
90











Ratios used in building Mortar acts like a glue for

Builders use ratios when mixing cement, sand and aggregate. Different ratios holding building materials
are used to make mortar for bricklaying, for making concrete floors and for such as brick or stone
making bridges. together. It is made up of
a thick mixture of water.
Example
• 2 times as much sand as cement makes a good mortar for bricklaying.
• 4 times as much aggregate as cement makes a strong concrete to use
as a floor.
• For maximum strength, the volume of the water in the mix should be
4 times the mass of the cement.

Batching by High strength cement Coarse sand Stone Approximate yield

Bucket 1 bucket 2 2 buckets 2 2 buckets 4 buckets

Wheelbarrow 2 bags (1 bag = 50 kg) 2 2 wheelbarrows 2 2 wheelbarrows 0.26 m

Per m 7.7 bags (1 bag = 50 kg) 0.63 m 0.63 m


UNIT 13 Ratio, Rate and Proportion

UNIT 14




Geometrical





Constructions





^ You will learn about:
thiS^ Drawing perpendicular and parallel lines

init j Measuring and drawing lines

Constructing triangles, squares, rectangles and polygons



The Interlace, in Singapore, was designed by Ole Scheeren to house 1040
apartments. Interlace consists of six-storey blocks staggered in different
angles, so that they do not form a straight line. Why do you think it was
designed this way? It looks like 31 cuboids stacked upon one another,
resembling Jenga blocks. The blocks are stacked up to four, near the centre,
to have a maximum of 24 floors. If you look from the top, the amount of
surface area for greenery increased by 30% from the original area of the
land the building is on. The Interlace won the World Building of the Year for
2015 having challenged the traditional architectural design mindset. Can you
identify parallel lines and perpendicular lines? Can you think of some clever
uses of geometrical shapes and lines employed in this design?

In this chapter
CHAPTER 14.1
Pupils should be able to:
• use the notation and
labelling conventions Drawing and Measuring
for points, lines, angles
and shapes
Perpendicular and
• use a ruler, set square
and protractor to draw
perpendicular lines and
parallel lines
• use a ruler, set square Parallel Lines
and protractor to
measure and draw
straight lines to the
^ RECALL
nearest millilitre

O Look at the figure below and name the pairs of perpendicular and
parallel lines.

a) is perpendicular to
Why do swimmers compete
b) is perpendicular to
along parallel lines?
c) is parallel to I I
Hint: Measure the length
of the red and blue lines in
the diagram below.
Q Using a ruler, draw
a) a line that is 17 mm long,
b) a line that is 3 cm 5 mm long and
c) a line that is 29 mm long.




Construction in geometry means to draw lines, angles and shapes accurately. Here
are some ways and words that we will be using in a geometrical drawing.

A convention is a standard way of doing something.

Point . .. Angle Shape

A A B A
B
B


r
Points are labelled This is line D (
using a capital segment AB.
► Figure 14.1 Labelling letter. This is BAC, or This is quadrilateral
conventions for points, This is point A. ^BAC ABCD.
lines, angles and shapes







UNIT 14 I Geometrical Constructions

Perpendicular lines and Prepare the following
materials for geometrical
parallel lines construction.
• sharp pencil
Perpendicular lines are straight lines that meet at a right angle. They are said to • eraser
be perpendicular to each other. The symbol 1 means 'is perpendicular to'. • ruler
• set square

Parallel lines are always the same distance apart. They will never meet no matter • protractor
how long they are drawn. They are said to be parallel to each other. The symbol // • compass
means 'is parallel to'. Arrow heads are used to show parallel lines.
Parallel lines Perpendicular lines








tL



The square indicates that line segment Try and Apply
The arrow heads indicate that line
AB is perpendicular to line segment CD.
segment AB is parallel to line segment CD.
We write this as AB 1 CD. 0 The diagram shows
We write this as AB H CD.
2 triangles. Name
s Figure 14.2 Notation for perpendicular lines and parallel lines 2 lines which are
perpendicular to AC.
Drawing perpendicular lines

In Stage 5, you learnt how to measure and draw right angles using a protractor.
Let us recall and learn to draw perpendicular lines using a protractor or a
set square.

Example 1

We can use a
Draw Z.xyz = 90°. Name a pair of perpendicular lines folded piece
of paper to
Solution check.
Place a protractor on line segment XY n
1 such that the centre of the protractor
is on point K Q Name all the pairs of
parallel lines in the
Adjust the position of the protractor
figure below.
such that its baseline is on line
segment XY.
A f
/ /
Mark a dot at the 90°-mark.
f
/
/ /
Remove the protractor and join the B C D
% dot to point Y. Label the point Z.

Example 2

O Use a ruler and a set square to construct AB1 CD.

■D L



.D C- D C- i:
0 I) •D C-
6 B
Use the pencil and Mark point B on line Place the set square Label the new line Draw a square to
the ruler to draw segment CD. on line segment CD segment AB. show that AB 1 CD.
line segment CD. such that the right
angle of the set /LABC and /.ABD
are right angles.
square is at point 6.
Draw a line segment
from point B.

Q Use a ruler and a protractor to construct AB 1 CD.




' > '
\
A \
■D C- ■D C- ■D C- r ■D
I • r • • 14 B
i . I . I . I . I B
Use the pencil and Mark point B on line Place the protractor Use the pencil and Draw a square to
the ruler to draw segment CD. on line segment CD the ruler to draw a show that 1 CD.
line segment CD. such that the centre line segment from B /LABC and /-ABD
of the protractor is to the dot. are right angles.
at B. Label the new line
Make a dot at the segment y4fi.
90°-mark.
Drawing parallel lines

You have learnt about parallel lines in Stage 5. In this section, you will learn to
construct parallel lines using a ruler and a set square.


Example 3

Construct a line MN parallel to line PQ such that they are 3 cm apart.

P
Q
Solution
Place a set square on
line PQ and a ruler
at the shorter side of M- N
the set square.
3 cm




Add in the arrow heads to
Slide the set square show that line MN is parallel
3 cm along the ruler. to line PQ.
E UNIT 14 Geometrical Constructions

Construct a line CD through point O, parallel to jlnevAS.







Solution

















Place a set square on line AB
and a ruler at the shorter side
of the set square. Slide the set square along the Add in the arrow heads to show
ruler. that line CD is parallel to line
Use the edge of the set square AB.
to draw the line CD through
point O.
Check My
Understanding
m
Q Construct a line PQ parallel to line AB such that they are 4 cm apart. >4-

Q Construct a line XZ through point Y,
parallel to line CD.









Q In the figure, M/V//OP and/WN1 Q/?.
a) On the figure, use the correct notation to show the parallel line segments.
b) On the figure, use the correct notation to show the perpendicular line segments.
c) Use a ruler. Measure the distance between MN and OP.
What does this tell you about the relationship between line MN and line OP?
d) Use a protractor to confirm that line OP is perpendicular to line QR.
Q Construct each of the following lines on a piece of blank paper.

a) EFIGH
b) JK1 LM such that WneJK is 60 mm long and line LM is 55 mm long.
c) UV/f WX
d) MNH OP such that line MN and line OP are each 50 mm long, and line MA/is 30 mm from line OP.

In this chapter

Pupils should be able to: CHAPTER 14.2
• use a ruler, set square
and protractor to
construct squares and Construct! hg Polygons
rectangles
• use a ruler, set square
and protractor to
construct a triangle 14.2.1 Constructing squares and
given two sides and the
included angle (SAS) rectangles
• use a ruler, set square
and protractor to
construct a triangle Constructing squares
given two angles and
You can use a ruler, a set square and a protractor to construct a square.
the included side (ASA)
Remember that a construction is an accurate drawing.
• use a ruler, set square
and protractor to
construct regular Example 1
polygons, given a side
and the internal angle
Construct square ABCD that has side lengths of 70 mm.
Solution
Draw a sketch to help you.
Use your ruler. Draw a line segment that is 70 mm long.
D
J L 1 This is one side of the square.
70 mm
"1 r 1 impiii iin|U(i
70 mm
si 6




RECALL Place the centre of your protractor
at one end of the line segment.
Use a protractor to draw
the following angles. Measure 90° and make
a construction mark.
a) 32° b) 90°
c) 141° d) 206°





Place the centre of your
5 protractor at the other
end of the line segment.

Measure 90° and make
a construction mark.










UNIT 14 I Geometrical Constructions

Use the construction marks to draw "3 Think and Share
4 another two sides of the square.
Each side should be 70 mm long.
How many squares and
E
rectangles can you see?
Hint: A square or rectangle
can be made up of more
than one block.




1- 8
E O
L...
Use your ruler to draw the last (III1
5" side of the square by joining
the ends of the other two
sides of the square you have
just drawn.
The last side of the square Squares and rectangles '
should be 70 mm long. are quadrilaterals.
They have four
straight sides.
Squares and rectangles
have four right angles.
Label the square ^SCD.
J L Opposite sides are
G
Write down the measurements parallel.
as shown. All sides in a square
are equal in length.
Draw squares to show that the 70 mm
The opposite sides of a
angles of the square are right
rectangle are equal in
angles.
length.


n r
70 mm
Check My Think and Share
Understanding
Bees can construct regular
hexagons without any
Q Construct a square with side lengths of 4.7 cm using a protractor. mathematical instruments.
Q Construct a square KLMN with sides that are 3.5 cm long. See if you can draw a
regular hexagon without
^ Use your set square to construct a square with side lengths of 5.4 cm.
using a protractor or ruler.

Constructing rectangles

You can use a ruler, a set square and a protractor to construct a rectangle.
Remember that a construction is an accurate drawing.


Example 2

Draw a sketch to help you. Draw a rectangle ABCD with length 9 cm and width 5 cm using a pencil, ruler
and protractor.
D C
J L Solution
5 cm
n r 1 5 cm
B
9 cm 9 cm
Draw a line AB that is 9 cm
long. This is the length of Draw a right angle by placing the centre
the rectangle. of the protractor at point B. Use your
ruler to measure 5 cm away from point
B. Make a mark and label it C. Line BC is
the breadth of the rectangle.

9 cm
C
+
J L
5 cm 5 cm 5 cm 5 cm

n r
9 cm 9 cm
Draw a line AD that is 5 cm long,
Join point D to point C. Label
parallel to line SC.
the sides and the corners of the
rectangle.
Example 3

Draw a sketch to help you. n
Construct rectangle PQRS that has side lengths of 40 mm and 60 mm.
Solution
j L
40 mm
Use your ruler to draw a line segment that
n r
60 mm Is 60 mm long. This is one side of the
rectangle.


Place the centre of your protractor
2 at one end of the line segment.


Measure 90° and make
a construction mark.



Place the centre of your protractor
? at the other end of the line segment

Measure 90° and make
a construction mark.



UNIT 14 I Geometrical Constructions

Use your ruler to draw the last side of
Use the construction marks to draw
4 another two sides of the rectangle. the rectangle by joining the ends of
the other two sides of the rectangle
Each side should be 40 mm long.
you have just drawn. The last side of
the rectangle should be 60 mm long.












Label the square PQRS. J L
Write down the measurements as shown.
Draw squares to show that the angles 40 mm
of the rectangle are right angles.
c ~l r
60 mm
Example 4


Construct a rectangle ABCD such that yAfi = 6 cm and BC-3 cm.
Draw a sketch to help you.
Solution
Draw a line y4S = 6 cm. ^ J L
6 cm
3 cm
n r
Using a protractor, draw 6 cm
7- a line perpendicular to
line AB from point A and
from point B.
6 cm


Widen the arms of
5 a compass to 3 cm and ^ arc 3 cm from A and point B. t.
tighten the hinge to Label the points where the
3 cm
lock the arms. arcs cut the perpendicular
lines D and C respectively.
n r
Draw the line DC to complete 6 cm
the rectangle ABCD. Draw
squares to show that the
angles of the rectangle are
right angles.
Check My
Understanding


O Construct a rectangle with side lengths of 3.6 cm and 4.5 cm using a protractor.
Q Use a set square to construct a rectangle with side lengths of 11 cm and 5.2 cm.
0 Construct a rectangle ABCD such that >46 = 8 cm and AD = 6 cm. Draw the diagonal AC
and measure its length.
Q*Challenge! Construct a rectangle whose diagonals are 5 cm long.

14.2.2 Constructing triangles


You have learnt the properties of the different types of triangles. Let us recall
some properties of the triangles to help us construct triangles. You will need a
ruler, a protractor and a compass.
Grouping triangles according to sides
Triangle A has 3 equal sides.
We name such a triangle an equilateral triangle.

Triangle D has 2 equal sides.
Triangle E also has 2 equal sides.
We name such triangles isosceles triangles.

Triangle B does not have any equal sides.
Triangles C and F also do not have any equal sides.
We name such triangles scalene triangles.



Grouping triangles according to angles
The angles of triangle A are all acute angles.
The angles of triangle D are also all acute angles.
We name such triangles acute-angled triangles.


Triangle C has an obtuse angle.
Triangle E also has one obtuse angle.
We name such triangles obtuse-angled triangles.

Triangle B has a right angle.
Triangle F also has a right angle.
We name such triangles right-angled triangles.


Constructing right-angled triangles


Example 5
Draw a sketch to help you.
Construct a triangle ABC such that AB = 5 cm, AC = 4 cm and ZCAS = 90°.

Solution
Draw a line AS = 5 cm. Using a compass, draw an
1 2. arc 4 cm from A.

5 cm











5 cm

UNIT 14 Geometrical Constructions

Using a protractor, draw a line that is Draw the line CB to complete
5 perpendicular to line AB from point A. 4" triangle ABC. Draw a square to
Label the point at which the line cuts show that aCAB is a right angle.
the arc C.



4 cm







5 cm
5 cm
Constructing isosceles triangles






Construct a triangle ABC such that AS = 5 cm, and LABC = /.CAB = 40".
Draw a sketch to help you.
Solution C
A B
Draw a line AB = S cm. 5 cm
1

5 cm
Using a protractor, construct /. = 40".














Using the protractor, construct A = 40°. Label the point at which the two lines meet C.










5 cm
5 cm
Observe that we can construct a triangle
• if we are given the lengths of two sides and the if we are given the sizes of two angles and
size of the angle between the two given sides. the length of the side between the two given
This is known as side-angle-side or SAS. angles. This is known as angle-side-angle or ASA.

side



side

Constructing scalene triangles


Example 7


Use a protractor and a ruler to make
an accurate drawing of triangle
ABC given the sketch shown.
50 mm







70 mm

Use your ruler to measure c
1 and draw a line segment iiii|fin im|(i({ imp iiii|iii[ mi|iiif (iii|i[i[ ini{iii{ m
70 mm long. OcmI 2 3 4I 5 el 7
Label this line segment SC.



Position your protractor
2- at point B and measure
an angle of 60°.
Draw a construction
mark at 60°.



Use your ruler to draw a
5 construction line from point B,
through the 60° construction mark.
Measure BA = 50 mm.
Label the end of the line
segments.








Draw line segment AC.
Label the triangle with the
given information.
50 mm







70 mm





UNIT 14 Geometrical Constructions

Example 8


Use your protractor and a ruler to draw triangle DEF, given that D = 25°,
f=35° and Df=6.5cm.



Draw a rough sketch of the
information given.
6.5 cm


Use your ruler to measure and
niipin na[riimm|Tnnmi[ini im|iiii [in|iiii inipiir m
^ draw line DF = 6.5 cm.
Ocul 2 3 4 5 6
Label the line segment DF.



Position the centre of your protractor
^ at point D and measure an angle of 25°.

Draw a construction mark at 25°.






Use your ruler to draw a construction
line from point D, through the
25° construction mark.






Position the centre of your protractor
^ at point F and measure an angle of 35'


Draw a construction mark at 35°.





Use your ruler to draw a construction
^ line from point F, through the
35° construction mark.





Draw and label the completed triangle.
7 Do not remove the construction lines.



6.5 cm

Check My
Understanding


Q Construct these triangles using a ruler and a protractor

a) ^ b)

38 mm 66 mm


60 mm 4 cm
44 mm


a) Make an accurate drawing of this triangle.
b) Measure the length line AC.

c) Measure ZfiyAC. 6 cm



8 cm

0 Construct a triangle PQR such that PC? = 5.5 cm, QR = 3.5 cm and LPQR = 90".
0 Construct a triangleXyZsuch that >7= 7 cm. yx = 4.5 cm and 7Xyz = 90°.
0 Construct an equilateral triangle y4BC with sides 4.5 cm.

O Construct an equilateral triangle JKL with sides 7 cm.
O Construct a triangle PQR such that 0/? = 9 cm and /LPQR = APRQ = 50".
Q Construct a triangle JKL such that J/C = 7 cm and AUK = ALKJ = 70°.
0*Challengel Make an accurate drawing of this triangle.











6.5 cm



o Journal Writing



MNO is a triangle. MN = 42 mm, NO = 6.4 cm and AMNO = 60°.
a) Construct triangle MNO.
Measure the length of side MO.
Measure the size of the other angles.
Based on your measurements in part (c), do you think you have
constructed the triangles accurately? Give a reason for your answer,
e) What type of triangle is triangle MNO? Give a reason for your answer.




Geometrical Constructions
n

14.2.3 Constructing regular polygons


Constructing quadrilaterals

Refer to pages 156-157 of Unit 6 to recall the properties of quadrilaterals to help
us construct polygons.


Example 9 Always draw a sketch of |
the geometric figure that |

Construct a parallelogram ABCD such that>4B = 7 cm, SC = 6 cm and you are constructing to |
AABC = 40^ help you. \
Solution

Draw a line AB-1 cm. A
7 cm
1

Using a protractor, draw /LA = 140" and ^B = 40°.











7 cm

Using a compass, draw two arcs 6 cm from points B and A.
^ Label the intersection points C and D. Draw the line CD to complete the

parallelogram.













7 cm

Add 2 sets of arrow heads in the drawing to show that line AB is parallel to
line DC, and line AD is parallel to line BC.

Example 10


Construct a trapezium PQRS such that PQ = 9 cm, ^QPS = 60°, /LPQR = 70° and the
height of the trapezium is 5 cm.

Solution

Draw a line PO = 9 cm. H-
^ P 9 cm


Using a protractor, draw /.P = 60° and ZQ = 70°.












9 cm

Using a ruler and set square, draw a line parallel to line PQ. Label the
5 intersection points R and 5.














square






















Draw the line RS to complete
^ the trapezium. Add in 2 arrow
heads to show that line
PQ is parallel to line RS.






9 cm

UNIT 14 Geometrical Constructions

Example 11


Construct a rhombus VJXYZ with diagonals l/l/V = 8 cm and XZ = 6 cm.

Solution
Draw a line WY = 8 cm. Label the points LVand Y.



8 cm
W

Using the compass, draw two arcs 4 cm (half the length of WY) from point
7- W, one above and the other below the line.
















8 cm
w













Without changing the position of the ends of the compass, draw two arcs

y from point Y. Draw a construction line through the two intersection points.
Mark the point where the two perpendicular lines meet, O.
77











n
w 0 Y

Use the compass to draw 2 arcs 3 cm away from point 0 on the vertical
line, one above and one below line VJY. This is half the length of diagonal
XZ of the rhombus.
7|\


~N



Notice that we are given
3 cm
the diagonal of the
rhombus to be 6 cm. We
need to half that for
length from centre to IV
vertex.
3 cm











Label the points of intersection X and Z. Join the points of intersection to
5* points \N and /to complete the rhombus.




























Check My
Understanding


Q Construct a parallelogram /ABCD such that AB = 6 cm, BC = 5 cm and ^ABC = 50*.
Q Construct a trapezium PQRS such that PQ = 7 cm, AQPS = 55°, /.PC?/? = 60° and the
height of the trapezium is 3 cm.
Q Construct a rhombus WXYZ with diagonals WY = 9 cm and XZ = 5 cm.






UNIT 14 Geometrical Constructions

Constructing regular polygons


Investigate!



Let us investigate the size of the internal angles of a pentagon,

a) Measure the sizes of and Zd in the regular pentagon. What is the sum
of the interior angles of the pentagon?
















/.a =
^c = zlc/ = zle =

Conclusion
In a regular polygon, all sides are of length and all angles are
of size.

b) We can also find the sum of the internal angles by dividing the polygon
into triangles.

3 X 180° = 540°

sum sum

sum



Conclusion
Each internal angle = (180° x ) 5




We can make use of the size of each internal angle of regular polygons when we
construct regular polygons.


Example 12


Construct a pentagon with sides of length 40 mm and internal angles of 108°.
Use your pencil and ruler to draw a line 40 mm long.



40 mm

Use your protractor to measure
an angle of 108°. Draw
construction mark at 108°.




Use your construction mark. Place the centre of your protractor
The size of the internal
4 on the second construction mark
angle of a regular Measure and draw a 40-mm
pentagon will always be line that passes through the Measure an angle of 108° and make
108° regardless of the first construction mark. Make a a third construction mark at 108°.
length of the sides. second construction mark.














40 mm









Repeat Steps 3 and 4 until you
have completed the 5
angles and 5 sides of
the pentagon.



Check My
Understanding


O Use a protractor to construct
a) an equilateral triangle with sides 42 mm and internal angles of 60°.
b) a regular hexagon with sides 5 cm and internal angles of 120°.
c) a regular octagon with sides 55 mm and Internal angles of 135°.
d) a regular pentagon with sides 62 mm and internal angles of 108°.
0 Construct a parallelogram ABCD such that AB = 8.5 cm, BC = 6.5 cm and AABC = 65'
Q Construct a rhombus WXYZ with diagonals WY = 9 cm and XZ=1 cm.

Q Draw a regular polygon with side lengths of 48 mm and internal angles of 120°.
What type of regular polygon have you drawn?
A Draw a regular hexagon with side lengths of 58 mm.








UNIT 14 Geometrical Constructions

fievision


Construct these parallel and perpendicular lines.
3) 5 cm 3 cm





4 cm 5 cm
5 cm
6.8 cm
n
4.3 cm
5.3 cm 3.6 cm



Q Construct a line RT through point S, parallel to line XV.


5*





Q Make an accurate drawing of the following triangles,
a) ^ b)



10 cm
5.4 cm






4 cm
O On the figure, use a coloured pen to
outline the parallelogram.
a) Label the parallelogram PQRS.
b) Use your ruler to measure the
lengths of line segments:
i) AB
ii) GH

iii) EF










Q Construct triangie PQR given that Q is a right angle, zi/? = 35° and QR = 72 mm.
@ Construct triangle DEF given that zlD = 38°, /.£= 47° and DE = 6.3 cm.

Q HIJK is a square with side lengths of 8 cm.
IJL is a right-angled isosceles triangle.
Make an accurate drawing of the diagram. 8 cm..
Hint: You need to calculate the missing
angles in triangle IJL first.


@*Challenge!
Construct a trapezium PQRS such that PQ = 9 cm, ^QPS = 48°, /LPQR = 64° and the height of the
trapezium is 5 cm.

^*Challenge!
Construct a trapezium KLMN, with a line of symmetry, such that KL = 9 cm, NM = 7 cm and with
a height of 5 cm.

(Ii)*Challenge!
Paul wants to build a regular pentagonal pyramid like the one shown below.













7 cm
Help Paul construct the net of a regular pentagonal pyramid.








Journal Writing

Q Construct a rectangle with sides 45 mm and 38 mm long respectively.

a) Draw the diagonals of the rectangle.
b) Measure the lengths of the diagonals.
c) Compare the lengths of the diagonals. Is this what you expected? Explain
Q Construct a square with sides 11 cm.
a) Draw the diagonals of the square.

b) Measure the lengths of the diagonals.
c) Compare the lengths of the diagonals. Is this what you expected? Explain












UNIT 14 Geometrical Constructions

Construction in geometry requires us to draw lines, angles and shapes accurately.
Useful tips for neat accurate geometrical constructions


Align the protractor baseline correctly.

so ^ 200
N ,V0 »» ,
Show all construction
lines clearly.






baseline centre baseline centre

Read all instruments from
Read instruments on the correct starting Hold all instruments firmly.
directly above.
scale and from the correct starting point.

outer scale
» M 100
oS> . \ \ \ 1 I / / / , ^

inner scale not ac accurate




To measure this angle, read from the Use a sharp pencil to draw light Draw single, smooth lines.
inner scale. construction lines.
V.

Remember to measure accurately
rd y
to 1 mm and 1°.




Notation and labelling conventions


Point Line Angle Shape Parallel lines i Perpendicular lines





point A line segment
AB
tL

ABC, or
^ABC quadrilateral AB H CD ASICD
ABCD

Parallel, intersecting and perpendicular lines









parallel lines intersecting lines perpendicular lines

Constructing parallel lines













Constructing perpendicular lines


/. V
n,
■D C L D Cr ■D c-


Constructing squares or rectangles

V

3 cm

n
6 cm
6 cm
Constructing triangles

Given side-angle-side or SAS Given angle-side-angle or ASA.
















5 cm 8 A 5 cm
5 cm 5 cm








UNIT 14 Geometrical Constructions

Internal angles in regular polygons

If a shape Is regular, all of Its angles are of the same size.

~lo°
108° 108

O
0
90°^ 120° 120


Equilateral triangle Square Regular pentagon Regular hexagon Regular octagon

Constructing regular polygons







40 mm A .''//









Mathematics Connect
J


Many engineers are engaged in structural engineering: from aerospace engineers who design
satellite structures, to civil engineers who design bridges and flyovers, to mechanical engineers
who design vehicle chassis and the placement of components inside computers and cell phones.
Geometric shapes, such as hexagons and triangles, play a very important role in strengthening
structures that are used to support the entire construction.
Structural engineers make use of connector plates to help strengthen connecting points on bridges
and buildings. A connector plate is most commonly shaped as a triangle, square or even a parabola.

A motorcycle frame design consists of two triangles
to support the wheels and seats. Mechanical
engineers design cranes that use triangles and
squares in their frames. Even satellites use these m
familiar and basic regular geometries.
The image on the right shows a geodesic dome
called the Montreal Biosphere. The structure of
geodesic domes is similar to the structure of soccer
balls and appears to be made up of a group of
pentagons and hexagons. But, if we break each of
those shapes down, we can see that they are made
up of triangles.

Measuring Time,






Area, Perimeter






and Volume





You wili learn about:
The relationships between units of time
The use of 12-hour and 24-hour clock systems

Interpreting timetables
Calculating time intervals
Abbreviations for and relationships between units of measurement
Deriving and using the formula for the area and perimeter of a
rectangle
Calculating the perimeter and area of compound shapes made from
rectangles
Deriving and using the formula for the volume of a cuboid
Calculating the surface area of cubes and cuboids from their nets











I. »•







I!, "P




The Dubai Shopping Mall aquarium h 51 m long,
20 m wide and 11m high. How much space does
the aquarium occupy in the mall? How much
water is needed to fill it to the brim?


UNIT ; b I Measuring Time, Area, Perimeter and Volume

CHAPTER 15.1

In this chapter

Time • know the relationships
Pupils should be able to:
between units of time
15.1.1 Reading and writing time in • understand and use
12-hour and 24-hour
the 24-hour notation clock systems
• interpret timetables
Let us recall telling time using the 24-hour notation. We can draw a diagram to • calculate time intervals
show the time line in one day. There are 12 hours from midnight to noon, and
12 hours from noon to midnight. This makes up 24 hours in one day.

midnight
midnight noon
12 1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 10 11 12
—I 1 1 1 \ 1 1 1 1 i 1 1 1 1 1 1 1 1 1 i 1 -I-
00 00 01 00 02 00 03 00 04 00 05 00 0600 07 00 08 00 09 00 10 00 11 00 12 CO 13 00 14 00 15 00 1600 17 00 18 00 19 00 20 00 21 00 22 00 23 00 00 00


We can see that
• 9.00 a.m. is written as 09 00, and
• 5.00 p.m. is written as 17 00.


When we write time
using the 24-hour clock
Write the time using the 24-hour notation. notation, we do not
a) 3.15 a.m. b) 10.47 a.m. c) 1.30 p.m. d) 9.24 p.m. need to write 'a.m.' or
'p.m.'. Instead, the time
Solution
is written using four
a) We write 3.15 a.m. in the 24-hour notation as 03 15. digits. The first two digits
b) We write 10.47 a.m. in the 24-hour notation as 10 47. represent the hours
c) We write 1.30 p.m. in the 24-hour notation as 13 30. and the last two digits
represent the minutes.
d) We write 9.24 p.m. in the 24-hour notation as 21 24.


0000 03 15 1047 1200 1330 21 24 00 00


^ RECALL


O Mandy woke up later than usual at 6.15 for work this morning. She usually woke up 45 min earlier.
At what time did she usually wake up for work?
@ Anthony went on a vacation to Vietnam from 12 September to 21 September. How long did his
vacation last in weeks and days?
O Express each of the following times in 24-hour clock notation.
a) 4.08 a.m. b) 10.25 a.m. c) 11.12 p.m.
O Express each of the following times in 12-hour clock notation. Write the times using a.m. or p.m.
a) 00 45 b) 09 48 c) 16 39
Q Mr Sanchez took a flight from Singapore to London. The duration of the flight was 13 h 35 min.
If he arrived in London at 05 50 the next day, at what time did Mr Sanchez leave Singapore?
Express your answers in 12-hour clock notation.

15.1.2 Converting between units of time


You have learnt to tell time in hours (h) and minutes (min). The second(s) is a unit
used to tell very small amounts of time. How are these units related?

1 hour (h) = 60 minutes (min) 1 min = 60 seconds (s)


Example 2


Convert the following to minutes.
a)2h b) 4 h 20 min c) 11 h 45 min

Solution
a) 2 h = 2 X 60 b) 4 h = 4 X 60
= 120 min = 240 min
4 h 20 min = 240 + 20
= 260 min

(c) 11 h = 11 X 60
= 660 min
11 h 45 min = 660+ 45
= 705 min

Example 3


Convert the following to seconds,
a) 3 min b) 5 min 4 c) 20 min 10 s
s
Solution
a) 3 min = 3 x 60 b) 5 min = 5 x 60
= 3 X 6 X 10 = 300 min
= 18 X 10 5 min 4 s = 300 + 4
= 180 s = 304s
c) 20 min = 20 x 60
= 1200 s
20 min 10 s = 1200+ 10
= 1210s




Convert
a) 420 min to h, b) 3000 s to min,
c) 400 min to h and min, and d) 725 sto min and s.

Solution
When converting minutes
to hours, or seconds to a) 420 min = 420+ 60 b) 3000 s = 3000 + 60
minutes, we divide by 60. = 420 + 6 + 10 = 50 min
= 70 + 10
= 7h


Measuring Time, Area, Perimeter and Volume

400 = 360 + 40 d) 725 = 720 + 5
360 min = 360 ■¥ 60 720 s = 720 + 60
= 6h = 12 min
400 min = 6 h 40 mIn 725 s = 12 min 5 s






Convert There are 60 minutes in
an hour.
a) 140 min to h, and b) 960 s to min.
0.5 h means 'half an
Solution
hour'. So, 0.5 h = 30
a) 140 min = 140+ 60 b) 960 s = 960 + 60 minutes.

= 14 + 6 = 16 min
= 2.5h

Check My

Understanding


o Write the time using the 24-hour notation.
a) 1a.m. b) 7 a.m. c) 4 p.m. d) 10 p.m.

© Write the time using the 24-hour notation.
a) 2.45a.m. b) 11.23a.m. c) 2.05p.m. d)n.59p.m.

e Find the duration of time,
a) From 00 10 to 09 40 b) From 08 00 to 13 50
c) From 15 15 to 19 00 d) From 17 30 to 23 45

o Convert the following to minutes,
a) 6 h b) 3 h 15 min c) 9 h 20 min d) 600 s
© Convert the following to seconds.
a) 8 min b) 7 min 24 s c) 30 min 2 s
© Convert 240 min to h.

© Convert 350 min to h and min.
o Convert 520 s to min and s.
© Fill in the blanks.
a) seconds in 5 minutes b) minutes in 3 hours

c) minutes in half an hour d) hours in half a day
e) minutes in ^ of an hour f) hours in | of a day

g) seconds in ^ of a minute h) hours in 4 ^ days
i) hours in 2 days j) seconds in 7^ minutes
k) minutes in 4^ hours I) hours in 10 days

Think and Share


Write your answer in the thought bubble using the information provided.


The chicKen has
Recipe a mass oC 2.75 kg. Simply add
Cook the
We will need to 2.75 to 1130. The chicken will
chicken for
take the chicken be ready at 14 05.
20 min per
out of the oven at 500 g.
1310.

















At what time do you think the
chicken will be ready?







Think and Share 15.13 Calculate tinne intervals


The Shepherd Gate clock Working out a time interval means being given two times and calculating the
is an analogue 24-hour amount of time between the two.
M clock in London. How do
we read the time on this
clock?
O An airplane left the airport at 08 45 and arrived at its destination at 16 48 on
the same day. How long was the flight?

Solution
You may use a timeline to help you.
From 09 00 to 16 00
= 7h
From 08 45 to 09 00 From 16 00 to 16 48
= 15 min = 15 min


10 11 12 13 14 15 16
Total flight time is 15 min + 7 h + 48 min = 7 h and 63 min
= 8 h and 3 min

The flight was 8 h 3 min long.



UIMIT 15 Measuring Time, Area, Perimeter and Volume

@ A movie starts at 3.45 p.m. and ends at 5.30 p.m. How long was the movie?

Solution
Write both times in the 24-hournotation. Subtract the minutes.
1 Start time: 15 45 ^ 17 30
End time: 17 30 -1545



As 45 is greater than 30, you need to convert an hour into minutes.
5 Reduce the hours by 1 hour, and add 60 minutes to 30 minutes.

16 90

-15 45



Subtract the hours.
T 16 90


-1545
1 45

The movie was 1 h 45 min long.

Example 7

Find the time duration from
a) 10 00 to 14 20, b) 0045 to 11 15,

c) 12 25 to 17 10, and d) 11 45 to 22 05.
Solution
a) Draw a time line from 10 00 to 14 20.

14 20
10 00
10 00 14 00 14 20 4 20

The time duration from 10 00 to 14 20 is 4 h 20 min.
b) Draw a time line from 00 45 to 11 15.
0 75

00 45
10 30
00 45 10 45 11 15
The time duration from 00 45 to 11 15 is 10 h 30 min.
Draw a time line from 12 25 to 17 10.
6 70
I^Kf
12 25
4 45
12 25 16 25 17 10
The time duration from 12 25 to 17 10 is 4 h 45 min.

d) Draw a timeline from 11 45 to 22 05.
1 60
7 7

-1145
10 20
11 45 21 45 22 05

The time duration from 11 45 to 22 05 is 10 h 20 min.


Check My
Understanding


Calculate the time intervals.
a) 09 49 to 11 15

b) 6.52 a.m. to 00 00
c) 10.54 a.m. to 5.15 p.m.
o A train left station A at 08 49 and arrived at station B at 11 13. How long
was the train ride?
e The school day starts at 7.30 a.m. and ends at 3.40 p.m. How long is the
school day?
o The time in Singapore is 8 hours ahead of London.
When it is 10 15 in London, what is the time in Singapore?

e Jane takes 30 min to walk to work from home. She has to leave home
at 7.20 a.m. to get to work on time. It takes Jane 1 of that time If she
drives to work, if Jane drives to work, what is the latest time she can
leave home to arrive at work on time? Write your answer in the 24-hour
notation.




o Journal Writing


Lexy wants to calculate the duration
from 12.57 p.m. to 16.05 p.m. She
uses her calculator to subtract the
start time from the end time as
shown.

Lexy writes the answer as 3 h 48 min.
a) Explain why her calculation is
not correct,
b) Work out the correct answer.











UNIT 15 Measuring Time, Area, Perimeter and Volume
I

15.1.4 Timetables


A timetable is a table that shows the times at which certain events occur. Times
are usually shown using a 24-hour clock. Examples include school, train, bus and
flight timetables. Timetables are a tool for planning. Timetables often show
departure times, and sometimes arrival times.

Example 8


This is part of a timetable for the tram at the Singapore Zoo.
Each column shows the departure time of a tram.


Departure times
1 Location Tram 1 Tram 2 Tram 3j .The first tram
leaves the main
1 Main entrance 08 45"*" 09 15 09 45
entrance at
\ Leopard trail 09 25 09 55 10 25 ' 08 45.

1 East Lodge trail 10 15 10 45 11 15
1 Forest Giants trail 11 GO 11 30 12 00
1 Elephant and rhino sanctuary 11 55 12 25 12 55

1 Creatures of the night enclosure 12 35^ 1305 13 35 The first tram
arrives at
Creatures of the
O How long does it take for Tram 1 to travel from the main entrance to the night enclosure
Creatures of the night enclosure? at 12 35.
O Ari wants to be at the Elephant and rhino sanctuary by 1.15 p.m. What is the
latest tram he can catch from the main entrance?
O Beth gets off Tram 1 at the East Lodge trail because she wants to spend some
extra time there. What Is the maximum amount of time Beth can spend at the
East Lodge trail if she wants to arrive at the Creatures of the night enclosure
at 13 35?

Solution
O The tram leaves the main entrance at 08 45 and arrives at Creatures of the
night enclosure at 12 35.
From 09 00 to 12 00
= 3h From 12 00 to 12 35
From 08 45 to 09 00
= 35 min
= 15 min
H 1 ^ =1 1 ^
7 8 9 10 11 12 13 14
Time Interval = 15 min + 3 h + 35 min = 3 h 50 min
@ 1.15 pm—>-13:15
The 09 45 tram arrives at the Elephant and rhino sanctuary at 12 55. Tram 3 is
the latest tram that Ari can take and still arrive by 1.15 p.m.
@ Tram 3 leaves East Lodge trail at 11 15 and arrives at the Creatures of the
night enclosure at 13 35. Beth arrived at the East Lodge trail at 10 15. She can
spend 1 h at the East Lodge trail.

Check My
Understanding



O Use the train timetable to answer the questions.

Train station Departure times
Crossroads West 10 30 11 15 11 45
Airport International 10 45 11 30 12 00

Junction International 11 30 - 12 45
Bay Beach 11 50 1205 13 05

a) If I catch the 12 45 train from Junction International station, how long does it take to get to Bay
Beach station?
b) How long does the 10 30 train from Crossroads West station take to travel to Bay Beach station?

c) Which train is the fastest to travel from Crossroads West station to Bay Beach station?
d) How many minutes after the first train does the second train depart from Crossroads West
station?
e) If I catch the 11 45 train from Crossroads West station, how many stops are there until I reach Bay
Beach station?
f) How many trains depart from Airport International station between 10 30 and 12 30?

© Look at the flight departure and arrival times. Then, answer the questions.

O Arrivals 9 Departures
From Scheduled Arrtval Status Flight Scheduled Status

MU245 Shanghai FVG 19:00 19:24 F2441 [-11 Cochin COK 20:00 Scheduled
EK3771+11 Bangkok BKK 19:00 19:00 Scheduled SQ495 [.f2] Singapore SIN 20:00 Scheduled
EK236 (+1) Chicago ORO 19:10 19:12 Scheduled FZ49 [-1] Muscat MOT 20:00 Scheduled
London LHR 19:10 19:12 Scheduled Bahrain BAH 20:25 Scheduled
EK44 1+2) Frankfurt FRA 19:15 19:24 6E62 Mumbal BOM 20:25 Cancelled

EK100 (+2) Rome FCC 19:20 19:20 Scheduled SV561 (+1] Riyadh RUH 20:30 Scheduled
EK102 [+l| Milan MXP 19:20 19:23 EK859 (+2) Kuwait City KWI 20:30
FZ814 Apha ARB 19:20 Scheduled MU246 Shanghai PVG 20:30
EK226 U1 San Francisco SFQ 19:25 19:28 SV551 [*1; Jeddah JED 21:00 Scheduled


a) What is the scheduled time of departure for the flight to Singapore?
b) What is the scheduled time of arrival for the flight from London?
c) Is the Shanghai flight expected to arrive on time?
d) John has booked a flight to Mumbai. What information can you give him about his flight?
e) Claire arrived from Bangkok and needs to catch a connecting flight to Jeddah.
How long will she have to wait at the airport?
f) How much later does the flight from Milan arrive after the Shanghai flight?
g) Does the flight from London depart earlier or later than the flight to Rome?



Measuring Time, Area, Perimeter and Volume

CHAPTER 15.2 In this chapter

Pupils should be able to:
• derive and use formula
Area and Perimeter for the area and
perimeter of a rectangle
• calculate the perimeter
and area of compound
Investigate! shapes made from
rectangles
• know the abbreviations
Each square represents 1 cm^. for and relationships
between square metres
(m^), square centimetres
D
(cm^) and square
B
millilitres (mm^)

^ RECALL



O Find the areas of the
following figures.
a) A figure that is
Complete the table. Rertangle A has been done as an example for you. made up of five 1-m
squares.
Rectangle A B C D E F G b) A figure that is made
up of nine 1-cm
Number of 10
squares.
squares in each
row (length of O Mr Fong wants to fence
one side, /) up his square garden.
The length of each side
Number of 4
of his garden is 2 m.
squares in each
Find the length of
column (length
fencing that he needs.
of the other
side, 6) O Mrs Steward buys a
rectangular plot of land.
length x length 40 The plot of land is
Area (cm^) 40 15 m long and 9 m wide.
What is the area of the
2 X (length + 28 plot of land?
breadth)
O the perimeter
Perimeter (cm) 28 and area of the figure.
(All lines meet at right
Is the shape a rectangle
angles.)
rectangle or a 14 cm
1 square?
3 cm


Formula for the Formula for the
7 cm
area of rectangle: perimeter of rectangle:

4 cm 4 cm

15.2.1 Perimeter


You have learnt that a rectangle has two pairs of equal sides and that a square
has four equal sides. The longer side of a rectangle is called its length and the
shorter side is called its width (or breath). The side of a square Is called its length.


5 cm


2 cm 3 cm




Length of the rectangle = 5 cm
Width of the rectangle = 2 cm
Length of the square = 3 cm
The perimeter of a figure is the total length around the edges of the figure.





Find the perimeters of the rectangle and the square.


8 cm b)




5 cm
6 cm




Solution
a) Perimeter of the rectangle = 8 + 8 + 5 + 5
= 2 X (8 + 5) Perimeter = 2 x (Length + Breadth)
= 2 X 13
= 26 cm
b) Perimeter of the square = 6 + 6 + 6 + 6

= 4x6 Perimeter = Length x 4
= 24 cm

Perimeter of a rectangle
= length + breadth + length + breadth

b
The perimeter is the sum = I + b + I +
of the lengths. = (l + b) + {I+ b)
= 2 X (/ +h)
We use units such as mm, length (/)
cm or m, instead of mm^ = 2{l + b)
cm^ and m^.

breadth (b)


Measuring Time, Area, Perimeter and Volume

If the rectangle is a square, then both sides are the lengths.
The formula for the perimeter of a square then becomes as follows:
length (/)
Perimeter of a square = length + length + length + length
=/+/+/+/
length (/)
= 4 X /
= 4/

Check My

Understanding


O Calculate the perimeter and area of each rectangle.


12 cm
Perimeter =
5 cm
Area =



b) cm
3 m
Perimeter
Area =



23 mm

Perimeter
23 mm
Area -



120 mm
d)
Perimeter =
7 cm
Area =



4 cm

e)
Perimeters
cm
Area =

© Find the perimeters of the rectangle and the square,
a) b)
10m

4 m 15 cm



O Find the perimeters of the rectangle and the square.

a) .. b)
24 cm

14.5 m
15 cm








15.2.2 Area

You have learnt that the area of a figure is the number of square units contained
within the figure. Let us learn to use formulas to find the areas of some
quadrilaterals.

Example 2


Find the areas of the rectangle and the square.
The length and breadth
must be in the same unit a) 14^ b)
before we can calculate
the area or perimeter. 10cm





Solution
a) Area of the rectangle = 14x8 Area = Length x Breadth

= 112m^
b) Area of the square = 10 x 10 Area = Length x Length
=100cm^
The area is 2-dimensional
Area of a rectangle = length of one side x length of the other side
since it is the product of
2 sides — the length and = l X b
breadth.

We use units such as
mm^ cm^ and m^. The breadth {b) length (/)
superscript 2 indicates the
2 dimensions. length (/)


breadth {h)
^3^ UiXJIT 15 I Measuring Time, Area, Perimeter and Volume

If the rectangle is a square, then both sides are the lengths.
The formula for the area of a square then becomes as follows:
length (/)
Area of a square = length x length
= /x /

length (/)



Find the missing side In each rectangle.
? m
a) b)


Area = 84 cm 7 cm Area = 72 m 9 m




Solution
a) Area of rectangle = 84 cm^ b) Area of rectangle = 72 m^
We can use the formula to
Length of rectangle = ? cm Length of rectangle = 9 m find the missing side.

Width of rectangle = 7 cm Width of rectangle = ? m
? X 7 = 84 9 X ? =72
? = 84 -r 7 ? = 72 9
= 12 = 8
The length of the rectangle The length of the rectangle
is 12 cm. is 8 m.

Example 4



O Calculate the breadth of © Calculate the length of © Calculate the length of
the rectangle. the rectangle. the square.

9 cm


25 m"
18 cm 4 mm 32 mm'



Solution Solution Solution
A=lx b A =lx b A = / X /
^B = 9xb 32 = 1 X A 25 = /'
l = V2S
= 2 = 8 = 5

The breadth is 2 cm. The length is 8 mm. The length is 5 m.

Check My

Understanding

O find the areas of the rectangle and the square,

s) 20.5 cm

30 m
9 cm




@ Find the areas of the rectangle and the square,

a) . 12cm b)
45 m
14 cm




© Find the area of each shape given that 1 small square is 1 unit

a) b)






O Find the area of each shape.

18 m
a) b) 7 cm

7 cm
10m




© Find the missing side in each rectangle,

a) . b) ? m


Area = 102 cm 6 cm Area = 108 m 12 m




©a) Calculate the b) Calculate the c) Calculate the
length of the breadth of the length of the
rectangle. rectangle. square.

9 cm



5 m 45 m' 72 cm" 144 mm'



^ UNIT 15 Measuring Time, Area, Perimeter and Volume

O Draw:
a) three different rectangles, each with an area of 12 cm^. Try and Apply ►
Write down the lengths of the sides.
a) Calculate the area of each
b) four different rectangles, each with an area of 30 cm^.
field.
Write down the dimensions of each rectangle.
80 m
Calculate the perimeter of each rectangle. i)
72 cm
O A painting has a rectangular frame 50 mm 20 m
wide. The outer dimensions of the framed
painting are 42 cm by 72 cm.
Calculate 42 cm il) 40 m
4
a) the total area of the painting and frame.
b) the area of the part of the painting that
we can see.
c) the area of the frame.
b) Fencing is charged per
d) the perimeter of the outer edge of the frame.
metre. Which field
e) the perimeter of the inner edge of the frame.
is cheaper to fence?
Complete the following
Converting units in area sentences.
The rectangle and the
Sometimes we need to convert from one unit of measurement of area, square both have the same
to another. 1 m . However,
1 cm
the perimeter of the
1 mm
1 m rectangle is
1 cm
than the perimeter of
1 mm
the square. Therefore, it
Area = 1 mm Area = 1 cm' Area = 1 m'
is cheaper to fence the
Converting cm^ to mm 2 field.
10 mm
1 cm
When converting cm^ to
1 cm 10 mm
mm^ multiply by 100.
When converting mm^ to
1 cm X 1 cm = 1 cm^ 10 mm X 10 mm = 100 mm'
cm^ divide by 100.
So, 1 cm^ = 100 mm^
Since 100 mm^ = 1 cm^
1 mm^ = 1 -r 100
= 0.01 cm^

Converting m^ to cm^
100 cm When converting m^ to
1 m
cm^ multiply by 10 000.
When converting cm^ to
1 m 100 cm
m^ divide by 10 000.




1 m X 1 m = 1 m^ 100 cm x 100 cm = 10 000 cm^
So, 1m^= lOOOOcm^

Example 5


Kane calculates the area of an envelope using his ruler and got an answer of
150 cm^ But the post office wants the area in in mm^ Help Kane convert his
answer from cm^ to mm^.

Solution
10mm - 1cm
To convert cm to mm, we multiply by 10.
10 mm X 10 mm
To convert cm^ to mm^ we multiply by 10^ because there are 100 mm^ in 1 cm'
= 1 cm X 1 cm
So, to convert 150 cm^to mm^ multiply by 100.
So, 100 mm^ = 1 cm^.
150 X 100= 15 000 mm^





Find the area of the rectangle in cm^. 8 mm




3 mm
Solution
Method 1
8 mm = 0.8 cm 3 mm = 0.3 cm
Area = 0.8 x 0.3
= 0.24 cm^
10 mm^= 0.10 cm^ f
S^m^= 0.24 cm^ | Method 2
To convert 24 mm^ to cm^ : Area = 8 mm x 3 mm
divide by 100. | = 24 mm^


= 0.24 cm^





Find the area of the square in m' 5 cm





5 cm


Solution
Method 1
5 cm = 0.05 m
Area = 0.05 x 0.05 = 0.002 5 m^
fyry^
0.001 0 m
Method 2
25 cm2= 0.002 5 m
Area = 5 cm x 5 cm
To convert 25 cm^ to m^
= 25 cm^
divide by 10 000.
= 0.002 5 m^




Measuring Time, Area, Perimeter and Volume

Check My
Understanding



Express
a) 19 cm^ in mm^ b) 270 cm^ in mm^ c) 8600 cm^ in mm^
d) 46 m^ in cm^ e) 150 m^ in cm^, f) 2300 m^ in cm^
g) 8460 mm^ in cm^ h) 500 mm^ in cm^ i) 64 mm^ in cm^
j) 100 000 cm^ in m^ k) 350 000 cm^ in m^ 1) 7400 cm^ in m^
m) 910 mm^ in m^ n) 4881 mm^ in m^and o) 2 160 000 mm^ in m^,

© The cellphone screen measures 75 mm by 150 mm.
150 mm
a) Find the area in mm^.
b) Find the area in cm^. 75 mm
c) Find the perimeter in mm.
d) Find the perimeter in cm.

e A square has a length 13 m. Find its area in cm^.

O A rectangle has a length 18 cm and width 7 cm. Find its area in m^.
© The front cover of a picture book is a square with a length 25 cm. Find the area of the front cover.

O A rectangular field has an area of 7000 m^. It has a length of 100 m. Find Its width in cm.





Areas and perimeters of

composite figures



A composite figure or a compound shape is a shape that is made up of two or Think and Share
more basic shapes. We can break down the figure into smaller basic shapes to 9
find its total area or perimeter. This
This ccompound shape can
be divided into rectangles in
In Stage 7, you will calculate the area and perimeter of compound shapes made
different ways.
up of rectangles.





O Look at the composite figure given, 15 cm
a) Calculate its perimeter.
b) Calculate its area.
8 cm
12 cm
Think of at least one other
12 cm
way to divide the compound
shape into rectangles.

3 cm

Solution
a) Perimeter =15 + 12 + 3 + 4 + 12 + 8
= 54 cm
b) There are two methods to calculate the area of this compound shape.

Method 2
Method 1
Divide the compound shape into rectangles using Draw dotted lines to form a complete
dotted lines. There is more than one way to do this. rectangle.

15 cm 15 cm
< >■ < ►



cm 8 cm
12 cm 12 cm


12 cm 12 cm
4 cm
'■3
3 cm 3 cm


Calculate the area of each rectangle. Calculate the area of the rectangle formed.
Then find the sum of the areas. Subtract the area of the unshaded rectangle.
15 cm
15 cm



8 cm
12 cm 12 cm


12 cm


3 cm
12 cm
Area of rectangle A = 15 x 8
= 120 cm^ 4 cm
Area of rectangle B = 4 x 3

= 12 cm^ Area of rectangle A = 15 x 12
Area of composite figure = 120 + 12 =180cm^

=132cm^ Area of rectangle B = 4 x 12
= 48 cm^

Area of composite figure = 180 - 48
= 132 cm^








3^ UNIT 15 Measuring Time, Area, Perimeter and Volume

Check My
Understanding


O This composite figure is made up of a square and a rectangle.
10 cm
Find the area of the figure.

12 cm 12 cm
8 cm

O Calculate the area of this 4 cm
compound shape.

5 cm
Use the length on the other
8 cm
side to calculate this missing
side length like this:
8 cm - 5 cm = 3 cm

10 cm
O Find the length of the missing sides and then calculate the perimeter and area of each composite figure
a) b) c)
7 m
in rm C
M > ■4 ^

2 cm e
4 cm
7 cm 5 cm 5 cm 6 m 5 m
d


3 cm
4 m
4 cm
a = c = . e =

b = d = f =
Perimeter = Perimeter = Perimeter =

Area = Area = Area =

O Calculate the perimeter and area of these compound shapes.
a) b)
7 cm 8 cm
9 cm
2 cm
8 cm
11 cm 7 cm
4 cm 4 cm


13 cm 12 cm
10 cm


Perimeter = Perimeter = Perimeter =

Area = Area = Area =

d) e) 8 cm f) 14m
5 cm
—I—
4.5 m
4 cm
8 cm 7 cm 12 m 9m ■•3 m
4 cm


5 cm



Perimeter = Perimeter = Perimeter
Area = Area = Area =

O^Challenge!
Calculate the perimeter and area of the compound shape.


3 cm

3 cm

4 cm

9 cm
4 cm







9 cm



Try and Apply!
21 cm
A printing shop cuts a small piece of cardboard out from a big sheet of
cardboard as shown.
O What is the length and breadth of the small piece of cardboard? 50 mm
Q What is the perimeter of the small piece of cardboard?

O What is the area of the small piece of carboard? 28 cm

O What is the area of the cardboard that is wasted?
O If the shop cuts 500 big sheets of cardboard, what is the total area
of cardboard that is wasted?

O The 500 small pieces of cardboard are needed to cover a surface 60 mm
with an area of 16.7 m^. Are there enough small pieces to cover
the surface? Show all your calculations.






UNIT 15 I MeasuringTlme, Area, Perimeter and Volume

CHAPTER 15.3
In this chapter


Pupils should be able to:
Volume of Cubes and • derive and use the
formula for the volume
Cuboids • of a cuboid
calculate volumes of
cuboids

In Stages 3 and 5, you learnt about cuboids and cubes. Let us recall some
properties of these solids. A 2D shape has two dimensions — length and width.
A solid is a 3D object with three dimensions — length, width and height.


Think and

What is a cube? What is a cuboid? What are some similarities and differences
between a cube and a cuboid?


A cube is a type of cuboid.
cube
: cuboid







Volumes of cuboids



Cuboids and cubes are solids.
The volume of a solid is the amount of space it occupies. Two faces of a solid are
joined by a line called an
This is a 1-cm cube. edge.
Each edge has a length of 1 cm. Two faces of a solid meet
Volume of one 1-cm cube 1 crn at a point called a vertex.
= 1 cm X 1 cm X 1 cm
1 cm
= 1 cubic centimetre 1 cm
= 1 cm^
We write 1 cubic
centimetre as 1 cm^ and
read it as '1 cm cubed'.

1 cm
The volume of the solid
= 4 X 1
= 4 cm^ . I:
1 cm
So, the volume of the solid is 4 cm^.
r

The solids below are made up of 1-cm cubes. Find the volume of each solid.

a) b)
^ ^ ^ ^




Try and Apply!

How many unit cubes are
there in this cuboid? Solid A Solid B

Solution
a) Number of cubes in Solid A = 7
^ ^ ^ ^ ^
Volume of Solid A = 7 cm^
b) Number of cubes in Solid B = 9
Volume of Solid 8 = 9 cm^


Deriving the formula of the volume of a cuboid

We can use 1-cm cubes to build cuboids. Let us look at the example below.
Each cube is a 1-cm cube.




2 cm


1 cm
4 cm
Number of cubes in 1 layer = 4x3 Number of cubes in 2 layers =12x2 Volume of the cuboid = 24 cm^.
= 12 =24



ThTnOn^nare We can calculate the volume like this;
4 X 3 X 2 = 24 cubes in the cuboid.
If there are 5 layers of
12 cubes in each layer, Volume of cuboid = length x breadth x height
what is the volume of the
cuboid formed? What is
the relationship between The length (/) of a cuboid is its longest edge.
the number of cubes in
The breadth (/;) or width of a cuboid is the
1 layer and the number height
edge that shows how wide the base is.
of layers?
The height (//) of a cuboid is the edge that length breadth
shows how high the cuboid is







UNIT 15 Measuring Time, Area, Perimeter and Volume