3
CHAPTER
Area and Volume
Introduction
Man needs measurement for many
tasks. Early records indicate that
man used body parts—such as
his hand and forearm—and his
natural surroundings as measuring
instruments. Later, the ‘imperial
system’ was introduced whereby
the units of length were inches,
feet, yards and miles. Today we
use the metric system, which was
introduced in the 18th century.
The metric system uses the basic
unit of the metre. This is a very
convenient system because all the
units are based on multiples of 10
and, hence, it is easy for conversion
purposes. All of these were studied
in some detail in Year 7.
Today many people are
interested in do-it-yourself
projects. Many hardware stores
offer free classes for people to
learn how to tile or carpet their
floors or paint the walls of their
house, as well as many other
interesting tasks. You may wish
to help your dad complete such
a project. In this chapter, we
will learn such things as how to
find the area required to tile or
carpet a floor, or how to calculate
the amount of paint required to
paint the walls of a house, or the
amount of water required to fill a
swimming pool.
In order to do this, we will
need to understand a few
concepts such as:
• area—the amount of surface
covering a closed shape
• volume—the amount of
material of which a solid is
made
• capacity—the amount of space
inside a container that can be
filled with a solid, liquid or gas
• a solid, which occupies space,
has a definite shape and can be
seen and touched.
More information on these
concepts will be dealt with in
detail in this chapter.
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AREA AND VOLUME
1 Draw each shape and highlight the
WA R M
U P
4 Calculate the area of each shaded section.
perimeter.
a
a
b
£xÊV“
ÓÇÊV“
{nʓ“
b
c
d
ÎÈʓ“
£Óʓ“
£nʓ“
c
Óxʓ“
2 Calculate the perimeter of each diagram.
a
£Îʓ“
b
£xʓ“
4 cm
4.3 cm
8.5 cm
c
Ónʓ“
d
Îʓ
d
35 mm
£Ê“
£äʓ
£ÈÊV“
Ӱxʓ
42 mm
nʓ
3 Find the area of the following.
a
9 cm
23 mm
15 cm
48 mm
c
d
nʓ“ £Óʓ“
17 cm
e
ÎÊV“
5 Convert the following to the units given
b
ÓÎʓ“
f
£xʓ
nʓ
£äʓ
in brackets.
a 10 cm (mm)
c 1.8 km (m)
e 420 cm (m)
g 5 cm2 (mm2)
i 12 m2 (cm2)
k 5 ha (m2)
m 600 mm2 (cm2)
o 70 000 m2 (ha)
b 5.5 m (cm)
d 50 mm (cm)
f 2500 m (km)
h 2.7 cm2 (mm2)
j
11.8 m2 (cm2)
l 3 km2 (ha)
n 50 500 cm2 (m2)
p 750 ha (km2)
area and volume
3A Special types of quadrilaterals
A quadrilateral is a figure that has four sides. The lengths of the sides and the size of the
angles in a quadrilateral determine its name.
This flow chart describes the special types of quadrilaterals.
µÕ>`Àˆ>ÌiÀ>
Two opposite
sides are
equal.
ÌÀ>«iÈՓ
One pair of
opposite sides
are parallel
Two adjacent
sides are
equal.
ÀiVÌ>˜}i
All four sides
are equal.
õÕ>Ài
ŽˆÌi
All four sides
are equal and
parallel.
Two opposite
sides are equal
and parallel.
À…œ“LÕÃ
«>À>iœ}À>“
Features of these special quadrilaterals
Shape
Name
Properties
Rectangle
•
•
•
•
Two opposite sides are equal.
Diagonals are equal.
Diagonals bisect (cut equally) each other.
The angle at each vertex is 90°.
Square
•
•
•
•
•
All sides are equal.
Diagonals are equal.
Diagonals bisect each other at right angles (90°).
The angle at each vertex is 90°.
The diagonals bisect each angle at the vertex.
Parallelogram
•
•
•
•
•
Two opposite sides are equal and parallel.
Diagonals are not equal.
Diagonals bisect each other.
Opposite angles are equal.
Adjacent angles are supplementary (add up to 180°).
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Shape
Name
Properties
Rhombus
•
•
•
•
•
All four sides are equal and parallel.
Opposite angles are equal.
Diagonals bisect each other at right angles (90°).
Diagonals bisect each angle at the vertex.
Adjacent angles are supplementary.
Kite
•
•
•
•
•
Two pairs of adjacent (next to) sides are equal.
One pair of opposite angles are equal.
Diagonals bisect each other at right angles (90°).
One diagonal bisects each angle at the vertex.
It is symmetrical about one diagonal.
Trapezium
•
•
•
•
One pair of opposite sides are parallel.
Two non-parallel sides are not equal.
Diagonals are not equal.
Diagonals do not bisect each other.
Isosceles
trapezium
•
•
•
•
One pair of opposite sides are parallel.
Two non-parallel sides are equal.
Diagonals are not equal.
Diagonals do not bisect each other.
A diagonal is the line joining two opposite non-consecutive vertices of a polygon.
A polygon is a closed figure made up of many (‘poly’) straight lines. You will learn more
about this later.
Area conversions
In Year 7, we learnt how to convert units in area. Recall the following:
3 100
km2
3 10 000
ha
4 100
3 10 000
m2
4 10 000
3 100
cm2
4 10 000
mm2
4 100
Remember
To convert from a larger unit to a smaller unit, you multiply.
To convert from a smaller unit to a larger unit, you divide.
Areas of composite shapes
Composite shapes are made up of more than one regular shape.
To find the area of a composite shape, find the area of each regular shape and add or
subtract them.
I N V E S T I G A T I O N Area of special quadrilaterals
1 Find the area of each triangle.
a
12 cm
Hence:
Area of the parallelogram
5 ______ 3 ______ units 2
height
10 cm
Area of triangle 5 _____________ cm
b
2
8 cm
base
e The area of a parallelogram can
be also obtained by the following
method.
5 cm
Area of triangle 5 _____________ cm2
c
iˆ}…Ì
L>Ãi
8 cm
12 cm
Area of triangle 5 _____________ cm2
2 Find the area of the shaded region in
each parallelogram:
a
ÇÊV“
£{ÊV“
Area of triangle 5 _____________ cm2
b
width = height
length = base
Area of parallelogram
5 area of rectangle
5 length 3 width 5 base 3 height
4 Find the area of the shaded region.
10 cm
a
7 cm
ÇÊV“
12 cm
£{ÊV“
Area of triangle 5 _____________ cm2
c Put both shaded regions together.
The area of the parallelogram
5 ______ 1 ______ cm2
d This area can be also obtained as
follows.
Area of parallelogram
5 __
​ 12 ​(14 3 ____) 1 __
​ 12 ​(14 3 ____) cm2
5 (14 3 ____) cm2 (since __
​ 12 ​1 __
​ 12 ​5 1)
5 ________ cm2
(14 cm is the base and 7 cm is the
height of the parallelogram.)
Area of triangle 5 _____________ cm2
b
10 cm
7 cm
12 cm
Area of triangle 5 _____________ cm2
Investigation continued next page
area and volume
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Investigation continued
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c Put both shaded regions together.
Area of the trapezium
5 _______ 1 _______ cm2
d Area of the trapezium can be also
obtained as follows:
Area of the trapezium
5
​ __12 ​3 12 3 ____ 1 __
​ 12 ​3 ____ 3 7 cm2
5 __
​ 12 ​3 7(____ 1 10) cm2
(12 cm and 10 cm are the parallel
sides and 7 cm is the height of the
trapezium.)
Hence:
Area of trapezium
5 __
​ 12 ​3 height 3 (_____ 1 _____)
a
(12 cm and 6 cm are the diagonals of
the rhombus.)
Hence:
Area of rhombus
5 __
​ 12 ​3 _____ 3 _____ units2
{ÊV“
£{ÊV“
h
Area of triangle 5 _____________ cm2
b
£{ÊV“
{ÊV“
b
Where a and b are the parallel sides
and h is the height (perpendicular
distance between them).
5 Find the area of the shaded region.
a
ÎÊV“
ÊV“
£Ó
Area of triangle 5 _____________ cm2
b
“
ÓÊV
£
ÎÊV“
Area of triangle 5 _____________ cm2
c Put both shaded regions together.
Area of the rhombus
5 _______ 1 _______ cm2
d Area of the rhombus can be also
obtained as follows.
Area of the rhombus
5 ​ __12 ​3 12 3 ____ 1 __
​ 12 ​3 _____ 3 3 cm2
5 __
​ 12 ​3 12(____ 1 ____) cm2
5 __
​ 1 ​3 ___ 3 ___ cm2
2
`Ó
8 Find the area of the shaded region.
a
5 __
​ 12 ​3 product of ________
Where d1 and d2
`£
are the diagonals
of the rhombus.
Area of triangle 5 _____________ cm2
Put both shaded regions together.
c Area of the kite
5 _______ 1 _______ cm2
d Area of the kite can be also obtained
as follows:
Area of the kite
5 ​ __12 ​3 14 3 ____ 1 __
​ 12 ​3 _____ 3 4 cm2
5 __
​ 12 ​3 14 (____ 1 ____) cm2
5 __
​ 12 ​3 ___ 3 ___ cm2
(14 cm and 8 cm are the diagonals of
the kite.)
Hence:
Area of kite
5 __
​ 12 ​3 _____ 3 _____units2
5 __
​ 12 ​3 product of ________
Where D1 and D2 are the diagonals
of the kite.
£ÊV“
ÓÊV“
area and volume
From the investigation, the area of each special type of quadrilateral can be summarised
as given in the table.
Name
1
Shape
Formula for area
Rectangle
Area of rectangle 5 l 3 w
where l is the length and w is the
width of the rectangle.
w
l
2
Square
Area of square 5 l 2
where l is the side of square.
l
l
3
Parallelogram
h
b
4
Rhombus
5
Kite
6
Trapezium
`£
£
Ó
a
h
b
`Ó
Area of parallelogram 5 b 3 h
where b is the base and h is the height
of the parallelogram.
Area of rhombus 5 __
​ 12 ​3 d1 3 d2
where d1 and d2 are the diagonals of
the rhombus.
Area of kite 5 ​ __12 ​3 D1 3 D2
where D1 and D2 are the diagonals of
the kite.
Area of trapezium 5 __
​ 12 ​3 h(a 1 b)
where a and b are the parallel sides
and h is the height of the trapezium.
Other types of quadrilaterals that you should know exist
are given here.
A concave quadrilateral is one in which one or more
diagonals are outside the quadrilateral.
A convex quadrilateral is one in which all of the diagonals are
inside the quadrilateral. All the special types illustrated above
are convex.
A cyclic quadrilateral has all the vertices on the circumference
of a circle.
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E X AM P L E S
Exa mple A1
Convert the following.
a 0.576 m2 to cm2
t o
b 600 000 m2 to ha
N e ed
k no w
Larger 3 smaller
Smaller 4 larger
$
$
c 350 000 cm2 to m2
Need
to do
★
Use the units of conversion.
a 0.576 m2 5 0.576 3 10 000 cm2
5 5760 cm2
2
b 600 000 m 5 600 000 4 10 000 ha
5 60 ha
2
c 350 000 cm 5 350 000 4 10 000 m2
5 35 m2
Exa mple A2
Find the area of:
a a rectangle with length 12.5 cm and width 5.6 cm
b a square with sides 4.8 cm
c a parallelogram with base 11 cm and height 7 cm.
t o
N e ed
k no w
a Area of rectangle
5l3w
b Area of square
5l3l
c Area of
parallelogram
5b3h
Need
to do
Use the formulae.
a Area of rectangle 5 l 3 w (length 3 width)
5 12.5 3 5.6 cm2 (Use your calculator.)
5 70 cm2
b Area of square 5 l 3 l (side 3 side)
5 4.8 3 4.8 cm2 (Use your calculator.)
5 23.04 cm2
c Area of parallelogram 5 b 3 h (base 3 height)
5 11 3 7 cm2
5 77 cm2
★
area and volume
Examples continued next page
E X AM P L E S
Exa mple A3
a Find the area of a rhombus with diagonals d1 5 7 cm and d2 5 13 cm.
b Find the area of a kite with diagonals D1 5 11 cm and D2 5 6 cm.
Need
to do
N e ed
t o k no w
• Area of rhombus
5 _​ 12 ​3 product of
diagonals
• Area of kite
5 _​ 12 ​3 product of
diagonals
a Area of rhombus5 _​ 12 ​3 product of diagonals
5 _​ 12 ​3 d1 3 d2
5 _​ 12 ​3 7 3 13 cm2
5 45.5 cm2
1
_
b Area of kite5 ​ 2 ​3 product of diagonals
5 _​ 12 ​3 D1 3 D2
5 _​ 12 ​3 11 3 6 cm2
5 33 cm2
Exa mple A4
8.5 cm
Find the area of a trapezium with parallel sides 11 cm and
8.5 cm, and with a perpendicular distance between them
(height) of 6.3 cm.
t o
6.3 cm
11 cm
Need
to do
N e ed
k no w
Area of trapezium
5 ​ _12 ​3 h(a 1 b),
where a and b are the
parallel sides and h is
the height.
★
Use the formula.
Area of trapezium5 _​ 12 ​3 h(a 1 b)
5 _​ 12 ​3 6.3(11 1 8.5) cm2
5 _​ 12 ​3 6.3 3 19.5 cm2 (Use your calculator.)
5 61.425 cm2
★
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Examples continued
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Exa mple A5
The area of a parallelogram is 77.5 cm2. If the base is 5.5 cm, find its height.
N e ed
t o k no w
Area of parallelogram
5 b 3 h,
where b 5 base and
h 5 height.
Hence:
h 5 __
​ A ​ b
Need
to do
★
Use the formula.
Height of parallelogram5 area 4 base
5 ____
​ 77.5 ​ (Use your calculator.)
5.5
5 14.09 cm (correct to 1 decimal place)
Exa mple A6
9 mm
Find the area of this shape. All measurements are in millimetres.
7 mm
9 mm
13 mm
Need
to do
N e ed
t o k no w
a Area of trapezium
5 _​ 12 ​3 h(a 1 b)
b Area of parallelogram
5b3h
★
Use the formulae and add the areas because the shape is made up of a
trapezium and a parallelogram.
a Area of trapezium5 _​ 12 ​3 h(a 1 b)
5 ​ _12 ​3 7 (9113) mm2
5 ​ _12 ​3 7 3 22 mm2
5 77 mm2
b Area of parallelogram5 base 3 height
5 13 3 9 mm2
5 117 mm2
Hence, area of shape5 (77 1 117) mm2
5 194 mm2
area and volume
Exa mple A7
£xʓ
A backyard is in the shape of a trapezium. There
is a square flowerbed and a triangular vegetable
garden. Aaron wants to sow some lawn seeds
around the garden. Find the area of the lawn.
(All measurements are in metres.)
t o
Ç°Îʓ
ΰxʓ
xʓ
£äʓ
Need
to do
N e ed
k no w
• Area of square
5l3l
• Area of triangle
5 _​ 12 ​3 b 3 h
• Area of trapezium
5 _​ 12 ​3 h(a 1 b)
★
Use the formulae to find the areas of the square and the triangle, and
add them. Then find the area of the trapezium and subtract the areas
of the flowerbed and the garden to give the area of the lawn.
Area of square5 l 3 l
5 3.5 3 3.5 m2 (Use your calculator.)
5 12.25 m2
Area of triangle5 _​ 12 ​3 b 3 h
5 _​ 12 ​3 5 3 3.5 m2
58.75 m2
 Area of square and triangle 5
(12.25 1 8.75) m2
5 21 m2
1
Area of trapezium5 _​ 2 ​3 h(a 1 b)
5 _​ 12 ​3 7.3(15 1 10) m2
5 _​ 12 ​3 7.3 3 25 m2
5 91.25 m2
 Area of lawn5 (91.25 2 21) m2
5 70.25 m2
Exercise 3A
1 Convert the following to the units given in brackets.
a 6500 cm2 (m2)
b 5 ha (m2)
d 3 m (mm )
e 13.7 m (cm )
f 0.0048 m2 (cm2)
g 30 cm2 (mm2)
h 3 m2 (cm2)
i
0.09 cm (mm )
m 0.0006 km2 (ha)
p 80 000 ha (km2)
k 50 000 m (ha)
2
j
2
2
c 0.0037 ha (m2)
2
2
2
2
n 2000 ha (km2)
0.007 m2 (mm2)
l 3 km2 (ha)
o 8 000 000 m2 (km2)
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2 Giving your answer correct to 2 decimal places, find the area of each of the following.
a base 5 7 mm, height 5 5.5 mm
b base 5 9 mm, height 5 7.5 mm
h
L
b
c l 5 11.7 cm, w 5 6 cm
d l 5 13.5 mm, w 5 5.7 mm
w
w
l
l
e side 5 7.3 cm
f side 5 6.7 mm
g b 5 11 mm, h 5 12 mm
h b 5 11.2 mm, h 5 0.73 cm
L
h
b
i
b 5 7.7 cm, h 5 3.3cm
j
d1 5 10 mm, d2 5 0.9 cm
d1
d2
L
k d1 5 7.8 m, d2 5 320 cm
l
d1 5 10 mm, d2 5 1.2 cm d1
d2
d1
d2
area and volume
m d1 5 11.3 mm, d2 5 7.9 mm
n d1 5 14 cm, d2 5 16 m
d2
d1
d1
d2
o d1 5 21 cm, d2 5 17 cm
p l 5 12.3 mm, w 5 11.7 mm
d1
w
d2
l
3 Find the area of each trapezium. Give your answers to two decimal places.
a
b
6 cm
5 cm
4.3 cm
7 cm
9 cm
13 cm
3.2 cm
11 cm
d
c
7.8 cm
e
4 cm
f
£x°Óʓ“
11.3 cm
6 cm
Ç°{ʓ“
3.4 cm
Èʓ“
10.3 cm
11 cm
4 For each of the following, give your answer to the nearest integer.
a Find b if the area is 4.5 m2 b Find h if the area is 9.6 m2
and the height is 150 cm.
£xäÊV“
and the base is 320 cm.
L
c Find a if the area is 680 cm2, the d Find the long diagonal if the area is
height is 17 cm and the base is 46 cm.
380 mm2 and the short diagonal is 19 mm.
>
`Ó
e Find h if area 5 560 mm2, f Find h if area 5 720 cm2 and base 5 32 cm.
a 5 30 mm and b 5 40 mm.
L
>
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5 Find the area of composite shape. Give your answer correct to 2 decimal places.
a Îʓ
b
c
Ӱ{ʓ
ΰÈÊV“
x°xÊV“
nʓ
™°Îʓ
£È°xʓ
Ç°Óʓ ££°xʓ
x°Èʓ
d {ʓ
e
Îʓ
f
£{ÊV“
£{ʓ
£Ç°xʓ
ÈÊV“
nÊV“
x°Îʓ
£ÓÊV“
£äʓ
6 Giving your answer correct to 2 decimal places, find the area of:
i the shaded section
ii the unshaded section.
a
b
£{ʓ
Óʓ
£Î°Óʓ
Óʓ
£Çʓ
£x°Çʓ
c
d
Óä°{ʓ
Óxʓ
£xʓ
ÓÓ°Èʓ
ÓÓʓ
£Óʓ
Ónʓ
7 Paving tiles measuring 375 mm by 375 mm are used to pave the area shown below.
a Find the area to be paved.
£{°xʓ
b Find the area of one tile.
c How many tiles are required to pave this area?
d Find the total cost of paving the area if the tiles
cost $32.95 per square metre, other materials such
as sand and crushed rock cost $268 with 10% GST
and labour costs $30 per square metre.
{ʓ
Çʓ
Ȱxʓ
area and volume
8 For her new home, Jacqueline selects ceramic floor tiles measuring 450 mm by 450 mm.
a How many tiles are required to tile three rooms with floors measuring 6 m by 4.5 m?
b If the tiles chosen cost $24.50 per square metre plus 10% GST and labour costs $25
per square metre, find the total cost of laying the tiles.
9 A rectangular lawn is surrounded by a concrete path, which is 1.5 m wide. The lawn
measures 22 m by 28 m.
a Draw a diagram and find the area of the lawn.
b Find the area of the concrete path.
10 A path that is 0.8 m wide is built around two flowerbeds,
as shown in the figure. The two flowerbeds are identical in
shape and size. Find:
a the width of a flowerbed
b the length of a flowerbed
c the area of a flowerbed
d the area of the path.
£xʓ
£™Ê“
3B Circles
A circle is a shape enclosed by a curved line. This curved line is its boundary.
In Year 7, we learnt that the boundary of a closed shape is called its perimeter and the
perimeter of a circle is called the circumference.
We also have learnt the formula for calculating the circumference of a circle.
Circumference of a circle 5 2pr
or
Circumference of a circle 5 pD, since D 5 2r
It would be interesting to know the different features of a circle.
Features of a circle
No.
Name of feature
Diagram
Definition
1
Centre
A point in the middle of the circle
equidistant from all points on the
circumference.
The angle at the centre of the circle is
360°.
2
Circumference
The boundary line enclosing the circle.
The perimeter of the circle.
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No.
Name of feature
Diagram
Definition
3
Radius
The line joining the centre of the circle
to any point on the circumference of
the circle.
4
Diameter
A line joining two points on the
circumference and passing through the
centre.
5
Chord
A line joining two points on the
circumference but not passing through
the centre.
6
Tangent
A line touching the circle at one point
and forming an angle of 90° with the
radius.
7
Major arc
Minor arc
Each is a fraction of the whole
circumference.
8
Major sector ¢
Minor sector
A sector is a fraction of the surface of
the whole circle. It is bounded by two
radii and an arc. The two radii make an
angle at the centre of the circle.
If θ is the angle at the centre, then a
fraction of the circle 5 ____
​ θ ​.
360
9
Semicircle
Half a circle bounded by the diameter
and an arc.
10
Quadrant
One quarter of a circle, bounded by
two radii and an arc. It is a sector with
the angle at the centre at 90°.
11
Concentric circles
Two or more circles having the same
centre.
area and volume
No.
Name of feature
12
Major segment ¢
(larger section)
Diagram
Definition
A segment is part of a circle bounded
by a chord and an arc.
Minor segment
(smaller section)
The length of an arc is a fraction of the whole circumference.
Length of arc 5 ____
​  ​ 3 2 3 p 3 r
360
Area of circles
An area is the amount of surface that covers a closed shape. To find the area of a circle,
complete the following investigation.
I N V E S T I G A T I O N Area of a sector
What you will need: cardboard, a pair of
compasses, a pencil and a ruler.
What to do
1 Draw a circle of radius 6 cm on a piece of
cardboard.
2 Cut this circle into eight sectors, as you
would do for a pizza.
3 Cut the last sector into half again.
2
1
5
6
3
4
b
a
7
4 Arrange the seven sectors alternately, as
shown in the diagram.
a
2
1
4
3
b
6
5
7
5 Attach the halves of the last sector to
either end to form a rectangle. Note that
this rectangle is made up of the whole
surface of the circle.
In general, if the radius of the circles is r cm,
then:
• length of the rectangle 5 half of the
_________ of the circle 5 p___
• width of the rectangle 5 _______ of the
circle 5 _____
• area of the rectangle 5 length 3 width
• area of the circle 5 ____ 3 ____
5 ________ (units)2
From the investigation, the area of the
circle 5 pr2 (units)2.
The area of a sector is a fraction of the
area of the whole circle. Hence:
Area of the sector 5 ____
​  ​ 3 p 3 r2
360
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Construction of 90°
By following these instructions, a right angle (a 90° angle) can be constructed using a ruler
and a pair of compasses.
Step 1
Step 2
Draw a line AB that measures 4 cm.
A
B
With A as centre and the radius a little
more than half of AB, draw a semicircle to
cut AB at C and AB extended at D.
A
D
Step 3
C
B
With C as centre and the radius a little
more than AC, draw an arc above the
semicircle.
A
D
C
With D as centre and using the same
radius, cut this arc at E.
B
Step 4
Draw a line joining A to E, using a ruler.
Angle EAB is 90°.
E
Step 5
A
D
C
B
Constructing a tangent to a circle
We know that a tangent touches a circle at one point and the radius makes a right angle with
the tangent at that point.
Draw a circle of radius 3 cm. Draw the radius
OA of the circle.
Step 2
O
A
Step 1
With A as centre and radius a little more than
half of OA, draw a semicircle to cut OA at B
and OA extended at C.
C
A
B
O
area and volume
With B as centre and radius a little more than
AB, draw an arc above the semicircle.
Step 3
C
With C as centre and using the same radius,
cut this arc at D.
Step 4
Step 5
A
B
O
D
C
Draw a line joining A and D. Extend the line.
Angle DAO is 90° and the line AD is the
tangent at the point A.
A
B
O
D
C
A
B
O
E X AM P L E S
Exa mple B1
a Find the circumference of a circle with radius 2.5 cm.
b Find the circumference of a circle with diameter 7.3 cm Give your answer correct to 2 decimal
places. (Use p 53.14.)
Circumference of a
circle 5 2pr or pD
(since D 5 2r).
Need
to do
Use the formulae.
a Circumference 5 2pr
5 2 3 3.14 3 2.5 cm2
5 15.7 cm2
b Circumference 5 pD
5 3.14 3 7.3 cm2
5 22.92 cm2
★
Examples continued next page
N e ed
t o k no w
93
94
Examples continued
oxford mathematics for victoria
Exa mple B2
a Find the radius of a circle with circumference 154 cm2.
b Find the diameter of a circle with circumference 125 mm2.
Give your answers to 2 decimal places. (Use p 5 3.14.)
t o
N ee d
k n o w
• C 5 2pr,
hence r 5 ___
​ C ​ 2p
C ​
• C 5 pD, hence D 5 __
​ p
Need
to do
★
Use the formulae.
C ​ a Radius5 ​ ____
(2p)
154
5 ____________
​ ​
(2 3 3.14) cm
5 24.52 cm
C ​
b Diameter5 __
​ p
5 ________
​ 125 ​ 3.14 mm
5 31.81 mm
Exa mple B3
a Find the area of the following circles with:
i radius 5 3.6 cm
ii diameter 5 9.8 cm
b Find the radius of a circle if the area is 167.33 cm2.
Need
to do
N ee d
t o k n o w
a i Area of circle 5 pr2
ii Radius 5 __
​ D ​ __
2
A ​ ​ b Radius of circle 5 ​ __
​ p
√ Use the formulae.
a i Area5 pr2
5 3.14 3 3.6 3 3.6 cm2
5 40.69 cm2
ii Radius 5 ___
​ 9.8 ​5
4.9 cm
2
Area5 3.14 3 4.9 3 4.9 cm2
5 75.39 cm2
__
A ​ ​ b Radius5 ​ __
​ p
√______
√ 5 ​ ______
​ 167.33 ​ ​ 5 7.3 cm
3.14
★
area and volume
Exa mple B4
a
For the following sectors, determine:
i the fraction of the circle
ii the length of the arc
iii the area of the sector.
À>`ˆÕÃÊrÊxÊV“
{än
270°
3 cm
Need
to do
N ee d
t o k n o w
★
Use the formulae.
a i Fraction of circle5 ____
​  ​ 360
40
____
5 ​ ​ 5 __
​ 1 ​
360 9
ii Length of arc5 ____
​  ​ 3 2 3 p 3 r
360
40
____
5 ​ ​ 3 2 3 3.14 3 5 cm
360
5 3.49 cm
iii Area of sector5 ____
​  ​ 3 p 3 r2
360
40
____
5 ​ ​ 3 3.14 3 5 3 5 cm2
360
5 8.72 cm2
b i Fraction of circle5 ____
​  ​ 360
3 ​ ____
5 ​ 270 ​5 ​ __
360 4
3 ​3 2 3 3.14 3 3 5 14.13 cm
ii Length of arc 5 ​ __
4
iii Area of sector 5 __
​ 3 ​ 3 3.14 3 32 5 21.20 cm2
4
Examples continued next page
i Fraction of a circle
5 ____
​  ​ .
360
ii The length of an arc is
a fraction of the whole
circumference.
iii The area of a sector is
a fraction of the area of
the whole circle.
b
95
96
Examples continued
oxford mathematics for victoria
Exa mple B5
For the following shapes, find:
ii the perimeter
ii the area.
a
b
72°
7 cm
9 cm
15 cm
N e ed
t o k no w
• Fraction of a circle
5 ____
​  ​ 360
• Length of arc of
semicircle is
C 5 2pr 4 2 5 pr
• Area of semicircle
2
5 ___
​ pr ​ 2
Need
to do
★
a i Find the length of the arc, and add the one radius to the other
three sides of the square.
Length of arc5 ____
​  ​3
23p3r
360
5 ____
​ 72 ​ 3 2 3 3.14 3 7 cm
360
5 8.792 cm
Perimeter of shape5 8.792 1 7 1 7 1 7 1 7 cm
5 36.792 cm ≈ 36.79 cm
ii Find the area of the sector and the area of the square and add
them.
​  ​ 3 p 3 r2 1 l2
Area of shape5 ____
360
5 ​ ____
​ 72 ​ 3 3.14 3 72 1 72 ​cm2
360
5 30.772 1 49 5 79.772 ≈ 79.77 cm2
b i Find the length of the arc of the semicircle and add it to the
length of the three sides of the rectangle.
Radius of semicircle 5 ___
​ 15 ​ 5 7.5 cm
2
Length of arc of semicircle5 pr
5 3.14 3 7.5 cm
5 23.55 cm
Perimeter of shape 5 23.55 1 9 1 15 1 9 5 56.55 cm
ii Find the area of semicircle and area of rectangle and add them.
3.14 3 7.52
Area of shape5 ​ __________
1 9 3 15
​ 2
5 88.3125 1 135 5 223.125
 223.13 cm2
( )
area and volume
Exa mple B6
Anneka jogs around the inner side of a running track,
whereas Calvin jogs around the outer side of the
Óäʓ
track.
ÓÓʓ
a What distance does Anneka jog?
b What distance does Calvin jog?
c What is the difference between the distances along the two edges of the track?
d Find the area of the track (ie the white area).
t o
Need
to do
N e ed
k no w
The circumference of
inner circle C 5 pD.
★
Find the circumference of inner circle and add this result to the inner
straight edges. Do the same for the outer circle and straight edges.
Circumference of inner circle C1 5 pD
5 3.14 3 15 m
5 47.1 m
Length of inner edge of track 5 47.1 1 22 1 22 m
a The distance Anneka jogs 5 91.1 m.
Circumference of outer circle, C2 5 3.14 3 20 m
5 62.8 m
Length of outer edge of track 5 62.8 1 22 1 22 m
b The distance Calvin jogs 5106.8 m.
c The difference in the distance along the two edges
5 (106.8 2 91.1) m
5 15.7 m
d To find the area of the track, we need to find the area of the inner
shape and outer shape and subtract the first from the second.
Area of inner shape 5
(3.14 3 7.52 122 3 15) m2
5 176.625 1 330 ≈ 506.63 m2
Area of outer shape 5 (3.14 3 102 122 3 20) m2
5 314 1 440 5 754 m2
 Area of track (white section) 5 754 2 506.63 5 247.37 m2
Exercise 3B
1 Name the marked features of these circles.
a
£xʓ
b
c
97
98
oxford mathematics for victoria
d
e
f
g
h
i
j
k
l
2 For the circles illustrated, calculate i the circumference, ii the area. Give your answers
correct to 2 decimal places. (Use p 5 3.14.)
a
b
3.2 cm
7.2 m
8 cm
d
c
e
Ó£°ÎÊV“
f
6.7 m
18.3 cm
3 Using the circumference given, determine:
i the diameter ii the radius.
Give your answers correct to 2 decimal places.
a 15 cm
b 75 cm
d 11.4 cm
e 13.7 cm
c 39 cm
f 275.7 cm
4 For each of the following sectors, determine:
i what fraction of the circle it represents
ii the length of arc (correct to 2 decimal points where necessary)
iii the perimeter of the shape (correct to 2 decimal places where necessary)
iv the area of the shape (correct to 2 decimal points where necessary).
a
b
c
180°
3.7 cm
9.4 cm
144°
7.2 cm
area and volume
d
e
27°
f
315°
225°
8.45 mm
3.9 cm
11.6 cm
5 For the following areas of circles, determine:
i the radius
ii the diameter
Give your answers correct to 2 decimal places.
a 100 cm2
b 54 cm2
2
d 256.7 cm e 324 cm2
c 18 cm2
f 576 cm2
6 For the following composite shapes, calculate:
i the perimeter
ii the area
Give your answers correct to 2 decimal places:
a
b
c
8.5 cm
39
m
m
21.4 mm
9.2 cm
32.6 mm
12 cm
d
e
f
72°
13 m
7.7 cm
18.9 cm
g
5.3 cm
h
10
cm
72°
8 cm
6 cm
5.7 cm
7 Determine the shaded area in each of the following, correct to 2 decimal places.
a
b
c
4 cm
6m
8m
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oxford mathematics for victoria
d
e
f
7.4 cm
4.4 cm
7m
12.6 cm
8 A tyre completes 642 revolutions to travel 1 km.
a Calculate the distance travelled by the tyre in one revolution.
b Calculate the radius of the car’s wheel, correct to the nearest centimetre.
9 A rectangular yard has a swimming pool surrounded by paving.
ÓÓ°xʓ
*œœ
x°xʓ
£x°xʓ
££°xʓ
a Determine the area occupied by the swimming pool. Give your answer correct to the
nearest m2. (Use p 5 3.14.)
b Correct to the nearest m2, determine the area to be paved.
c If a paving tile measures 375 3 375 mm, find the number of tiles required to pave the
remaining yard.
d If each tile costs $25.50 and the labour cost of paving is $30 per square metre, find
the total cost of paving. Include 10% GST on both costs.)
10 Two hatboxes have the same base area. One has a rectangular base with a width of
11.5 cm, and the other has a circular base of radius 15.4 cm. Calculate the length of the
rectangular base.
11 Flowerbeds are designed inside a rectangular
garden, as shown in the figure. The rest of the
area is covered with lawn.
a Find the area of all the flowerbeds.
b Find the area of the lawn.
ΰÓʓ
££°Îʓ
{°Îʓ
£ä°xʓ
12 Semicircles are drawn on each side of rightÊV“
angled triangles, as given in the diagram.
Ó{ÊV“
a Calculate the area of each semicircle.
b In each case, find a relationship between
ÇÊV“
the large semicircle and the two smaller
semicircles.
c If quadrants are drawn on the sides, find the relationship
between the large quadrant and the two smaller quadrants.
25
c
m
Óx
100
7 cm
24 cm
area and volume
101
13 Construct a tangent at any point on a circle with:
a radius 5.5 cm
b radius 7 cm
c radius 6.3 cm
Homework
Sheet 3.1
3C Surface area of solids using nets
A solid is anything that takes up space. Dotted lines are used to give the impression that the
object is three-dimensional—that it has three dimensions: length, width and height.
Examples of solids are shown.
cube
cylinder
cone
The flat or curved section of a solid is called the surface or face.
A cube has six faces and each is in the shape of a square.
A cylinder has two flat surfaces in the shape of circles and one curved surface when
opened out has the shape of a rectangle.
A cone has one flat surface in the shape of a circle and one curved surface that when
opened out is in the shape of a sector of a circle.
We will learn more about these solids later.
Types of solids
Polyhedrons
Any solid that is made up of all flat surfaces is called a polyhedron (the plural form is
polyhedra).
Each flat surface is called a face and is in the shape of a polygon
A polygon is a closed shape made up of straight lines. Some examples are the triangle,
the square and the hexagon.
The corner point in a polyhedron is called the vertex.
i`}i
The intersection of two flat surfaces is called an edge.
In the diagram, ABEF is a face and A, B, C, etc are vertices (the
ÛiÀÌiÝ
plural form of vertex).
v>Vi
Length CG is an edge.
Platonic solids
There are five regular polyhedra that are called Platonic solids. They are named after the
Greek philosopher, Plato. These solids have been known for at least 3000 years.
In a regular polyhedron:
• all faces are congruent (the same size and shape)
• all edges have the same length
• all vertices have the same number of edges meeting at them
• all faces are regular polygons.
102
oxford mathematics for victoria
The five regular polyhedra are described in this chart.
Tetrahedron
• 4 triangular faces
• 3 triangular faces meeting at a vertex
Cube
• 6 square faces
• 3 square faces meeting at a vertex
Octahedron
• 8 triangular faces
• 4 triangular faces meeting at a vertex
Dodecahedron
• 12 pentagonal faces
• 3 pentagonal faces meeting at a
vertex
Icosahedron
• 20 triangular faces
• 5 triangular faces meeting at a vertex
Prisms
A prism is a solid having two congruent faces, being the rest of the faces are rectangles. The
congruent faces are the same shape as the cross-section. A cross-section is a ‘slice’ of the
solid and is cut parallel to its congruent face.
Both congruent faces are called the base of the prism. The base can be any polygon. The
name of the prism depends on the shape of its base.
If the rest of the faces are rectangles, then the prism is called a right prism.
Shown here are some prisms.
cylinder
rectangular prism
hexagonal prism
cube
triangular
prism
area and volume
Some examples of prisms are shown here.
cylinder
cuboid or rectangularbased prism
cross-section (slice)
cross-section (slice)
cube
cross-section (slice)
or
triangular-based prism
hexagonal-based prism
cross-section (slice)
cross-section (slice)
ˆÀÀi}Տ>À‡L>Ãi`Ê«ÀˆÃ“
VÀœÃÇÃiV̈œ˜Ê­ÃˆVi®
Pyramids
A pyramid is a solid with a base in the shape of a polygon; the rest of the faces are triangles
meeting at a point called the apex or vertex. The cross-sections of a pyramid are the same
shape as the base but are of different sizes. A tapered solid that has its apex above the centre
of the base is called a right pyramid or right cone.
A cone is a solid that has a circular base and a curved surface that tapers to a point called
the apex.
Vœ˜i
ÀiVÌ>˜}Տ>À‡L>Ãi`
«ÞÀ>“ˆ`
õÕ>Ài‡L>Ãi`Ê«ÞÀ>“ˆ`
VÀœÃÇÃiV̈œ˜Ê­ÃˆVi®
VÀœÃÇÃiV̈œ˜Ê­ÃˆVi®
VÀœÃÇÃiV̈œ˜Ê­ÃˆVi®
103
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oxford mathematics for victoria
The sphere and the hemisphere
Other solids are the sphere and the hemisphere.
A sphere is a solid that has one curved surface. It is like a ball. A cross-section passing
through the centre is a circle. It has many identical cross-sections that pass through its centre.
A hemisphere is half a sphere, much like half an orange. It has one flat surface in the shape of
a circle and one curved surface.
Solid
Sphere
Diagram
cross-section 1
cross-section 2
Hemisphere
Cross-sections through the solid
cross-section 1
cross-section 2
cross-section 1
cross-section 2
cross-section 1
cross-section 2
Surface area using the net of a solid
A net of a solid is a diagram showing all its faces. If a box in the shape of a cube is opened
out and laid flat, it will look like this diagram.
The diagram is the net of the cube. It shows six square faces.
The total surface area or, simply, the surface area, is the sum of the areas of the six
squares. If the side of the cube is 3 cm, then:
Surface area of cube 5
6 3 area of a square
5 6 3 32 5 54 cm2
To find the surface area of a solid using its net:
• draw the net of the solid
• find the area of each face
• add the areas of the faces.
area and volume
E X AM P L E S
a
Exa mple C1
b
For each solid:
ii draw the cross-section
ii identify the number of faces.
N e ed
k no w
• The cross-section is
a slice cut parallel to
the congruent base.
• The number of
faces is the number
of surfaces that can
be seen.
Need
to do
a i ii There are 2 pentagonal faces and 5 rectangular
faces: 7 faces.
ii There are 2 triangular faces and 3 rectangular b i faces: 5 faces.
48 cm2
Exa mple C2
Find the surface area of the rectangular prism, using a net.
t o
N e ed
k no w
Surface area is the sum
of the areas of the faces
that make up the net.
★
96 cm2
32 cm2
Need
to do
Draw the net.
Add the areas of all the faces.
★
32 cm2
96 cm2
48 cm2
96 cm2
48 cm2
32 cm2
The surface area of the rectangular prism 5 2(96) 1 2(48) 12(32) 5 192 1 96 1 64
5 352 cm2
Examples continued next page
t o
105
106
Examples continued
oxford mathematics for victoria
3 cm
Exa mple C3
Find the surface area of the rectangular prism, using its net.
8 cm
Need
to do
N e ed
t o k no w
The net has two square
faces that are the same
and four rectangular
faces that are the same.
• Area of square 5 l2
• Area of rectangle
5l3w
Draw the net.
Find the area of each face
and add them.
• Area of square bases
5 2(3)2 5 18 cm
• Area of rectangular
faces 5 4(8 3 3)
5 96 cm2
★
8 cm
3 cm
The surface area of the rectangular prism 5
2(3)2 1 4(8 3 3)
5 (18 1 96) cm2
5 114 cm2
Exa mple C4
xÊV“
Find the surface area of the triangular prism, using its net.
{ÊV“
£ÓÊV“
ÈÊV“
Need
to do
N e ed
t o k no w
The net of the
triangular prism is made
up of two triangles
that are the same and
two rectangles that are
the same. The third
rectangle is different.
Draw the net.
Find the areas of the
two triangles and
three rectangles and
add them.
★
{ÊV“
xÊV“
xÊV“
ÈÊV“
£ÓÊV“
£ÓÊV“
area and volume
Exa mple C4
Continued
Need
to do
( ★
)
• Area of two triangles 5 2​ _​ 12 ​3 6 3 4 ​5 24 cm2
• Area of two small rectangles 5 2(5 3 12) 5 120 cm2
• Area of larger rectangle 5 6 3 12 5 72 cm2
Surface area of the triangular prism 5
2​ _​ 12 ​3 6 3 4 ​1 2(5 3 12) 1(6 3 12)
5 (24 1 120 1 72) cm2
5 216 cm2
( )
Exa mple C5
9.54 cm
Find the total surface area of a rectangular pyramid.
8.74 cm
6.5 cm
9 cm
Need
to do
N e ed
t o k no w
The net of a rectangular
pyramid is made up of
four triangles and one
rectangle.
★
Draw the net.
Find the area of each face
and add them.
9.54 cm
• Area of two small triangles
5 2​ _​ 12 ​3 6.5 3 8.74 ​
( 8.74 cm
)
5 56.81 cm2
6.5 cm
• Area of two larger triangles
5 2​ _​ 12 ​3 9 3 9.54 ​
( 9 cm
)
5 85.86 cm2
• Area of rectangular base
5 9 3 6.5 5 58.5 cm2
Surface area of the rectangular pyramid
( )
( )
5 2​ _​ 12 ​3 6.5 3 8.74 ​1 2​ _​ 12 ​3 9 3 9.54 ​1 (9 3 6.5) cm2
5 (56.81 1 85.86 1 58.5) cm2
5 201.17 cm2
107
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oxford mathematics for victoria
Exercise 3C
1 For each solid, find the number of:
i faces
ii edges
iii vertices
a
b
c
2 From these diagrams, identify those that are:
i prisms
ii pyramids
a
b
c
d
e
f
g
h
i
j
k
l
m
n
o
p
q
r
s
t
3 Sketch a cross-section of each solid parallel to its base.
a
b
c
area and volume
d
e
g
h
f
4 For each net:
i calculate the surface area
ii draw the shape of the solid formed.
a
25 mm2
70 mm2 70 mm2
b
ÓÇ°xÊV“Ó
ÓÓΰ{ÊV“Ó
70 mm2
ÓÇ°xÊV“Ó
25 mm2
c
d
52.5
90.8 cm2
90.8 cm2
each 35.6 cm2
150.6 cm2
cm2
90.8 cm2
52.5 cm2
90.8 cm2
e
f
28 mm2
44
mm2
77
mm2
44
mm2
each is 19.3 m2
77
mm2
28 mm2
5 For each prism:
i draw the net
a
ii find the surface area.
36 cm2
b
top 108 cm2
side 70 cm2
36 cm2
front 80 cm2
109
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oxford mathematics for victoria
c
sloping face
350 cm2
bottom 170 cm2
d
È`iÊÓääʓ“Ó
Ϝ«ˆ˜}Êv>Vi
ÓÓ{ʓ“Ó
vÀœ˜ÌÊ
ÇÓʓ“Ó
LœÌ̜“Ê£Çxʓ“Ó
triangular base 54 cm2
e
f
49 cm2
Ϝ«ˆ˜}Êv>ViÊÎÓʓÓ
sides 112.7 cm2
È`iÊ{nʓÓ
L>ÃiʜvÊVœ“«œÃˆÌi
Å>«iÊʙnʓÓ
g
h
each face 76 cm2
LœÌ̜“ÊxÈʓÓ
45.8 mm2
150 mm2
92 cm2
6 For each prism:
i draw the net
ii find the surface area.
a
b
10 cm
9 cm
4 cm
11 cm
6 cm
12 cm
c
d
3 cm
4 mm
5.2 cm
8.3 mm
12.5 cm
6.2 mm
e
f
30 mm
1.7 m
40 mm
35 mm
153 m
72 mm
25 mm
g
170 m
25 m
h
26 cm
24 cm
24 m
32 m
7m
8 cm
10 cm
area and volume
i Ç°ÓÊV“
j
£Ó°Óʓ
n°Îʓ
™°ÇÊV“
£{°Îʓ
£Î°ÇÊV“
££°ÎÊV“
£Î°{ʓ
7 For each pyramid:
i draw the net
a
ii find the surface area.
b
£{ÊV“
£Ó°ÎÊV“
£ÓÊV“
nÊV“
£Î°xÊV“
Ç°ÓÊV“
c
d
Ç°nÊV“
£{°Îʓ
™ÊV“
Çʓ
>Ài>ʜvÊL>ÃiÊrÊÇn°ÈʓÓ
8 Find the surface areas of the solids formed by these nets.
a
b
nÊV“
£äÊV“
xÊV“
c
d
Çʓ
Èʓ
£{°xÊV“
ΰÇÊV“
111
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oxford mathematics for victoria
9 For each prism:
i find the area of the front face
ii find the total surface area.
a
b
™°Îʓ
ÇÊV“
Ӱ{ʓ
£Óʓ
™ÊV“
™°Îʓ
££ÊV“
£Î°xʓ
£nÊV“
xʓ
c
ÇÊV“
d
Îʓ
x°nʓ
nʓ
{ÓÊV“
£{ʓ
f Ignore the inside surface area of this prism.
e
£Ó°nÊV“
n°xÊV“
Îʓ
ÓÇ°nÊV“
x°ÓÊV“
10 a A square prism has an area of 348 cm2. The square face has a side of 6 cm. Find the
other dimension.
b How many rectangular boxes measuring 12 cm by 16 cm by 5 cm could be covered,
with no overlapping, with 6 m2 of plastic wrap?
3D Surface area of solids using
formulae
As seen in the previous section, the surface area of a solid can be obtained by using the net
that makes up the outer surfaces of the solid.
Using this idea, we can obtain formulae to find the surface areas of certain solids.
The surface area of a cube is made up of six identical squares.
If the side of a cube is x cm, then the total surface area of a cube
5 6 3 area of one square.
Total surface area of a cube 5 6x2
area and volume
The surface area of a cuboid or rectangular prism is made up of three pairs of congruent
rectangles; that is, a total of six rectangles.
Consider the rectangular prism and its net.

The areas of the three pairs of rectangles are:
l 3 w 5 lw
w 3 h 5 wh
Ü
Ü
h 3 l 5 hl

Ü
The surface area of the rectangular prism 5 2lw 1 2wh 1 2hl.
Surface area of a rectangular prism 5 2(lw 1 wh 1 hl)
The cylinder has two congruent circular surfaces and one curved surface. When opened
out, the curved surface forms a rectangle, and its length is the circumference of the circular
base. The width of the rectangle is the height of the cylinder, as seen in the diagrams below.
P
ÊrÊÓÊÊÀ
iˆ}…ÌʜvÊVޏˆ˜`iÀ]ʅÊrÊÜ
As seen from the net, the curved surface area of the cylinder 5 area of the rectangle 5 lxw
5 2pr 3 h.
Curved surface area of a cylinder 5 2prh
The total surface area of the cylinder is made up of the areas of
the two circles and the curved surface area. So the total surface
area of the cylinder 5 2prh 1 2pr2.
Remember
Area of a circle 5 pr2
Total surface area of a cylinder 5 2pr (h 1 r)
It is not possible at this stage to find the formulae for the surface area of every type of solid.
If we need to find the surface area of any other type of solid, we must find the area of all
the surfaces exposed (ie that can be seen) and add them.
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oxford mathematics for victoria
E X AM P L E S
Exa mple D1
Find the surface area of a cube with sides 4.3 cm.
Need
to do
N e ed
t o k no w
Surface area of a cube
5 6x2.
★
Use the formula.
Surface area 5
6x2
5 6(4.3)2 cm2
5 110.94 cm2
Exa mple D2
Find the total surface area of a rectangular prism with length 7.3 cm,
width 6.2 cm and height 2.5 cm.
N e ed
t o k no w
Surface area of a
rectangular prism
5 2(lw 1 wh 1 hl),
Need
to do
Use the formula.
Surface area 5
2(lw 1 wh 1 hl)
5 2(7.3 3 6.2 1 6.2 3 2.5 1 2.5 3 7.3) cm2
5 2(45.26 1 15.5 1 18.25) cm2
5 2 3 79.01 cm2
5 158.02 cm2
7 cm
Exa mple D3
Find the curved surface and the total surface area of a cylinder with
radius 7 cm and height 10.5 cm. (Use p 5 3.14.) Give your answer
correct to 2 decimal places.
N e ed
t o k no w
• The curved surface
area of a cylinder
5 2prh
• Total surface area
of a cylinder 5
2pr(h 1 r).
★
Need
to do
Use the formulae.
Curved surface area 5
2prh
5 2 3 3.14 3 7 3 10.5 cm2
5 461.58 cm2
Total surface area 5
2pr(h 1 r)
5 2 3 3.14 3 7(10.5 1 7) cm2
5 2 3 3.14 3 7 3 17.5 cm2
5769.3 cm2
10.5 cm
★
area and volume
Exa mple D4
5 cm
Find the surface area of the triangular prism illustrated.
3 cm
11 cm
8 cm
Need
to do
N e ed
t o k no w
The triangular prism
has two congruent
triangular surfaces
and three rectangular
surfaces.
To find the surface area
of composite shapes, we
find and add the areas of
all the surfaces that can
be seen.
★
Find the area of the two triangles and the area of the rectangles and
add them.
h • Area of triangle5 _____
​ b 3 ​
2
3 5 _____
​ 8 3 ​
5 12 m2
2
• Area of side rectangle5 l 3 w
5 11 3 5 5 55 m2
• Area of bottom rectangle 5 11 3 8 5 88 m2
Total surface area of triangular prism 5 (2 3 1212 3 55 1 88) m2
5 (24 1 110 1 88) m2
5 222 m2
Exa mple D5
nʓ
Ç°Îʓ
̜«
Find the surface area of this composite solid.
`i
}ÊÈ
̈˜
Ï>˜
Èʓ
xʓ
È`i
vÀœ˜Ì
£x°Óʓ
£{ʓ
Need
to do
N e ed
t o k no w
Find the area of all the faces and add them.
[ ]
1
_
5 2​[ ​ 2 ​(14 1 8) 3 6 ]​
• Area of trapezium faces5 2​ _​ 12 ​(a 1 b)h ​
5 1(22) 3 6 5 132 m2
• Area of front and back rectangles5 2lw
5 2 3 14 3 5 5 140 m2
• Area of two slanting rectangles 5 2 3 15.2 3 7.3 5 221.92 m2
• Area of two side rectangles 5 2 3 15.2 3 5 5 152 m2
• Area of top rectangle 5 8 3 15.2 5 121.6 m2
• Area of bottom rectangle 5 14 3 15.2 5 212.8 m2
Hence, the surface area of the composite solid
5 (132 1 140 1 221.92 1 152 1 121.6 1 212.8) 5 980.32 m2.
Examples continued next page
The front face is made
up of a trapezium and
a rectangle. There are
two such faces. The
rest of the faces are
rectangles.
★
115
116
Examples continued
oxford mathematics for victoria
Exa mple D6
A shed has to be painted with a protective substance to
withstand the harsh weather conditions.
a Find the surface area that needs to be painted. The door
need not be painted with this substance.
b If the cost of a 5 L can of paint is $57.95 and one can
covers 6 m2, find the number of cans required to
complete the job, and the cost of painting the whole
shed.
t o
N e ed
k no w
{°Èʓ
Óʓ
ΰÓʓ
Ӱxʓ
££°xʓ
™äÊV“
ΰx“
Need
to do
The surface to be
painted is made up of
two triangles and a
number of rectangles.
★
Exercise 3D
1 Use the formula to calculate the surface area of each cube.
a
b
c
11 cm
d
0.6 m
15.7 cm
0.45 m
2 Use the formula to calculate the surface area of each rectangular prism. Give your answer
to the nearest square metre.
a
30 cm
b
7m
74 cm
100 cm
6.3 m
4.1 m
area and volume
3 Calculate the surface areas of the following triangular pyramids using the appropriate
formula for each face.
a
b
26 m
12 cm
10 cm
17 m
18 cm
15 cm
32 m
24 m
c
d
42.6 cm
18.5 cm
32 cm
17.3 cm
52.8 cm
28 cm
22.3 cm
14.5 cm
e
f
12.7 cm
18 cm
11.3 cm
7.2 cm
14.6 cm
42 cm
28 cm
4 For each cylinder, use the appropriate formulae to find:
i the curved surface area
ii the total surface area
Give your answers to the nearest cm . Use p 5 3.14.
2
a
b
ÇÊV“
c
££°ÎÊV“
Ç°nÊV“
È°nÊV“
£nÊV“
££°ÇÊV“
d
28.4 m
e
f
7.5 m
0.75 m
0.32 m
1.25 m
0.32 m
5 Calculate the surface areas of these composite solids.
a
b
£xÊV“
3m
1.5 m
£ÓÊV“
£ÎÊV“
££ÊV“
3.2 m
£äÊV“
2m
£{ÊV“
ÓxÊV“
Ignore the inside surface area of this solid.
2.5 m
117
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oxford mathematics for victoria
c
d
15 cm
£Óʓ
9.5 cm
4 cm
£{ʓ
5 cm
Ó{ʓ
18.5 cm
6 The walls and ceiling of a bedroom are to be painted.
a Find the surface area of the four walls (excluding
the door and the windows) and the ceiling. The
dimensions of the door are 2.1 m 3 90 cm and
each window is 1.8 m 3 90 cm. Give your answer
to the nearest square metre.
b Calculate the cost of painting if a 3 L can costing
$52.95 covers an area of 7 m2.
2.6 m
4.8 m
3.2 m
7 The diagram shows a tent. Calculate the smallest
amount of fabric required to make this tent if an
extra 6% allowance is required for seams.
3m
2.2 m
3.2 m
2.5 m
8 What is the external surface area of the illustrated cardboard
aluminium foil roll?
£°xÊV“
Homework
Sheet 3.2
{xÊV“
3E Volume of solids
The volume of a solid is the amount of space it occupies, or the amount of material that
makes up the solid. It is measured in cubic units.
In order to understand this concept, we could carry out an investigation.
I N V E S T I G A T I O N 1 a Calculate
the volume
of this layer
of cubes.
Volume of rectangular prisms
1 cm
5 cm
4 cm
b What will be the volume of the
rectangular prism formed by a
stack of:
i 3 layers?
ii 5 layers?
iii 10 layers?
area and volume
2 This rectangular
prism is formed
by stacking layers
of cubes on top
ÈÊV“
of one another.
a How many
cubic
centimetres
{ÊV“
ÎÊV“
are there in
the bottom
layer?
b How many layers are in the stack?
c What is the volume of this prism?
3 Repeat Question 2 for the rectangular
prisms given below.
a
2 cm
6 cm
2 cm
c
3 cm
4 cm
2 cm
d
3 cm
6 cm
5 cm
4 a Using the results of questions 2 and
3, write in words and symbols a
rule for calculating the volume of a
rectangular prism.
b Discuss with other groups and modify
your rule if you wish to do so.
b
4 cm
5 cm
3 cm
From the investigation above, it is observed that:
Volume of a rectangular prism 5 l 3 w 3 h
where l 5 length of a rectangular prism
w 5 width of a rectangular prism
and h 5 height of a rectangular prism.
Calculating volumes for other solids
In the investigation above, we observed that the first layer is the base of the rectangular
prism. This is also referred to as the cross-section of the prism. We learnt in the last section
that a prism has uniform cross-sections if the slices cut parallel to the base are the same.
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oxford mathematics for victoria
Hence: Area of base 5 Area of cross-section 5 l 3 w.
So, the volume of a rectangular prism 5 l 3 w 3 h 5 area of base 3 height.
Hence, for any prism that has a uniform cross-section as a base, the general formula to find
its volume is:
Volume of prism 5 area of base 3 height
If a solid has a base that is an irregular shape, the
volume is found by using the general formula for a prism.
The area of the base will be supplied.
Note
Sometimes the prism might be in a
horizontal position. In such cases the
height will be its ‘length’, but we still
use the word ‘height’ in the formula.
Formulae to find the volume of prisms
If the base area is not given, it should be found using the
appropriate formula. In the previous section we studied the
area of various regular polygons, which will help us in this section.
Cube
Area of base of a cube of side x 5 x2 (unit)2
Height of cube 5 x (unit)
Hence: Volume of cube 5 area of base 3 height
x
5 x2 3 x 5 x3
x
x
Volume of cube 5 x3
Triangular prism
Area of (base) triangle 5 __
​ 12 ​3 b 3 h
Height of triangular prism 5 H
Hence:
Volume of triangular prism 5
_
​ 12 ​3
b3h3H
Cylinder
Area of (base) circle 5 pr2
Height of cylinder 5 h
Hence:
Note
h
H
r
h and H stand
for different
parts of the
prism.
h
Volume of cylinder 5 pr h
2
Other prisms
It is not possible to get formulae for the volumes of all the different types of prisms.
However, it is possible to find volumes by using the general formula for the volume of
a prism.
If the base of the prism is a composite shape, then the area of this base is found by
calculating the area of each regular shape and adding or subtracting them.
area and volume
Volume of tapered solids
Tapered solids are solids with faces that meet at a point. Examples include the cone and the
pyramid.
I N V E S T I G A T I O N olumes of solids and
V
pyramids
You will need: cardboard, sticky tape and
scissors.
What to do
1 Construct and cut out the nets of the open
cube and the open square pyramid below
using the information on the diagrams.
The cube and the pyramid have the same
base and height measurements. Use sticky
tape to make the objects.
Fill the pyramid with sand and see
how many pyramids of sand it takes to fill
the cube.
Construct others of your own and test
them.
Fill the cone with sand and see how
many cones of sand it takes to fill the
cylinder.
Construct others of your own and
test them.
94 mm
open cylinder
15 mm
open cone
216°
œ«i˜
VÕLi
20 mm
Óxʓ“
open
pyramid with
square
base
30.6 mm
20 mm
25 mm
2 Construct and cut out the nets of the
open cylinder and the open cone using
the information on the diagrams. The
cylinder and the cone have the same
circular base and height measurements.
Use sticky tape to make the objects.
It is observed that the cube is filled with
three times the amount of sand in the
square pyramid and the cylinder is filled
with three times the amount of sand in
the cone. This is provided that the solids
have the same base and height.
Hence: The volume of the tapered solid
5 __
​ 13 ​(volume of the prism).
Volume of tapered solid 5
_
​ 13 ​3 area of base 3 height of pyramid
Hence:
Volume of square pyramid 5 ​ _13 ​x2h
Volume of cone 5 _​ 13 ​pr2h
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oxford mathematics for victoria
Composite solids
The volume of a composite solid is found by calculating the volume of each solid contained
in it and then adding or subtracting them.
E X AM P L E S
Exa mple E1
Find the volume of this prism.
7 cm
6 cm
11 cm
N e ed
t o k no w
Volume of a rectangular
prism 5 l 3 w 3 h,
where
length of prism 5 11 cm
width of prism 5 6 cm
height of prism 5 7 cm.
Need
to do
★
Use the formula.
Volume of prism 5
l3w3h
5 11 3 6 3 7 5 462 cm3
Exa mple E2
Find the volume of these prisms. The area of each base is given.
a b
40 cm2
t o
12 cm
N e ed
k no w
a Area of base
5 40 cm2,
height 5 12 cm
b Area of base
5 72 cm2,
height 5 15 cm
72 cm2
15 cm
Need
to do
Use the formula.
a Volume of rectangular prism 5
area of base 3 height
5 40 3 12 5 480 cm3
b Volume of pentagonal prism 5
area of base 3 height
5 72 3 15 5 1080 cm3
★
area and volume
Exa mple E3
3.2 cm
Using p 5 3.14, calculate the volume of the following
prisms:
a
b
c
11.5 cm
£xÊV“
£ÇÊV“
7 cm
Give your answers correct to 2 decimal places.
Need
to do
N e ed
t o k no w
• Volume of cube 5 x3
• Volume of triangular
prism 5 _​ 12 ​3 b 3 h
3H
• Volume of cylinder
5 pr2h
★
Use the formulae.
a Volume of cube 5 x3 5 (7)3 5 343 cm3
b Volume of triangular prism5 _​ 12 ​3 b 3 h 3 H
5 (0.5 3 17 3 10 3 15) cm3
5 1275 cm3
2
c Volume of cylinder5 pr h
5 (3.14 3 (3.2)2 3 11.5) cm3
 369.77 cm3
Exa mple E4
Find the volume of the trapezoidal prism.
11 m
17 m
4m
• The base is a
trapezium with
parallel sides
measuring 17 m
and 11 m.
• The perpendicular
distance between
them is 4 m.
• Height of prism is
12 m.
Need
to do
★
Use the appropriate formulae.
Area of (base) trapezium5 _​ 12 ​3 h(a 1 b)
5 0.5 3 4(17 1 11) 5 56m2
[ Volume of trapezoidal prism 5
area of base 3 height
5 56 3 12 5 672 m3
Examples continued next page
N e ed
t o k no w
12 m
123
124
Examples continued
oxford mathematics for victoria
Exa mple E5
Calculate the volume of this prism.
5m
14 m
11 m
15 m
N e ed
t o k no w
Need
to do
• The base of the
prism is made of
a rectangle and a
triangle.
• Height of the prism
5 15 m.
Find the area of the triangle and the rectangle and add them to give
the area of the base of the prism.
★
Area of base5 _​ 12 ​bh 1 lw
( )
5 ​ _​ 21 ​3 11 3 5 1 11 3 14 ​m2
5 (27.51 154)5 181.5 m
[ Volume of prism 5
area of base 3 height
5 181.5 3 155 2722.5 m3
The volume of this solid could also be found by finding the volumes
of the rectangular prism and triangular prism and adding them.
2
Exa mple E6
A rectangular piece of metal measuring 115 cm by 144 cm is rolled along its longer side to form a hollow
cylinder. Find the volume, correct to 2 decimal places, of the resulting cylinder. (Use p 5 3.14.)
115 cm
115 cm
144 cm
144 cm
N e ed
t o k no w
• The width of the
rectangle is the
circumference of
the base of the
cylinder.
• The height of
the cylinder is
the length of the
rectangle.
Need
to do
Radius of cylinder5 ____
​ C ​ (2p)
5 _________
​ 115 ​ ≈ 18.31 cm
(2 3 3.14)
[ Volume of cylinder 5
pr2h
5 (3.14 3 18.312 3 144)
5 151 589.39 cm3
★
area and volume
Exa mple E7
£ÓÊV“
a
Find the volume of the following solids.
b
£ä°xÊV“
ÇÊV“
{ÊV“
N e ed
t o k no w
Need
to do
★
Use the formulae.
Side of square base
5 4 cm
Height of pyramid
5 12 cm
Radius of circular base
of cone 5 7 cm
Height of cone
5 10.5 cm
a Volume of square pyramid5 _​ 13 ​x2h
5 _​ 13 ​3 42 3 12
5 64 cm3
b Volume of cone 5 _​ 13 ​pr2h
5 _​ 13 ​3 3.14 3 72 3 10.5
5 538.51 cm3
Exercise 3E
1 Find the volume of each solid, given the area of the base. Give your answers to 2 decimal
places where necessary.
a
b
42 cm2
10 cm
d
c
54.6 cm2
e
f
5m
15 ha
44
m2
10 m
g
Èʓ
£Óʅ>
ÎÓ°ÇÊV“Ó
11 cm
™°ÎÊV“
xÈ°nʓÓ
£{ʓ
125
126
oxford mathematics for victoria
2 Using appropriate formulae, find the volume of each solid. Give your answers correct to
2 decimal places. (Use p 5 3.14.)
a
55 mm
b
c
12.3 cm
12.5 cm
4 cm
11.6 cm
9 cm
7.6 cm
d
e
3.5 cm
140 mm
f
9.4 cm
175 mm
24 m
30 m
8 cm
7m
g
h
14.6 cm
ΰÇÊV“
£ÓÊV“
££Çʓ“
146 mm
3 Find the volume of each solid. Give your answers correct to 2 decimal places.
(Use p 5 3.14.)
a
b
17 cm
60°
7 cm
18 cm
5 cm
12 cm
6 cm
c
d
£È°ÇÊV“
££°xÊV“
n°ÎÊV“
Óä°xÊV“
7.6 cm
17.3 cm
25 mm
e
f
7.7 cm
144 mm
18.6 cm
7.2 cm
140 mm
9.6 cm
area and volume
4 Find the volume of each tapered solid. Give your answers correct to 2 decimal places.
Use p 5 3.14.
a
12.5 cm
b
c
£ÓÊV“
11 cm
{ÊV“
7.5 cm
££ÊV“
d
3 cm
e
7.6 cm
f
™°ÎÊV“
115 mm
6.7 cm
{°ÓÊV“
7.4 cm
8.2 cm
nÊV“
5 Find the volume of each composite solid. Give your answers correct to 2 decimal places.
Use p 5 3.14.
a
b
ÎÊV“
10.5 cm
ÈÊV“
£nÊV“
8.6 cm
11.6 cm
ÈÊV“
c
d
12 cm
2.5 cm
3.7 cm
8.3 cm
4.8 cm
7.5 cm
8.7 cm
6 A cylinder of diameter 180 mm and height 210 mm is melted and cast into rectangular
bars measuring 20 mm 3 30 mm 3 40 mm. How many bars can be produced?
7 A rectangular sheet of cardboard measuring 24 m 3 36 m is rolled on its shorter end to
form a cylinder. Find:
a the radius of the cylinder
b the volume of the cylinder.
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3F Capacity
Capacity is the measure of space inside a solid container. The space can be filled with a fluid
(liquid or gas).
Hence, capacity is the volume of fluid that a solid container can
Remember
hold. The unit of capacity depends on the units for volume.
1000 mL 5 1 L
The units for capacity are the millilitre (mL), litre (L), kilolitre (kL)
1000 L 5 1 kL
and megalitre (ML). The basic unit for capacity is the litre.
1000 kL 5 1 ML
For easy conversion, a diagram is given here.
3 1000
3 1000
ML
kL
4 1000
3 1000
L
mL
4 1000
4 1000
Some familiar capacities are given in this table.
Item
Estimate of capacity
Medicine glass
25 mL
Cup
250 mL
Milk carton
1L
Petrol tank
65 L
Hotwater system
170 L
50 m swimming pool
1500 kL
Dam
10 ML
Reservoir
1000 ML
Volume conversion
Using the unit of conversion for length, we can find the unit of conversion for volume.
We know that
1 cm 5 10 mm
and
1 cm 3 1 cm 3 1 cm 5 10 mm 3 10 mm 3 10 mm
Hence:
1 cm3 5 1000 mm3
We know that
and
Hence:
1 m 5 100 cm
1 m 3 1 m 3 1 m 5 100 cm 3 100 cm 3 100 cm
1 m3 5 1 000 000 cm3 5 106 cm3
area and volume
A conversion diagram for easy calculation is given here.
3 1 000 000
m3
3 1000
cm3
mm3
4 1 000 000
4 1000
Capacity/volume conversion
If we fill a cube of sides 1 cm with water and pour the water into
a measuring cup, it will read as 1 millilitre (1 mL). The volume of
water is the same as the volume of the cube—that is, 1 cm3.
Hence:
1 cm3 5 1 mL
1 mL
1 cm
3 both sides by 1000, 1000 cm3 5 1000 mL
1000 cm3 5 1 L (since 1000 mL 5 1 L)
3 both sides by 1000, 1000 3 1000 cm3 5 1000 L (1 000 000 cm3 5 1 m3)
1 m3 5 1000 L
or
1 m3 5 1 kL (since 1000 L 5 1 kL)
E X AM P L E S
Exa mple F1
N e ed
t o k no w
1 m3 5 1 000 000 cm3
1 cm3 5 1000 mm3
c 7500 mm3 to cm3
d 675 000 cm3 to m3
Need
to do
Use the conversion diagram.
a 0.045 m3 5 (0.045 3 1 000 000) cm3
5 45 000 cm3
3
b 2.8 cm 5 (2.8 3 1000) mm3
5 2800 mm3
c 75 000 mm3 5 (75000 4 1000) cm3
5 7.5 cm3
3
d 675 000 cm 5 (675 000 4 1 000 000) m3
5 0.675 m3
★
Examples continued next page
Convert the following.
a 0.045 m3 to cm3
b 2.8 cm3 to mm3
129
130
Examples continued
oxford mathematics for victoria
Exa mple F2
Convert the following.
a 4600 mL to L
b 320 kL to L
t o
c 1500 kL to ML
Need
to do
N e ed
k no w
1000 mL 5 1 L
1000 L 5 1 kL
1000 kL 5 1 ML
d 3.25 L to mL
★
Use the conversion diagram.
a 4600 mL 5 (4600 4 1000) 5 4.6 L
b 320 kL 5 (320 3 1000) 5 320 000 L
c 1500 kL 5 (1500 4 1000) 5 1.5 ML
d 3.25 L 5 (3.25 3 1000) 5 3250 mL
Exa mple F3
Find the capacity of containers with the following volumes.
a 4000 cm3
b 3.7 m3
c 4570 mm3
t o
Need
to do
N e ed
k no w
1 cm3 5 1 mL
1 m3 5 1000 L
1 m3 5 1 kL
1 cm3 5 1000 mm3
★
a 4000 cm3 5 4000 mL 5 (4000 4 1000) 5 4 L
b 3.7 m3 5 (3.7 3 1000) 5 3700L 5 3.7 kL
c 4570 mm3 5
(4570 4 1000) cm3 5 4.47 cm3
5 4.57 mL
Exa mple F4
6.4 cm
17.5 cm
Find the capacity of the given containers.
32 cm
5 cm
17 cm
18 cm
N e ed
t o k no w
Need
to do
• Volume of triangular
prism 5 _​ 12 ​bhH
• Volume of
rectangular prism
5 lwh
• 1 cm3 5 1 mL
• 1 L 5 1000 ML
Use the formulae.
a Volume of triangular prism5 _​ 12 ​3 b 3 h 3 H
5 (0.5 3 17.5 3 6.4 3 5) cm3
5 280 cm3 5 280 mL
5 (280 4 1000) 5 0.280 L
b Volume of rectangular prism 5 l 3 w 3 h
5 (17 3 18 3 32)5 9792 cm3
5 9792 mL
5 (9792 4 1000)5 9.792 L
★
area and volume
Exa mple F5
£{ÊV“
a Find the capacity of the cylindrical can.
b If one mug holds 250 mL of liquid, how many mugs must be used to fill the can?
£nÊV“
(Use p 5 __
​ 22
​.)
7
Need
to do
N e ed
t o k no w
Use the formulae.
a Volume of cylindrical can5 pr2h
• Volume of a cylinder
5 pr2h
14
• r 5 __
​ 2 ​5 7 cm,
( ★
)
5 ​ __
​ 22
​3 72 3 18 ​5 2772 cm3
7
 Capacity of can 5 2772 mL 5 2.772 L
b Number of mugs used 5 (2772 4 250)  11 mugs
• h 5 18 cm
• 1 cm3 5 1mL
Exa mple F6
A fish tank measuring 30 cm by 25 cm by 15 cm is filled
with water to a depth of x cm. If the volume of the water is
9 L, find the depth of water in the tank.
t o
£xÊV“
ÝÊV“
ÓxÊV“
ÎäÊV“
Need
to do
N e ed
k no w
★
Convert to same units:
Volume of water in tank 5 9 L 5 9000 mL 5 9000 cm3
Form an equation to solve for the unknown, x.
Volume of tank5 volume of water
30 3 25 3 x5 9000
750x5 9000
x 5 9000 4 7505 12 cm
 Depth of water 5 12 cm
The volume of the
rectangular tank to
a depth of x cm 5
volume of water in the
tank.
Exercise 3F
1 Convert each of the following to the unit given.
a 5 cm3 5 ________mm3
b 9 000 000 cm3 5 ________m3
c 0.83 cm 5 ________mm d 2.86 m3 5 ________kL
e 575 cm3 5 ________L
f 683 L 5 ________cm3
g 4 m 5 ________cm 5 ________L
h 6000 L 5 ________m3 5 ________cm3
3
3
i
3
3
50 000 cm35 ________m3 5 ________kL j 20.5 kL 5 ________L 5 ________cm3
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oxford mathematics for victoria
2 For each container, find:
i the volume in cubic centimetres
ii the capacity in millilitres
iii the capacity in litres.
a
b
Çʓ
15 cm
Óxʓ
£xʓ
c
d
£ääʓ“
£ÇÊV“
£xÊV“
ΰxÊV“
ÓäÊV“
3 A 24 cm circular cake tin has a depth of 7 cm. Find, in millilitres, the volume of cake
mixture it can hold if it is three-quarters full.
4 A concrete slab measuring 3.5 m by 2.7 m by 1.6 m is to be laid.
a Find the volume of concrete used to lay this slab.
b If the cost of concrete is $180 per cubic metre, find the total cost of laying the slab.
5 A fridge has the following internal dimensions: height 5 125 cm,
width 5 90 cm and depth 5 50 cm. Find the capacity of the fridge
in litres.
6 A swimming pool, as shown in the diagram, has a length of 20 m and a width of 8 m. The
depth is 1.2 m at the shallow end and 2.4 m at the deep end.
a Name the shape of the cross-section of the pool.
b Find the volume of the pool.
£°Óʓ
c Find the capacity of the pool in kilolitres.
d If the pool is filled to a depth of 90 cm, find
the volume of water in kilolitres.
Óäʓ
nʓ
Ӱ{ʓ
7 The volume of the block of wood is the same
as the capacity of the tea light.
a Calculate the volume of the block of wood.
b Hence, find the radius of the tea light.
4 cm
3.6 cm
8 A triangular hollow prism holds a chemical solution to a level of x cm.
x cm
a Find the volume of the chemical solution in terms of x.
Homework
Sheet 3.3
b If the volume of the solution is 147 cm3, find the value of x.
7 cm
10 cm
area and volume
Language
Adjacent
Side-by-side, or next to each other.
Capacity
The amount of space inside a container that can
be filled with a fluid (liquid or gas).
Circumference
The distance around a circle; the boundary line of
a circle.
Platonic solids
Solids for which all faces are the same shape
and size, all edges are equal length, all vertices
have the same number of edges meeting, and
all faces are regular polygons. There are five
Platonic solids—cube, regular tetrahedron, regular
octahedron, regular dodecahedron, regular
icosahedron.
Composite figures (solids)
Shapes that are made up of two or more regular
shapes, which can be either plane figures (flat
surface) or solids.
Polyhedron
A solid having only faces that are polygons (plane
shapes). The term comes from the Greek words
‘poly’, meaning ‘many’ and ‘hedron’, meaning
‘face’. The plural of polyhedron is polyhedra.
Congruent
Identical in size and shape, able to be exactly
placed one on another.
Polygon
A many-sided closed plane figure (eg triangle,
square).
Cross-section
The flat surface seen when a solid is cut through. It
is a slice cut parallel to the base of the solid.
Quadrilateral
A figure that has four sides. The length of its sides
and size of its angles determine its name and
shape.
Diagonal of a plane figure
A line segment joining two non-consecutive (not
side-by-side) vertices in a plane figure.
Diagonal of a polyhedron
A line segment joining two vertices not on the
same plane (flat surface).
Edge
The line along which two flat surfaces of a solid
meet or join.
Face
A flat (plane) surface in a solid.
Hemisphere
Half a sphere. It is made up of a curved surface
(​ __12 ​ sphere) and a flat surface (circle).
Line segment
A part of a line that has two definite end points.
Net
A flat figure or shape that is made up of all the flat
surfaces (faces) of a solid so that it could be folded
to form the solid.
Parallel (lines/planes)
Lines or planes that never intersect. They are
equidistant at every point.
Right prism
A solid made up of two congruent faces (same
shape and size), each called the base, the rest of
the faces being rectangles, which are called the
‘lateral’ faces.
Solid
Any object that occupies space and has three
dimensions; the three dimensions are length,
width and height.
Sphere
A solid in which every point on its surface is the
same distance from its centre—it is ball-shaped.
Surface area
A sum of the areas of all the flat (plane) surfaces
that make up a solid. It is the area of the net of the
solid.
Symmetrical
Identical with a reflection about an axis.
Vertex
A point in a solid shape where three or more flat
surfaces meet.
Volume
The amount of space occupied by a solid. It is also
the amount of material the solid is made of.
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S K I L L S
A N D
C ON C E P T S
special quadrilaterals
l
The following chart lists the different types of formulae for the area of special quadrilaterals.
Name
Shape
Formula for area
Rectangle
Area of rectangle 5 l 3 w
where l is the length and w is the
width of the rectangle.
w
l
Square
Area of square 5 l2
where l is the side of the square.
l
l
Parallelogram
h
b
Rhombus
`£
Kite
`Ó
Area of rhombus 5 __
​ 12 ​3 d1 3 d2
where d1 and d2 are the diagonals of
the rhombus.
Area of kite 5 __
​ 12 ​3 D1 3 D2
where D1 and D2 are the diagonals of
the kite.
£
Ó
Area of trapezium 5 __
​ 12 ​3 h(a 1 b)
where a and b are the parallel sides
and h is the height of the trapezium
(the distance between the parallel
lines).
a
Trapezium
Area of parallelogram 5 b 3 h
where b is the base and h is the
height of the parallelogram.
h
b
area conversion
3 100
km2
ha
4 100
3 10 000
3 10 000
m 2
4 10 000
3 100
cm2
4 10 000
To convert from a larger unit to a smaller unit, you multiply.
l To convert from a smaller unit to a larger unit, you divide.
l
mm2
4 100
areas of composite shapes
l
Composite shapes are made up of more than one regular shape.
To find the area of a composite shape, find the area of each regular shape and then add or subtract them.
circles
Name of feature
Definition
Centre
A point in the middle of the circle equidistant from all points on
the circumference.
The angle at the centre of the circle is 360°.
Circumference
The boundary line enclosing the circle.
The perimeter of the circle.
Radius
The line joining the centre of the circle to any point on the
circumference of the circle.
Diameter
A line joining two points on the circumference and passing
through the centre.
Chord
A line joining two points on the circumference but not passing
through the centre.
Tangent
A line touching the circle at one point and forming an angle of
90° with the radius.
Major arc
Each is a fraction of the whole circumference.
Minor arc
Major sector
Minor sector
A sector is a fraction of the surface of the whole circle. It is
bounded by two radii and an arc. The two radii make an angle at
the centre of the circle.
If  is the angle at the centre, then:
fraction of the circle 5 ____
​  ​.
360
Semicircle
Half a circle bounded by the diameter and an arc.
Quadrant
One quarter of a circle, bounded by two radii and an arc. It is a
sector with the angle at the centre at 90°.
Concentric circles
Two or more circles having the same centre.
Major segment
A segment is part of a circle bounded by a chord and an arc.
Minor segment
l
The length of an arc of a circle is a fraction of
the whole circumference.
Length of an arc 5 ___
​  ​ 3 2 3 p 3 r
360
Area of a circle 5 pr2 (units)2
l
The area of a sector of a circle is a fraction of
the area of the whole circle.
Area of a sector 5 ___
​  ​ 3 p 3 r2
360
Skills and concepts continued next page
area and volume
135
136
Skills and concepts continued
oxford mathematics for victoria
prisms
l
l
l
l
l
l
l
l
l
l
A prism is a solid having two congruent faces,
the rest of the faces being rectangles.
The congruent faces are the same shape as the
cross-section.
A cross-section is a ‘slice’ of the solid and is
cut parallel to its congruent face or base.
One of the congruent faces is called the base
of the prism. The base of the prism can be any
polygon. The name of the prism depends on the
shape of its base.
A net of a solid is a diagram showing all its faces.
To find the surface area of a solid using its
net:
1 draw the net of the solid
2 find the area of each face
3 add the area of all the faces.
The total surface area of a cube 5 6x2.
The surface area of a rectangular prism
5 2(lw 1 wh 1 hl).
The curved surface area of a cylinder 5 2prh.
The volume of a rectangular prism 5 l 3 w 3 h.
The general formula to find the volume of a
rectangular prism is:
Volume of prism 5 area of base 3 height
If a solid has a base that is an irregular shape,
the volume is found by using the general formula
for a prism. The area of the base will be given.
The volume of a cube 5 x3.
The volume of a triangular prism 5 _​ 12 ​3 b 3 h
3 H.
The volume of a cylinder 5 pr2h.
tapered solids: pyramid and cone
l
l
l
The volume of a tapered solid
5 _​ 13 ​(volume of the prism)
5 _​ 13 ​3 area of base 3 height of pyramid
Volume of square pyramid 5 _​ 13 ​x2h
The volume of a cone 5 _​ 13 ​pr2h.
capacity
Capacity is the measure of space inside a solid
container. The space can be filled with a fluid
(liquid or gas).
Hence, capacity is the volume of fluid that a
solid container can hold. The unit of capacity
depends on the units for volume.
l The units for capacity are the millilitre (mL), litre
(L), kilolitre (kL) and megalitre (ML). The basic
unit for capacity is the litre.
1000 mL 5 1 L
1000 L 5 1 kL
1000 kL 5 1 ML
l
capacity/volume conversion
3 1000
ML
3 1000
kL
4 1000
L
4 1000
3 1 000 000
m3
3 1000
mL
4 1000
3 1000
cm3
4 1 000 000
mm3
4 1000
1 cm3 5 1 mL
1 m3 5 1000 L
1 m3 5 1 kL (since 1000 L 5 1 kL)
area and volume
Chapter review
1 Complete the following conversions.
28 cm
a
a 3.8 cm2 5 _______mm2
32 cm
b 158 000 mm2 5 _______cm2
8 cm
c 2.5 m2 5 _______cm2
12 cm
d 40 000 cm2 5 _______m2
e 80 ha 5 _______km2
b
c
11 cm
f 4.8 ha 5 _______m
2
25 cm
g 400 mm3 5 _______cm3
h 3800 cm3 5 _______m3
i
j
k
l
m
n
2.8 cm3 5 _______mm3
1.6 m3 5 _______cm3
750 mL 5 _______L
2.7 kL 5 _______L
3 000 000 cm3 5 ______m3 5 _____kL
400 L 5 _______m3 5 _______cm3
25 cm
d
Area of square 5 121 cm2
18 cm
11 cm
29 cm
2 Calculate the area of each shape. Give
your answer correct to 2 decimal places.
a
4 Find the area of each composite shape.
75 cm
a
82 cm
20 cm
24.8 cm
80 cm
b
22 cm
17.4 cm
115 cm
b
140 cm
c
29.7 cm
11.5 cm
210 cm
15.5 cm
145 cm
c
d
18 cm
7 cm
8 cm
22.6 m
12 cm
45.5 m
3 Find the area of the shaded section for
each of the following shapes. Give your
answer correct to 2 decimal places.
(Use p 5 3.14.)
6 cm
d
21.9 cm
4.8 cm
7.6 cm
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oxford mathematics for victoria
5 Using p 5 3.14 and giving your answer
d
{È°ÎÊV“Ó
correct to 2 decimal places, calculate for
each shape:
i the perimeter ii the area.
a
35 cm
ʓ
£Èʓ
nÈ°È nÈ°È nÈ°È nÈ°È nÈ°È
V“Ó V“Ó V“Ó V“Ó V“Ó
b
£Óʓ
Óä
138
42 cm
c
d
7 For each solid:
i draw the net
75°
9 cm 18 cm
24.6 cm
ii find the surface area.
a
b
32 cm
6 For each net:
9 cm
7.5 cm
i name the solid you would get from it.
ii find its surface area.
a
36 cm2
8.6 cm
5.2 cm
21 cm
12 cm
c
d
7.5 cm
60 cm2
120
cm2
10 m
60 cm2
12.5 m
11.8 cm
11.2 m
14.3 m
b
8 Using p 5 3.14, find the surface area
81 cm2
108 cm2 108 cm2 108 cm2 108 cm2
of each solid using the appropriate
formulae. Give your answer correct to
2 decimal places.
a
81
b
6 cm
cm2
21 cm
18 cm
14 cm
c
75 cm2
60
75 cm2
17 cm
c
d
cm2
75 cm2
7 cm
ÓäÊV“
ÎxÊV“
6 cm
8 cm
area and volume
9 Find the volume of each solid, given the
e
area of its base.
{°xʓ
a
x°xÊV“
b
ÊrÊ{™°ÇÊV“Ó
ÊrÊ£ÓÇ°xÊV“Ó
ΰ{ʓ
£{ÊV“
c
Îʓ
d
f
x°xÊV“
A = 22.5 m2
A = 156.6 m2
4.9 m
14.3 m
£äÊV“
ÈÊV“
11 A cylindrical container with base radius
10 Using p 5 3.14 and the appropriate
formulae, and giving your answers
correct to 2 decimal places, find for
each solid:
i the volume
ii the capacity in litres.
a
1.5 m contains water to a level of 90 cm.
If 60 L of water is added, find the new
level of the water in the container.
12 A conical glass vase holds 375 mL of
liquid when full. If the radius of its base is
4.5 cm, find the height of the vase.
13 A driveway, as shown, is to be concreted.
The cost of concrete is $180 per cubic
metre. How
much will it cost £°nʓ
ÓÇän
to concrete the
Ӱ{ʓ
driveway, which is
1.8 m wide, to a
depth of 15 cm?
Èʓ
£Çʓ
£Óʓ
£{ʓ
b
£ÇÊV“
£™°nÊV“
Σ°xÊV“
14 A swimming pool is designed as illustrated.
a Calculate the capacity of the pool in
c
kilolitres.
b If it costs $0.15 per cubic metre per
fortnight to clean the pool, what will it
cost per year?
c If water is filled to 80 cm below the
top of the pool, how many kilolitres
are required to completely fill it?
ÎÇ°xÊV“
ÓxÊV“
d
È°ÎÊV“
£x°ÓÊV“
xxʓ
Ó°Îʓ
Ç°ÈÊV“
{°™ÊV“
££°xÊV“
£nʓ
Îäʓ
£°nʓ
139
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oxford mathematics for victoria
15 A large puddle of rainwater has a surface
area of 0.06 ha and is 30 cm deep. Find
the volume of water (in kilolitres) in the
puddle.
16 A garden is in the shape of a trapezium,
as shown. Flowerbeds are prepared and
the rest of the garden is filled with red
gum mulch.
™°xʓ
{°xʓ
x°xʓ
™äÊV“
£°Óʓ
a How many cubic metres of mulch is
required if it is spread to a height of
15 cm?
b If mulch costs $45 per cubic metre
and 10% GST is charged per cubic
metre, find the cost of mulch
required.