Chap 09

An architect who designs a

building must work within

certain constraints or

specifications, such as the

area the building will cover.

If a one-bedroom unit is to

cover an area of 60 m2, draw

three possible floor plans for

this unit, showing the outline

of the outer walls and their

dimensions.

In this chapter we find out

how to measure perimeter,

area and volume.

9

Kitchen Dining Lounge

WC

BathroomBedroom

Scale 1:200

Measurement

382 M a t h s Q u e s t 8 f o r V i c t o r i a

READY?areyou

Are you ready?Try the questions below. If you have difficulty with any of them, extra help can be

obtained by completing the matching SkillSHEET. Either click on the SkillSHEET icon

next to the question on the Maths Quest 8 CD-ROM or ask your teacher for a copy.

Rounding to one decimal place

1 Round each of the following to 1 decimal place.

a 26.321 56 b 0.7264 c 59.338 017

Measuring angles with a protractor

2 Measure the size of each angle with a protractor.

a b c

Measuring the length of a line

3 a Measure the length of this line to the nearest mm.

b Measure the length of this line in cm.

Multiplying and dividing by powers of 10

4 Calculate each of the following.

a 32 ÷ 100 b 0.04 × 1000.

Converting units of length

5 Convert each of the following measurements.

a 34 mm to cm b 1.6 m to mm c 4500 cm to km

Area of squares, rectangles and triangles

6 Find the area of each of the following shapes.

a b

Volume of cubes and rectangular prisms

7 Find the volume of the following solid.

9.1

9.2

9.3

9.4

9.5

9.8

1.3 cm

4.5 cm

4.6 cm

3.4 cm

9.14

12.5 cm

3 cm

3 cm

C h a p t e r 9 M e a s u r e m e n t 383

IntroductionThe concept of measurement is

encountered by all of us every day.

We start off every morning thinking

what time is it? and how long is it

before we have to leave home? We

also hear of comments such as how

much milk is there? how much sugar

do you take? how heavy is this? what

size are these shoes? and so on.

There is an infinite number of ques-

tions that have to do with quantities

such as distance, volume, area,

height, mass, angle, time, speed and

so on.

Accuracy of measurementWhen an architect draws a plan or when an engineer designs a structure, the measure-

ments have to be very accurate. Similarly, when athletic tracks or swimming pools are

designed, the measurements have to be accurate as some races are won or lost by hun-

dredths of a second. The instruments used to measure these times also need to be

extremely accurate. When following a recipe for cooking, accuracy is important, but we

do not need to be accurate to the hundredth of the required mass or volume.

In practical situations where measurements are taken, accuracy depends on the

appropriateness of the instrument used, the accuracy of the instrument and the accuracy

with which a person reads the measurements.

The appropriateness of the instrumentThe proper choice of measuring instrument is very important.

For example, if 20 g of butter is needed for cooking, you will

obviously not use bathroom scales calibrated in kilograms.

You need kitchen scales. But if a 40 kg bag of mulch is

delivered to your place and you have a suspicion that the bag

is under weight, you will most likely use the bathroom scales.

If you are required to measure the thickness of your hair for

a science experiment, you will obviously not use a ruler, or

even a calliper. You could use a micrometer or a microscope.

The accuracy of an instrumentSome instruments are more accurate than others. For

instance, to measure 20.0 mL of a liquid accurately, a beaker

is very inaccurate. A measuring cylinder is better than a

beaker but may not be accurate enough for a chemist who is

preparing a cough mixture for a baby. A burette or a pipette is

a more accurate instrument to use.A micrometer


Pipettes being used to make accurate measurements

Accuracy of readingTo obtain the most accurate reading, the eye level of the person reading the measurement

should be placed at the appropriate point. Consider the diagrams below, showing dif-

ferent positions of the person’s eye while reading the liquid level in a measuring cylinder.

In diagram 1, the person’s eye is below the liquid level and so a higher (than real)

value will be recorded. In diagram 3, a viewer is looking at the liquid from above; thus

a lower value will be recorded. These incorrect values are the result of parallax error.

The reading shown in diagram 2 should give the most accurate result, as the eye is level

with the surface of the liquid.

You are required to measure the distance between the front door of your classroom

and the door to the principal’s office. The instruments available are: a metre ruler,

a 30-metre measuring tape and a trundle wheel.

1 Comment on the accuracy (or inaccuracy) of the three instruments being used.

2 Working in groups of three, measure the distance between a point in your

classroom and the principal’s office (or any other agreed distance) using a metre

ruler, a measuring tape and a trundle wheel.

3 Compare and comment on the results.

COMMUNICATION Measuring distance

Value maybe read

as 22 mL.

Actualvalue

= 20 mL

Value maybe read

as 18 mL.

1. 2. 3.


Accuracy of measurement is also assured if several measurements are done, where

possible, and an average value worked out instead of a one-off measurement.

Comment on the accuracy of the results recorded by the three judges positioned at

points A, B and C relative to the finish line for a race, as shown below.

To do this activity you will need to work in a group of five.

1 Each person in the group is to independently measure the height of every other

member of the group. Do not discuss your answers with any other person in the

group.

2 Complete the table below using all of the results obtained.

3 Discuss the reasons why different results were obtained.

THINKING Judging the winning result

C

B

Finish line

A

SkillSHEET

9.1

Roundingto onedecimal

place

EXCEL Spreadsheet

Roundingand

significantdigits(DIY)

COMMUNICATION Measuring height

NameGreatest height

measuredLeast height

measuredAverage of the

four results


Estimation and approximation in measurement

Approximation and estimation are commonly used in

expressing values for quantities in measurement. For

instance, we often hear people saying ‘the super-

market is 10 minutes away’. It does not neces-

sarily mean that we would require exactly

10 minutes to get there. Lots of factors, such as

the speed at which we drive, road conditions,

traffic conditions, and so on, can influence the

duration of the journey.

Similarly, many distances are estimated or

approximated. The ability to approximate or esti-

mate is a useful skill to develop. Approximations

usually involve rounding values to a certain

number of decimal places.

Distance, time, angles, height, mass, etc. are

commonly estimated. Sports commentators always

estimate the distance to the posts and the angle at

which the kick has to be taken during a football match.

Estimation of the speed at which vehicles are driven is

also fairly common. All these estimations are based on some

previous experience or knowledge or a reference point. For

example, to estimate a mass, an idea of 1 kg mass is used as a comparison or reference

point. An understanding of the size of the basic units, such as kilograms and metres, is

essential to enable successful estimation. When estimating, the purpose is to get an

approximate idea of a measurement; we are not expected to be overly accurate. For

example, when estimating the height of a house we may be expected to be within a

metre of the actual height and not within 1 millimetre.

1 Working with a friend, make the following estimations:

a height of a door

b width of your desk or table

c mass of a stone

d the distance a football has been kicked

e the diameter of a clock

f length of your classroom

g length of your pen.

2 Find the actual value for each item above by taking an appropriate measurement.

3 Compare the actual measurements with your estimations and comment on the

result.

COMMUNICATION Estimating measurements


Estimation and approximation in measurement

1 Estimate the angle labelled x in each of the following diagrams. Measure the angle and

comment on the accuracy of your estimation.

a b c

d e

Estimate the angle labelled x in the diagram below. Measure the angle and comment on

the accuracy of your estimation.

THINK WRITE

A 90° angle is easy to imagine.

The angle marked x is about half

of 90°.

Estimation: x ≈ 45°

Measure angle x, using a

protractor.

Measurement: x = 41°

Compare the actual size of angle x

with your estimation and comment

on the result.

45° closely estimates the size of the angle

shown (the exact value is 41°) with an error

of 4°.

x

1

2

3

1WORKEDExample

1. Approximations are usually done in measurement by rounding numbers to a

certain number of decimal places.

2. Estimation of measurements is needed in real-life situations and is often based

on previous experience or knowledge.

3. In some cases, a certain level of accuracy is expected.

remember

9A

SkillSHEET

9.2

Measuringangles witha protractor

Mathcad

Estimation andapproximation

WORKED

Example

1

x x

x

x x


2 Estimate the lengths of each of the following. Measure the lengths using a ruler and a

piece of cotton or thread. Discuss the accuracy of your estimation in each case.

a b c d

3 Without actual measuring, draw lines with the following lengths:

a 10 cm b 30 cm.

Now measure the lengths of the lines, using a ruler to check your estimation.

4

Which of the following is the best estimation for the width of a normal classroom door?

A 20 cm B 98 cm C 50 cm D 220 cm E 150 cm

5 Estimate the body measurements of two friends and yourself and complete the table

below. Take measurements to check your estimation skills.

6 Estimate each of the following household expenses for your home. Verify your esti-

mations by asking to see the bills.

Money spent over a month on:

a electricity

b the telephone

c groceries

d buying clothes

e petrol.

7 Estimate the number of reams of photocopy paper used in your school per week. Your

maths teacher or office staff can help verify your estimation.

8 Estimate the petrol consumption of your family car or a friend’s car for a 100 km

journey (or any other distance that suits you). Check the actual petrol consumption of

the car and compare it with your estimation.

Measurement

My own Friend A Friend B

Estimated Actual Estimated Actual Estimated Actual

Handspan

Height

Mass

Arm length

Shoulder length

SkillSH

EET 9.3

Measuring the length of a line

multiple choice


ErrorWhen estimating, as seen in the previous section, we notice differences between the

measurement and the actual values. This difference is called the error. For example, if

a line is estimated to have a length of 5 cm when the actual length is 4.8 cm, there is an

error of 0.2 cm.

The significance of the error depends upon the actual length involved. For instance,

an error of 0.2 cm is very significant if the actual length is only 2 cm while it may not

be significant when the actual length is 5 m. The significance of the error is seen when

the error is compared with the actual value.

Absolute relative error =

Absolute percentage error = × 100%

= absolute relative error ¥ 100%

Note: The calculation for the absolute relative error is performed in a pair of vertical

bars. This absolute value is also called the modulus and means that you ignore the sign

of the number. For example, |–7 | = 7 and |7 | = 7.

As mentioned earlier, the error involved in measurement is due to:

• inappropriateness of measuring instruments

• human error in reading or measuring

• inaccuracy of instruments.

Errors can be minimised by using the right instrument and taking a lot of care when

making any measurement.

estimated value actual value–

actual value----------------------------------------------------------------------------


actual value----------------------------------------------------------------------------

Maya measured the distance from the doorway of the classroom to the centre of the

netball court. Her measurement was 58.5 m. If the actual distance is 59.4 m, find:

a Maya’s error

b the absolute relative error, correct to 3 decimal places

c the absolute percentage error.

Continued over page

THINK WRITE

a Calculate the error. This is the

difference between the value obtained

as a result of measurement and the

actual value.

Note: −0.9 indicates the estimated value

was below the actual value. The error

however is recorded as 0.9 m.

a Error = estimated value − actual value

Error = 58.5 − 59.4

Error = −0.9

Error = 0.9 m

2WORKEDExample


When a measurement is taken it will be taken to a certain degree of accuracy. A

rounding error will be a part of every measurement. For example, if measuring a

person’s height, we usually take the measurement to the nearest centimetre. The

maximum error in a measurement is half of the degree of accuracy. That means, if a

measurement is given correct to the nearest centimetre the maximum error is 0.5 cm. If

a person’s height is given as 167 cm, the actual height of the person will be between

166.5 cm and 167.5 cm.

Maximum error in a measurement = of the degree of accuracy.

THINK WRITE

b Find the absolute relative error. This is

the ratio of the error to the actual value.

b Absolute relative error

=

=

= 0.015 151 515 …

= 0.015 (correct to 3 decimal places)

c To calculate the absolute percentage

error, multiply the absolute relative

error obtained in part b by 100%.

c Absolute percentage error

= absolute relative error × 100%

= 0.015 × 100%

= 1.5 %


acual value------------------------------------------------------------------------

0.9–

59.4----------

1

2---

The height of a building is given as 64 metres

correct to the nearest metre. Find:

a the maximum error

b the limits between which the actual height of

the building will lie.

THINK WRITE

a Determine the degree of accuracy and

calculate the maximum error.

Note: The maximum error is half the

degree of accuracy (nearest metre).

a The degree of accuracy is 1 m.

Maximum error = × 1 m

Maximum error = 0.5 m

b Calculate the lower limit; that is,

subtract the maximum error from the

given measurement.

b Lower limit

= measurement – the maximum error

= 64 − 0.5

= 63.5 m

1

2---

1

3WORKEDExample


Tolerance

Tolerance of a quantity is the amount by which a quantity may vary from its normal

value. It provides a socially and legally acceptable level of measurement. A measure-

ment is accepted if it is within normal measurement ± tolerance, otherwise it is

rejected. If a supermarket is selling 500 g of sugar, it does not necessarily mean that

all the bags are exactly 500 g. Consumer Affairs Victoria will not prosecute

businesses for a slight variation, provided the variation is within the national standard

tolerance limits.

THINK WRITE

Calculate the upper limit; that is, add

the maximum error to the given

measurement.

Upper limit

= measurement + the maximum error

= 64 + 0.5

= 64.5 m

Answer the question. The actual height of the building is between

63.5 m and 64.5 m.

2

3

Ashleigh completed a 100 m sprint in 12.25 s, correct to 2 decimal places. What limits does

this measurement actually lie between?

THINK WRITE

Determine the degree of accuracy and

calculate the maximum error.

Note: The maximum error is half the

degree of accuracy.

The measurement has been given correct to

2 decimal places. Therefore, the degree of

accuracy is 0.01 s.

Maximum error = × 0.01 s

Maximum error = 0.005 s

Calculate the lower limit; that is,

subtract the maximum error from the

given measurement.

Lower limit

= measurement − the maximum error

= 12.25 – 0.005

= 12.245 s

Calculate the upper limit; that is, add

the maximum error from the given

measurement.

Upper limit

= measurement + the maximum error

= 12.25 + 0.005

= 12.255 s

Answer the question. The actual measurement lies between 12.245 s

and 12.255 s.

1

1

2---

2

3

4

4WORKEDExample


A supermarket advertises 500 g bags of garlic

with a tolerance of 5 g.

a What is the range of the accepted mass of a bag

of garlic?

b Two particular bags have a mass of 496 g and

492 g respectively. Will these bags of garlic be

accepted or rejected?

THINK WRITE

a State what the accepted mass is.

Note: With a tolerance of 5 g, the 500 g

bags can be 500 ± 5 g.

a Accepted mass = 500 g ± 5 g

Calculate the lowest accepted mass by

subtracting tolerance from the normal mass.

Lowest accepted mass = 500 g − 5 g

= 495 g

Calculate the highest accepted mass by

adding tolerance to the normal mass.

Highest accepted mass = 500 g + 5 g

= 505 g

State the limits of the acceptable mass. 495 g ≤ accepted mass ≤ 505 g

b Compare each given mass with the accepted

mass range and answer the question.

ii 496 g is within the limits of the accepted

mass.

b

ii The 496 g bag is accepted as it lies

within the accepted mass range.

ii 492 g is below the limits of the accepted

mass.

ii The 492 g bag is rejected as it falls

below the accepted mass range.

1

2

3

4

5WORKEDExample

1. The difference between the measurement and the actual value is called the error.

2. Error involved in measurement can be due to inappropriateness of the

instrument, human error or inaccuracy of the instrument.

3. Absolute relative error =

4. Absolute percentage error = × 100%

Absolute percentage error = absolute relative error × 100%

5. Maximum error in a measurement = of the degree of accuracy

6. Actual measurement = measurement ± the maximum error

7. Tolerance of a quantity is the amount by which a quantity may vary from its

normal value.


actual value------------------------------------------------------------------------


actual value------------------------------------------------------------------------

1

2---

remember


Error

1 Sima took five measurements, as shown in the following table. Alongside Sima’s

results the actual values of the measurements are given. For each measurement,

calculate:

i Sima’s error

ii the absolute relative error

iii the absolute percentage error.

2 The studs in a building are supposed to be 1.5 m apart. The distance between two

studs is 1.7 m. Calculate:

a the absolute relative error b the absolute percentage error.

3 Elena, playing golf, estimated that she has a 2 m putt. She actually has a 2.3 m putt.

What is the absolute percentage error involved in Elena’s estimation?

4 A measurement is taken to the nearest centimetre. What is the maximum error in the

measurement?

5 State the maximum error when a measurement is taken:

a to the nearest metre

b to the nearest 10 millimetres

c in centimetres, correct to 1 decimal place

d in metres, correct to 2 decimal places.

6 Jodie is measured as being 156 cm tall, correct to the nearest centimetre. Find:

a the maximum error

b the limits between which her height actually lies.

7 Zdenka completed a 100-m

sprint in 15.36 s, correct to

2 decimal places. Between

what limits does this

measurement actually lie?

8 Between what limits does a

measurement actually lie

when a measurement is

given as:

a 45 m, correct to the

nearest metre?

b 18.5 cm, correct to

1 decimal place?

c 480 km, correct to the

nearest 10 km?

d 29.36 m, correct to

2 decimal places?

Sima’s measurement a 16.5 cm b 26 mL c 28.76 cm d 2.7 kg e 30 s

Actual value 16 cm 25 mL 30 cm 2.5 kg 32 s

9BWORKED

Example

2

Mathcad

Error

WORKED

Example

3

WORKED

Example

4


9

The absolute relative error in the measurement of a height is 0.25. If the measured

height is 70 cm the actual height is:

A 14 cm B 350 cm C 87.5 cm D 17.5 cm E 70.5 cm

10 A wholefood shop advertises 1 kg bags of lentils with a tolerance of 10 g.

a What is the range of the accepted mass of a bag of lentils?

b Two particular bags have a mass of 992 g and 890 g respectively. Will these bags

of lentils be accepted or rejected?

11

Note: There may be more than one correct answer.

Boxes of matches are accepted if they have a minimum of 47 sticks. The advertised

number of sticks is 50. Which of the following statements is correct?

A The maximum number of sticks accepted is 55.

B The tolerance is 5 sticks.

C The tolerance is 3 sticks.

D A box containing 54 sticks is rejected.

E The tolerance is 6 sticks.

12 A family pizza has a normal diameter of 30 cm. Find the minimum and maximum

accepted diameters for the pizza if the tolerance is 2 cm.

13 A chocolate factory has a tolerance of 2 mm on the

length of chocolate bars it makes. None of

the chocolate bars may be less than 198 mm.

a What is the normal length of a chocolate

bar produced by this factory?

b What is the longest bar of chocolate

expected from this factory?

14 Fill in the missing values in this table:

Minimum size Maximum size Normal size Tolerance

20 g ± 2 g

35 mm 40 mm

100 mL ± 5 mL

28 m ± 3 m

multiple choice

WORKED

Example

5

multiple choice


PerimeterUnits of length When we wish to find out how long something is

or to measure the distance between two points,

metric units of length are used. These units are

based on the metre (symbol m).

The metric units of length are shown below:

When converting to a larger unit, divide.

When converting to a smaller unit, multiply.

The ability to work confidently with metric units

is an essential skill in many trades and professions

such as architecture, fashion design, carpentry,

interior design, engineering and the retail trade.

÷ 10

millimetres(mm)

centimetres(cm)

÷ 100

metres(m)

÷ 1000

× 10 × 100 × 1000

kilometres(km)

Complete the following metric length conversions.

a 1.027 m = cm b 0.0034 km = m

c 76 500 m = km d 3.069 m = mm

THINK WRITE

a Look at the conversion table. To convert

metres to centimetres, we need to

multiply by 100. So, move the decimal

point two places to the right.

a 1.027 × 100 = 102.7 cm

b To convert kilometres to metres, we

need to multiply by 1000. So, move the

decimal point three places to the right.

b 0.0034 × 1000 = 3.4 m

c To convert metres to kilometres, divide

by 1000. This can be done by moving

the decimal point three places to the left.

c 76 500 ÷ 1000 = 76.5 km

d Look at the conversion table. To convert

metres to millimetres, we need to multiply

by 100 and then by 10. This is the same as

multiplying by 1000, so move the decimal

point three places to the right.

d 3.069 × 100 × 10 = 3069 mm

6WORKEDExample


Finding the perimeter

The perimeter of a shape is the total distance around that shape.

For example, the perimeter of a football field would be found by measuring the

distance around the boundary fence. The perimeter of a basketball court could be found

by measuring the length of each side of the court and adding all the side lengths.

Find the perimeter of each of the shapes below.

a b c

THINK WRITE

a Make sure that all the measurements

are in the same units and add them

together.

a P = 21 + 15 + 34

= 70

Write the answer in words, including

the units.

The perimeter of the shape shown is 70 mm.

b Notice that the measurements are not

all in the same metric units. Convert

to the smaller unit (in this case

convert 7.3 cm to mm).

b 7.3 cm = 73 mm

Add the measurements. P = 45 + 17 + 28 + 73

= 163


the units.

The perimeter of the shape shown is

163 mm.

c Make sure that given measurements

are in the same units.

c

Determine the lengths of the

unknown sides and label them on the

diagram.

Add the measurements. P = 45 + 18 + 15 + 11 +15 + [40

− (18 + 11)] + [45 − (15 + 15)] + 40

P = 45 + 18 + 15 + 11 +15 + 11 + 15 + 40

P = 170 mm


the units.

The perimeter of the shape shown is

170 mm.

21 mm 15 mm

34 mm

28 mm

17 mm

7.3 cm

45 mm

18 mm

45 mm

15 mm

11 mm

15 mm

40 mm

1

2

1

2

3

1 18 mm

45 mm

15 mm

11 mm

15 mm

40 mm

40 – (18 + 11)

45 – (15 + 15)

2

3

4

7WORKEDExample


Some basic shapes have a formula for perimeter.

The perimeter of a rectangle is given by the formula

P = 2(l + w), where l is the length of the rectangle and

w is its width.

The perimeter of a square is given by the formula P = 4l,

where l is the side length of the square.

l

ww

l

l

ll

l

Find the perimeter of a rectangular block of land that is 20.5 m long and 9.8 m wide.

THINK WRITE

Draw a diagram of the block of land

and write in the measurements.

Write the formula for the perimeter of a

rectangle.

P = 2(l + w)

Substitute the values of l and w into the

formula, and calculate.

P = 2 × (20.5 + 9.8)

P = 2 × 30.3

P = 60.6 m

Write the worded answer with the

correct units.

The perimeter of the block of land is 60.6 m

1 20.5 m

9.8m9.8 m

20.5 m

2

3

4

8WORKEDExample

A rectangular billboard advertising country Victoria has a perimeter of 16 m. Calculate its

width if the length is 4.5 m.

Continued over page

THINK WRITE

Draw a diagram of the rectangular

billboard and write in the

measurements.

P = 16 m

Write the formula for the perimeter of a

rectangle.

P = 2(l + w)

= 2l + 2w

1

ww

4.5 m

2

9WORKEDExample


Perimeter

1 Fill in the gaps for each of the following.

a 20 mm = cm b 13 mm = cm

c 130 mm = cm d 1.5 cm = mm

e 0.03 cm = mm f 2.8 km = m

g 0.034 m = cm h 2400 mm = cm = m

i 1375 mm = cm = m j 2.7 m = cm = mm

k 0.08 m = mm l 6.071 km = m

m 670 cm = m n 0.0051 km = m

THINK WRITE

Substitute the values of P and l into

the formula, and solve the equation:

(a) subtract 9 from both sides

(b) divide both sides by 2

(c) simplify if appropriate.

16 = 2 × 4.5 + 2w

16 = 9 + 2w

16 − 9 = 9 − 9 + 2w

7 = 2w

=

3.5 = w

w = 3.5

Write the worded answer with the

correct units.

The width of the rectangular billboard is 3.5 m.

3

7

2---

2w

2-------

4

1. The metric units of length are millimetres (mm), centimetres (cm), metres (m)

and kilometres (km).

2. Use the table below to convert metric units of length.

3. When converting to a larger unit, divide.

4. When converting to a smaller unit, multiply.

5. The perimeter of a shape is the total distance around that shape.

6. The perimeter of a rectangle is given by the rule P = 2(l + w).

7. The perimeter of a square is given by the rule P = 4l.

÷ 10

millimetres(mm)

centimetres(cm)

÷ 100

metres(m)

÷ 1000

× 10 × 100 × 1000

kilometres(km)

remember

9C

SkillSH

EET 9.4

Multiplying and dividing by powers of 10

SkillSH

EET 9.5

Converting units of length

WORKED

Example

6


2 Chipboard sheets are sold in three sizes. Convert each of the measurements below into

centimetres and then into metres:

a 1800 mm × 900 mm

b 2400 mm × 900 mm

c 2700 mm × 1200 mm.

3 A particular type of chain is sold for $2.25 per metre. What is the cost of 2.4 m of this

chain?

4 Fabric is sold for $7.95 per metre. How much will 4.8 m of this fabric cost?

5 The standard marathon distance is 42.2 km. If a marathon

race starts and finishes with one lap of Stadium Australia,

seen at right, which is 400 m in length, what distance is

run on the road outside the stadium?

6 Maria needs 3 pieces of timber of lengths 2100 mm,

65 cm and 4250 mm to construct a clothes rack.

a What is the total length of timber required, in metres?

b How much will the timber cost at $3.80 per metre?

c

7 Find the perimeter of the shapes below.

a b c

d e f

g h i

j k l

8 Find the perimeter of a basketball court, which is 28 m long and 15 m wide.

9 A woven rectangular rug is 175 cm wide and 315 cm long. Find the perimeter of the

rug.

EXCEL Spreadsheet

Lengthconversions

Mathcad

Lengthconversions

Cabri Geometry

Perimeterof a

rectangle

WORKED

Example

7

3 cm

4 cm

1 cm

5 cm 40 mm

31 mm 35 mm

GC program

– TI

Measurement

GC

program–

Casio

Measurement

SkillSHEET

9.6

Substitutioninto a

formula

1 cm

2 cm

1.5 cm

6 cm

3 cm

2 cm 60 mm

5 mm

11 mm

5.0 cm

4.5 cm

2.0 cm

9 mm

1.5 cm

14 mm

29 mm

4 m530 cm

330 cm

0.6 m

36 cm

346 cm

2.4 m

WORKED

Example

8


10 A line is drawn to form a border 2 cm from each edge of a piece of A4 paper. If the

paper is 30 cm long and 21 cm wide, what is the length of the border line?

11 A rectangular paddock 144 m long and 111 m wide requires a new three-strand wire

fence.

a What length of fencing wire is required to complete the fence?

b How much will it cost to rewire the fence if the wire cost $1.47 per metre.

12 A computer desk needs to have table edging.

If the edging cost $1.89 per metre, find the cost of the

table edging required for the desk.

13 Calculate the unknown side lengths in each of the given shapes.

a b c

Perimeter = 30 cm Perimeter = 176 cm Perimeter = 23.4 m

14 The rectangular billboard has a perimeter of 25 m. Calculate its width if the length is

7 m.

15 The ticket at right has a perimeter of 42 cm.

a Calculate the unknown side length.

b Olivia wishes to decorate the ticket by placing a gold line

along the slanted

sides. How long is the line on each ticket?

c A bottle of gold ink will supply enough ink to draw 20 m of line. How many

bottles of ink should be purchased if 200 tickets are to be decorated?

Circumference

The circumference is the length of the outer boundary of a circle.

The purpose of this activity is to establish a relationship

(if it exists) between the diameter of a circle and its

circumference.

You will need:

Chalk, a 20 m length of string and a measuring tape.

1250 mm

2.1

m 134 c

m

2.1 m

SkillSH

EET 9.7

Solving equations

8 cm

x cm

STOP

x c

m

5.8 m

4.9 m

WORKED

Example

9

11 cm

WorkS

HEET 9.1

COMMUNICATION The diameter of a circle and its circumference — any connection?

Cir

cum

ference

Diameter

Radiu

s


Finding the circumferenceAs a result of the previous activity, you should have found that the ratio of the circum-

ference to the diameter is about 3 for any circle. That is, the length of the circumfer-

ence is about 3 times the length of the diameter.

In fact, the circumference or perimeter of any circle is always about 3.141 59 times

its diameter. The ancient Greeks discovered this ratio and the Greek letter π

(pronounced ‘pie’) is used to represent the number.

The circumference of a circle is given by the formula C = πD, where C is the

circumference and D is the diameter of a circle.

The diameter of any circle is twice as long as its radius; that is, D = 2r, so another

way to write the formula for the circumference of a circle is C = 2πr, where C is the

circumference and r is the radius.

What to do:

1 Working with a partner in a flat area such as

an outdoor basketball court, draw 4 large

circles with different diameters. Keep the

string taut and walk around your partner,

drawing a chalk line as you go.

2 Record the length of the diameter of each

circle into a table, like the one below.

3 Now take the string and lay it carefully around the circumference of the circle.

4 Stretch this length of string into a straight line and use the measuring tape to

find the circumference of each circle. Record your results in the table.

5 Add more values to your table, by drawing four smaller circles in your

workbook with a pair of compasses or a template. Measure the circumference

using a piece of string and a ruler.

6 Calculate the ratio for each circle.

7 Do you observe any patterns? Can you describe the relationship between the

diameter of a circle and its circumference? Write a short statement that will

summarise your findings.

Chalk inloop

C

D----

Circle Diameter (D) Circumference (C)

1

2

3

4

5

6

7

8

C

D----


Note: π cannot be expressed as an exact fraction. It is somewhere between 3 and

3 . Expressed as a decimal, it begins 3.141 592 6535 . . . and goes on forever, with no

repeating pattern of numbers.

For problem solving purposes, 3.14 is a good approximation of the value of π. Alter-

natively, a calculator can be used. (Scientific or graphics calculators usually have a

button, labelled π, which gives a more accurate approximation of this special number.)

1

7---

10

71------

Find the circumference of each of the following circles, giving answers i in terms of π

ii correct to 2 decimal places.

a b

THINK WRITE

a ii Write the formula for the circumference

of a circle.

Note: Since the diameter of the circle is

given, use the formula that relates the

circumference to the diameter.

a ii C = πD

Substitute the value D = 24 into the

formula.

C = π × 24

Write the answer and include the

correct units.

C = 24π cm

ii Write the formula for the circumference

of a circle.

ii C = πD

Substitute the values D = 24 and

π = 3.14 into the formula.

C = 3.14 × 24

Evaluate and include the correct units. C = 75.36 cm

b ii Write the formula for the circumference

of a circle.

Note: Since the radius of the circle is

given, use the formula that relates the

circumference to the radius.

b ii C = 2πr

Substitute the value r = 5 into the

formula.

C = 2 × π × 5

Write the answer and include the

correct units.

C = 10π m

ii Write the formula for the circumference

of a circle.

ii C = 2πr

Substitute the values r = 5 and π = 3.14

into the formula.

C = 2 × 3.14 × 5

Evaluate and include the correct units. C = 31.40 m

24 cm5 m

1

2

3

1

2

3

1

2

3

1

2

3

10WORKEDExample


History of mathematicsARCHIMEDES OF SYRACUSE (c. 287–212 B C)

During his life . . .

Lighthouse of Alexandria built.

Hannibal crosses the Alps.

Archimedes was a Greek mathematician.

He lived in the city of Syracuse in Sicily and

devoted his entire life to research and

experiment. He wrote books about

mathematics and mechanics and was a great

inventor. One of his most famous

achievements was to determine that the value

of π (the ratio between the diameter and

circumference of a circle) is between 3 and

3 . He did this by using a large circle that

he cut into 96 sections.

He was the first to realise the enormous

power that can be exerted by levers and

pulleys. Archimedes said that if he had a lever

long enough he could move the Earth. He

would, however, need to stand somewhere else

other than on the Earth to do it. One of his

inventions, the hydraulic screw, is still used in

parts of the world today. This simple pump is

used to lift water from a lower to a higher

level. It consists of a large screw inside a

cylinder. One end is placed in water and the

screw is turned. As it revolves, water rises up

the spiral threads of the screw.

One of his most important discoveries,

Archimedes’ Principle, is said to have been

made while he was having a bath. As

Archimedes got into the tub one day, he

noticed that the water rose higher up the

sides. He got out of the bath and ran through

the streets shouting ‘Eureka!’ which is Greek

for ‘I have found it!’ Archimedes’ Principle

(or the buoyancy principle) states that the

apparent loss in weight of a body totally or

partially immersed in a fluid is equal to the

weight of the fluid displaced.

When the Romans attacked Syracuse,

Archimedes helped develop machines such as

catapults to defend the city. After three years

the Romans eventually won the war. It is said

that Archimedes was concentrating on a

problem and sketching geometric figures

when interrupted by a Roman soldier. ‘Don’t

disturb my circles!’ he exclaimed. The soldier

drew out his sword and killed him.

A large crater on the moon, more than

80 kilometres wide, is named after him.

Questions

1. What range of values did Archimedes

find for π ?

2. What did Archimedes claim he could

move if he had a lever long enough?

3. What is the device known as

Archimedes’ screw used for?

4. What was Archimedes doing just

before he was killed?

5. Where is the crater Archimedes and

how big is it?

Research

1. Look on the Internet for information

about π and decide how accurate

Archimedes’ value was compared to

the values known now and the values

known then.

2. What are levers and pulleys used for

now? Is there a limit to how big they

can be made?

10

70------

10

71------


Sometimes the outside boundary of a shape includes straight and curved sections. The

curved section may represent a fraction of a full circumference, for example, a half-

circle. In such cases, break the outside boundary up into its curved and straight sec-

tions. The length of each individual part can be calculated, and the length of all the

parts can be added together to give the perimeter.

To enter the value of π, find the button marked Ÿ. First

press the button marked and then press ; that

is, [ p ]. This gives you π. To find the circumfer-

ence of the circle with radius 4 cm, press 2 × [ p ]

× 4 and then press . You should obtain the result

as shown in the screen opposite. Round your answer to

a sensible number of decimal places. The circumference

of a circle with a radius of 4 cm is approximately

25.13 cm.

Find the perimeter of the shape below, correct to 2 decimal places.

THINK WRITE

Identify the parts that constitute the

perimeter of the given shape.

P = circumference + straight-line section

Write the formula for the circumference

of a circle.

Note: If the circle were complete, the

straight-line segment shown would be its

diameter. So the formula that relates the

circumference to the diameter is used.

P = πD + straight-line section

Substitute the values D = 12 and π =

3.14 into the formula.

P = × 3.14 × 12 + 12

To find the perimeter of the given

shape, halve the value of the

circumference and add the length of the

straight section.

P = 18.84 + 12

Evaluate and include the correct units. P = 30.84 cm

12 cm

11

2---

21

2---

31

2---

4

5

11WORKEDExample

Graphics CalculatorGraphics Calculator tip!tip! Calculations involving π

CASIO

Calculationsinvolving π

2 nd Ÿ

2 nd

2 nd

ENTER


Circumference

1 Find the circumference of each of these circles, giving answers i in terms of π


a b c

d e f

2 Find the circumference of each of the following circles, giving answers i in terms of π


a b c

d e f

1. The radius (r), diameter (D) and circumference (C)

of a circle are shown at right.

2. The circumference of a circle, C, is given by the

formula C = π D, or C = 2π r, where D is the

diameter and r is the radius of a circle.

3. π represents the ratio of the circumference of a

circle to its diameter, that is, .

4. The numerical approximation for π is 3.14.

Cir

cum

ference

Diameter

Radiu

s

C

D----

remember

9D

Cabri Geometry

Circumference

Mathcad

Circumference

GC program

– TI

Measurement

GC pro

gram– Casio

Measurement

WORKED

Example

10a

2 cm10 cm

7 mm

0.82 m 7.4 km 34 m

WORKED

Example

10b

4 m

17 mm

8 cm

1.43 km

0.4 m10.6 m


3 Choose the appropriate formula and find the circumference of these circles.

a b c

d e f

4 Find the perimeter of each of the shapes below. (Remember to add the lengths of the

straight sections.)

a b c

d e f

g h i

5

The circumference of a circle with a radius of 12 cm is:

6

The circumference of a circle with a diameter of 55 m is:

A π × 12 cm B 2 × π × 12 cm C 2 × π × 24 cm

D π × 6 cm E π × 18 cm

A 2 × π × 55 m B π × m C π × 55 m

D π × 110 × 2 m E 2 × π × 110 m

77 km 6 m 48 mm

1.07 m

31 mm400 m

WORKED

Example

11

10 cm

16 mm

24 m

11 mm

20 cm

18 cm

1.4 m

1.2 m

50 m

48 m

75 cm

30 cm

multiple choice

multiple choice

55

2------


7 In a Physics experiment, students spin a metal weight around on the end of

a nylon thread. How far does the metal weight travel if it completes 10

revolutions on the end of a 0.88 m thread?

8 A scooter tyre has a diameter of 32 cm. What is the circumference of the tyre?

9 Find the circumference of the seaweed 10 Find the circumference of the Ferris

around the outside of this sushi roll. wheel shown below.

11 Calculate the diameter of a circle (correct to 2 decimal places where appropriate) with

a circumference of:

a 18.84 m b 64.81 cm c 74.62 mm.

12 Calculate the radius of a circle (correct to 2 decimal places where appropriate) with a

circumference of:

a 12.62 cm b 47.35 m c 157 mm.

13 Calculate the radius of a tyre with a circumference of

135.56 cm.

14 Calculate the total length of metal pipe needed to assemble

the wading pool frame shown at right.

15 Nathan runs around the inside lane of a circular track that has a radius of 29 m. Rachel

runs in the outer lane, which is 2.5 m further from the centre of the track. How much

longer is the distance Rachel runs each lap?

8 m8 m

40 cm

r = 1.4 m

MA

TH

S Q

UEST

CHALL

EN

GE

CHALL

EN

GE

MA

TH

S Q

UEST

1 In Around the world in eighty days by Jules Verne, Phileas Fogg boasts that

he can travel around the world in 80 days or fewer. This was in the 1800s,

so he couldn’t take a plane. What average speed is needed to go around

the Earth at the equator in 80 days? Assume you travel for 12 hours each

day and that the radius of the Earth is approximately 6390 km.

2 Liesel’s bicycle covers 19 m in 10 revolutions of her bicycle wheel while

Jared’s bicycle covers 20 m in 8 revolutions of his bicycle wheel. What

is the difference between the radii of the two bicycle wheels?

r = 0.88 m

Weight

Nylon thread

7 m

22��

7

A

22 m 2.75 m 44 m 2.512 m 31.4 m 2.75 m 2.75 m

22��

7

N

7 m

O

22��

7

m

S

22��

7

14 m

D

0.1 m

P

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7

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B

22��

7

1 m

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7

1 m4�—�7

T3��

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3 m

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1.5 m

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0.628 m 22 m 37.68 m 31.4 m

9.42 m 44 m 4.71 m 3 m 31.4 m 44 m 9.42 m m11�—�14

1�—�7

5.5 m 31.4 m

15.7 m 88 m 31.4 m 44 m 22 m 2.75 m9.42 m3 m1�—�7

15.7 m 2.75 m5.652 m

5.5 m 9.42 m 31.4 m 3 m 31.4 m 22 m 5.5 m

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22 m

2 m5�—�14 22 m 37.68 m 50.24 m 44 m

6 m2�—�7

44 m 4.396 m 2.75 m 88 m

5.652 m 5.5 m 9.42 m

Calculate the perimeters (circumferences) of all thecircles using the values of given. Each answer and the letter

inside the circle gives part of the puzzle code.π

=π

=π=π

=π

=π

=π=π=π

=π=π

3.14 =π 3.14 =π 3.14 =π 3.14 =π

3.14 =π3.14 =π

3.14 =π

3.14 =π3.14 =π3.14 =π



Area of rectangles and trianglesMany buildings we see around us are constructed from

rectangles and triangles. For such shapes, we often need to

calculate their surface area.

The area of a shape is the amount of flat surface

enclosed by the shape.

For example, the frame of a window goes around its

perimeter, while the glass represents its area.

Area is measured in units based on the square metre, as

shown in this conversion chart.

Recall 10 000 m2 = 1 hectare

Recall 10 000 m2 = 1 ha

Note: The conversion is the square of the equivalent linear conversion.

Area of a rectangle

The area of a rectangle can be found using the formula A = l × w, where l is the

length and w is the width of the rectangle.

A special case of a rectangle is a square. Since l = w for a square, the formula

becomes A = l × l or A = l2.

squaremillimetres

(mm2)

squarecentimetres

(cm2)

squaremetres

(m2)

squarekilometres

(km2)

× 10002

10002 = 1 000 000 100

2 = 1 0 000 102 = 1 00

× 1002

× 102

÷ 102

÷ 1002

÷ 10002

Complete the following metric conversions.

a 0.081 km2 = ______ m2 b 19 645 mm2 = ______ m2

THINK WRITE

a Look at the metric conversion chart. To

convert square kilometres to square

metres, multiply by 1 000 000; that is,

move the decimal point 6 places to the

right.

a 0.081 km2 = 0.081 × 1 000 000 m2

0.081 km2 = 81 000 m2

b Look at the metric conversion chart. To

convert square millimetres to square

metres, divide by 1 000 000; that is,

move the decimal point 6 places to the

left.

b 19 645 mm2 = 19 645 ÷ 1 000 000 m2

19 645 mm2 = 0.019 645 m2

12WORKEDExample


Area of a triangle

The area of a triangle, A, is given by the formula: A = bh,

where b is the base and h is the height of the triangle.

The base and the height of the triangle are

perpendicular (at right angles) to each other.

Find the area of a rectangle with dimensions shown below.

THINK WRITE

Write the formula for the area of a

rectangle.

A = l × w

Identify the values of l and w. l = 8 and w = 5.6

Substitute the values of l and w into the

formula and evaluate. Include the

appropriate units.

A = 8 × 5.6

A = 44.8 cm2

8 cm

5.6 cm

1

2

3

13WORKEDExample

1

2---

h

b

Find the area of each of these triangles, in the smaller unit.

a b

THINK WRITE

a Write the formula for the area of a

triangle.

a A = bh

Identify the values of b and h. b = 7.5, h = 2.8

Substitute the values of b and h into

the formula.

A = × 7.5 × 2.8

Evaluate. Remember to include the

correct units (cm2).

A = 3.75 × 2.8

A = 10.5 cm2

7.5 cm

2.8 cm

1.8 m

55 cm

11

2---

2

3

1

2---

4

14WORKEDExample


Area of rectangles and triangles

1 Complete the following metric conversions.

a 0.53 km2 = ______ m2 b 235 mm2 = ______ cm2

c 2540 cm2 = ______ mm2 d 542 000 cm2 = ______ m2

e 74 000 mm2 = ______ m2 f 3 000 000 m2 = ______ km2

g 98 563 m2 = ______ ha h 1.78 ha = ______ m2

i 0.987 m2 = ______ mm2 j 0.000 127 5 km2 = ______ cm2

THINK WRITE

b Write the formula for the area of a

triangle.

b A = bh

Convert measurements to cm. 1.8 m = 1.8 × 100 cm

1.8 m = 180 cm

Identify the values of b and h. b = 180, h = 55

Substitute the values of b and h into

the formula.

A = × 180 × 55

Evaluate. Remember to include the

correct units (cm2).

A = 90 × 55

A = 4950 cm2

1

1

2---

2

3

4

1

2---

5

1. The area of a shape is the amount of flat surface enclosed by the shape.

2. Area is measured in units based on the square metre, as shown in this

conversion chart.

Recall 10 000 m2 = 1 hectare

Recall 10 000 m2 = 1 ha

Note: The conversion is the square of the equivalent linear conversion.

3. The area of a rectangle is given by the formula A = l × w, where l is the length

and w is the width of a rectangle.

4. The area of a square is given by the formula A = l 2.

5. The area of a triangle may be found using the rule A = bh, where b is the base

and h is the height of a triangle.

squaremillimetres

(mm2)

squarecentimetres

(cm2)

squaremetres

(m2)

squarekilometres

(km2)

× 10002

10002 = 1 000 000 100

2 = 1 0 000 102 = 1 00

× 1002

× 102

÷ 102

÷ 1002

÷ 10002

1

2---

remember

9E

Cabri Geometry

Area ofa rectangle

SkillSHEET

9.8

Areaof squares,

rectangles andtriangles

WORKED

Example

12


2 Find the area of each of the rectangles below.

a b c

d e f

3 Find the area of each of the squares below.

a b c

Questions 4 and 5 relate to the diagram at right.

4

The height and base respectively of the triangle are:

A 32 mm and 62 mm B 32 mm and 134 mm C 32 mm and 187 mm

D 62 mm and 187 mm E 134 mm and 187 mm

5

The area of the triangle is:

A 2992 mm B 2992 mm2C 5984 mm D 5984 mm2

E 6128 mm2

6 Find the area of the following triangles in smaller units.

a b

c d

WORKED

Example

13

4 cm

9 cm

25 mm

45 mm

3 m

1.5 m

27 km

45 km

50 cm

5 m

2.1 cm

16 mm

GC pr

ogram– TI

Measurement

GC pr

ogram– Casio

Measurement

5 mm

16 cm

2.3 m

187 mm

134 mm32 mm

62 mm

multiple choice

multiple choice

Cabri

Geometry

Area of a triangle

EXCE

L Spreadsheet

Area of a triangle

Mat

hcad

Area of a triangle

WORKED

Example

14a

37 mm

68 mm

40.4 m

87.7 m

85.7 mm

231.8 mm

184.6 cm

1.9 m


e f

7 Zorko has divided his vegetable patch, which is in

the shape of a regular (all sides equal) pentagon,

into 3 sections as shown in the diagram at right.

a Calculate the area of each individual section,

correct to 2 decimal places.

b Calculate the area of the vegetable patch, correct

to 2 decimal places.

8 Find the area of the triangle used to rack up the

billiard balls at right.

9 The pyramid at right has 4 identical triangular faces

with the dimensions shown. Calculate:

a the area of one of the triangular faces

b the total area of the 4 faces.

10 Georgia is planning to create a feature wall in her

lounge room by painting it a different colour. The

wall is 4.6 m wide and 3.4 m high.

a Calculate the area of the wall to be painted.

b Georgia knows that a 4 litre can of paint is sufficient to cover 12 square metres of

wall. How many cans must she purchase if she needs to apply two coats of paint?

11 Calculate the base length of the give-way sign at

right.

12 a Calculate the width of a rectangular sportsground

if it has an area of 30 ha and a length of 750 m.

b The watering system at the sportsground covers

8000 square metres in 10 minutes. How long

does it take to water the sportsground?

0.162 m

142.8 mm

22.7 m

WORKED

Example

14b8.89 m

13.6

8 m

5.23 m

16.3

1 m

28.5 cm

24.7 cm

200 m

150 m

GAME time

Measurement

— 001

45 cm

A = 945cm2


Use π = 3.14 in the following questions.

1 True or false? The conversion of 10 cm to millimetres is 100 mm.

2

The rule for finding the circumference of a circle is:

A π × r B π × 2 C π × D D π × r2E 2 × π × r2

Use the diagram at right for questions 3 and 4.

3 What is the perimeter?

4 What is the area?

5 Calculate the circumference of a circle with radius 3 cm.

6 Calculate the perimeter of the shape at right.

7 Draw a triangle with an area of 6 cm2.

8 Calculate the area of a rectangle with length 8 cm and width 6 cm.

9 What is the total area of 2 square tiles each with a length of 4.5 cm?

10 Find the area of a triangle with base length of 6.5 cm and height 10 cm.

1

multiple choice

4 cm

13 cm

5 m

MA

TH

S Q

UEST

CHALL

EN

GE

CHALL

EN

GE

MA

TH

S Q

UEST

1 A square and an equilateral triangle have

the same perimeter. The side of the triangle

is 3 cm longer than the side of the square.

How long is the side of the square?

2 One playing card is placed over another as

shown. Is the top card covering half, less

than half or more than half of the bottom

card? Explain your answer.


Area of a parallelogramAlthough parallel lines appear to come together in the

distance, parallel lines never meet. In mathematical

figures parallel lines are marked with arrows.

A parallelogram is a quadrilateral (four-sided figure)

with two pairs of parallel sides.

There are a number

of methods we can

use to find the area

of a parallelogram.

Try both of these to

obtain a formula.

Method 1

The parallelogram

shown at right has

been drawn onto

1 cm grid paper.

1 Count the number of squares contained inside the parallelogram.

2 What is the area of each square?

3 Hence, what is the area of the parallelogram?

4 What is the length of the side marked b?

5 What is the height, marked h, of the parallelogram?

6 Calculate the value of h × b and compare this to the value obtained for the area

in part 3.

7 Write a formula for the area of a parallelogram.

Method 2

1 Trace the outline of the

parallelogram at right

onto a sheet

of plain paper.

2 Using a set square or

ruler, draw in the line

labelled h. It must be

perpendicular

(at right angles) to the base of the parallelogram, which is labelled b.

3 Cut out the parallelogram.

4 Cut out the shaded triangle.

5 Fit the triangle onto the other end of the parallelogram to form a rectangle.

6 Explain how you can find the area of a parallelogram. Write a formula for the

area of a parallelogram.

THINKING Area of a parallelogram

b

h

h

b


You may like to investigate parallelograms further by opening the Cabri Geometry file

‘Area of a parallelogram’ on the Maths Quest 8 CD-ROM.

Finding the area of a parallelogram

The area of a parallelogram, A, is given by the formula

A = bh, where b is the length of the base and h is the

height of the parallelogram.

Note: The base and the height of a parallelogram are always perpendicular to each

other.

Area of a parallelogram

1 Find the area of the parallelograms shown below.

a b c

d e f

Cabri

Geometry


h

b

Find the area of the parallelogram shown.

THINK WRITE

Write the formula for the area of a parallelogram. A = bh

Identify the values of b and h. b = 13, h = 6

Substitute 6 for h and 13 for b. A = 13 × 6

Multiply the numbers together and include the

correct units.

= 78 cm2

6 cm

13 cm

1

2

3

4

15WORKEDExample

1. A parallelogram is a quadrilateral with two pairs of parallel sides.

2. The area of a parallelogram is given by the formula A = bh, where b is the

length of the base of the parallelogram and h is its vertical height.

3. The base and the height of any parallelogram are perpendicular to each other.

remember

9F

SkillSH

EET 9.9

Substitution into area formulas

Cabri

Geometry


WORKED

Example

1511 mm

25 mm 120 m

200 m

32 cm

20.5 cm

2.4 mm

4.6 mm

1.8 m

1.5 m

75 mm

32 mm


g h i

2 Find the area of gold braid, needed to make the four military stripes shown.

3 What is the area of the block of land in the

figure at right?

4

Which statement about a parallelogram is false?

A The opposite sides of a parallelogram are parallel.

B The height of the parallelogram is perpendicular to its base.

C The area of a parallelogram is equal to the area of the rectangle whose length is the

same as the base and whose width is the same as the height of the parallelogram.

D The perimeter of the parallelogram is given by the formula P = 2(b + h).

E The area of a parallelogram is given by the formula A = bh.

5 The base of a parallelogram is 3 times as long as its height. Find the area of the

parallelogram, given that its height is 2.4 cm long.

6 A designer vase has a square base of side length 12 cm and four identical sides, each of

which is a parallelogram. If the vertical height of the vase is 30 cm, find the total area

of the glass used to make this vase. (Assume no waste and do not forget to include the

base.)

7 a Find the length of the base of a parallelogram whose height is 5.2 cm and whose

area is 18.72 cm2.

b Find the height of a parallelogram whose base is 7.5 cm long and whose area is

69 cm2.

8 The length of the base of a parallelogram is equal to its height. If the area of the

parallelogram is 90.25 cm2, find its dimensions.

Mathcad

Areaof a

parallelogram

GC program

– TI

Measurement

GC

program–

Casio

Measurement

2.8 m

6.2 m

72 m

68 m

70 m

1.6 m

5.3 m

5.3 m

6 cm

2.1 cm

27 m

76 m

multiple choice


Area of a circle

Finding the area of a circle

The area of a circle, A, can be found using the formula

A = πr2, where π is a constant with a value of

approximately 3.14 and r is the radius of the circle.

1 Use a pair of compasses or template to draw a circle with a radius of about

8 cm.

2 Divide your circle into 20 sectors as shown in the diagram below. (Use your

protractor. Each sector makes an angle of 18° at the centre.)

3 Colour the sectors as shown. Since the circumference of the circle is given by

C = 2πr, the distance around the curved part of the coloured section is half of

this, or πr.

4 Cut out the sectors and arrange them as shown.

5 The shape closely resembles a rectangle, so its approximate area can be found

using the formula A = lw. Since the length of this ‘rectangle’ is πr and the width

is r, the area of the rectangle and, hence, the circle, is given by the rule

A = πr × r

= πr2

Note: If the circle is divided into smaller sectors, the curved sides of the sectors

become straighter and, hence, the shape is closer to a perfect rectangle.

DESIGN Area of a circle

πr

r

r


Using a calculator to find the area of a circleA scientific calculator can make your working out simpler. For example, to find the

area of a circle with a radius of 3.1 m, follow this calculator sequence:

Check this sequence, using your calculator. Your calcu-

lator display should show the number 30.1907054.

(This answer needs to be rounded to a reasonable

number of decimal places, say, 2.)

On a TI-graphics calculator, you would press

[p], enter 3.1, press and then press to obtain

the value of the area which can be seen in the screen at

right.

Find the area of each of the following circles.

a b

THINK WRITE

a Write the formula for the area of a circle. a A = πr2

Substitute 20 for r and 3.14 for π. A = 3.14 × 202

Evaluate (square the radius first) and

include the correct units.

= 3.14 × 400

= 1256 cm2

b Write the formula for the area of a circle. b A = πr2

We need radius, but are given the

diameter. State the relation between the

radius and the diameter.

D = 18; r = D ÷ 2

Halve the value of the diameter to get the

radius.

r = 18 ÷ 2

= 9

Substitute 9 for r and 3.14 for π. A = 3.14 × 92

= 3.14 × 81

Evaluate (square the radius first) and


= 254.34 cm2

20 cm

18 cm

1

2

3

1

2

3

4

5

16WORKEDExample

p × 3 . 1 x2

=

2nd

x2 ENTER

1. The area of a circle is given by the formula A = πr2, where r is the radius of a

circle and π has an approximate value of 3.14.

2. The radius of a circle, r, is equal to a half of its diameter, D: .rD

2----=

remember


Area of a circle

1 Find the area of each of the following circles.

a b c

d e f

2 Find the area of:

a a circle of radius 5 cm b a circle of radius 12.4 mm

c a circle of diameter 28 m d a circle of diameter 18 cm.

3 Annulus is the Latin word for ring. An annulus is the shape formed between two

circles with a common centre (called concentric circles). To find the area of an annulus,

calculate the area of the smaller circle and subtract it from the area of the larger circle.

Find the area of the annulus for the following sets of concentric circles.

9G

SkillSH

EET 9.9

Substitution

into area

formulas

EXCE

L Spreadsheet

Area of

a circle

Mat

hcad

Area of

a circle

Cabri

Geometry

Area of

a circle

WORKED

Example

16

GC pr

ogram– Casio

Measurement

GC pr

ogram– TI

Measurement

12 cm

2.5 km

1.7 m

0.7 cm

58 cm

8.1 mm

An annulus is the shaded area between

the concentric circles.

r1

r2

r1 = radius of smaller circle

r2 = radius of larger circle

Areaannulus = πr22 – πr1

2


a b c

4 Find the area of the following shapes.

a b c

d e f

g h i

5 Find the minimum area of aluminium foil that could be used to cover the top of the

circular tray with diameter 38 cm.

6 What is the area of material in a circular mat of diameter 2.4 m?

7 How many packets of lawn seed should Joanne buy to sow a circular bed of diameter

27 m, if each packet of seed covers 23 m2?

8 A landscape gardener wishes to spread fertiliser on a semicircular garden bed that has

a diameter of 4.7 m. How much fertiliser is required, if the fertiliser is applied at the

rate of 20 g per square metre?

4 cm

2 cm

7 cm

41 cm

50 m

81 m

20 cm1 cm

16 mm

4.2 m

10 cm

42 cm

6 cm

5 cm

7.5 cm 2.5 cm

3 cm

WorkS

HEET 9.2


Area of a trapeziumA trapezium is a quadrilateral (a four-sided figure) with one pair of parallel sides.

The following figures are all trapeziums.

1 Trace two copies of the trapezium below onto plain paper.

2 Cut out the two trapeziums carefully and then paste them together to form a

parallelogram as shown.

3 Recall that the area of a parallelogram is given by the formula: A = bh. So the

area, A, of this parallelogram is: A = (a + b) × h.

4 This parallelogram is made of two trapeziums, so the area of each trapezium is

half the area of the parallelogram; that is, the area, A, of each trapezium is:

A = (a + b) × h.

5 You can investigate whether this relationship holds for different trapeziums by

opening the Cabri Geometry file ‘Area of a trapezium’ on the Maths Quest 8

CD-ROM.

Cabri

Geometry

Area of a trapezium

THINKING Area of a trapezium

a

h

b

b

b

h

a

a

a + b

1

2---


Generally:

The area of a trapezium, A, is given by the formula A = (a + b) × h, where a and

b are the lengths of the parallel sides and h is the height of the trapezium.

Note: The height of the trapezium is always perpendicular to each of its parallel sides.

1

2---

a

b

h

Find the area of the trapezium at right.

THINK WRITE

Write the formula for the area of the

trapezium.

A = (a + b) × h

Identify the values of a, b and h.

Note: It does not matter which of the

parallel sides is a and which one is b,

since we will need to add them

together.

a = 10, b = 6 and h = 4

Substitute the values of a, b and h into

the formula.

A = × (10 + 6) × 4

Evaluate (work out the brackets first)

and include the correct units.

A = × 16 × 4

A = 32 cm2

11

2---

2

31

2---

41

2---

17WORKEDExample

6 cm

4 cm

10 cm

The area of a trapezium is given by the formula A = (a + b) × h, where a and b

are parallel sides and h is the height of a trapezium.

The height of the trapezium is always perpendicular to the parallel sides.

1

2---

a

b

h

remember


Area of a trapezium

1 Find the area of each of the following trapeziums.

a b c

d e f

2

Which of the following is the correct way to calculate the area of the trapezium shown?

3 A dress pattern contains these two pieces.

Find the total area of material needed to make both pieces.

4 A science laboratory has four benches with

the dimensions shown at right.

What would be the cost of covering all four

benches with a protective coating which costs

$38.50 per square metre?

5 Stavros has accepted a contract to concrete and edge the yard,

the dimensions of which are shown in the figure at right.

a What will be the cost of concreting the yard if concrete

costs $28.00 per square metre?

b The yard must be surrounded by edging strips, which cost

$8.25 per metre. Find:

ii the cost of the edging strips

ii the total cost of materials for the job.

A × (3 + 5) × 11 B × (3 + 5 + 11)

C × (11 − 3) × 5 D × (11 + 5) × 3

E × (3 + 11) × 5

9H

SkillSH

EET 9.9

Substitution into area formulas

Mat

hcad

Area of a trapezium

GC pr

ogram– TI

Measurement

GC pr

ogram– Casio

Measurement

WORKED

Example

17 3 cm

2 cm

6 cm

9 m

4.5 m

6 m

5.0 m

3.5 m

3.0 m

18 mm

14 mm

25 mm

0.9 cm

2.4 cm

8.0 cm

50 m

48 m

80 m

multiple choice

3 cm

5 cm

11 cm

1

2---

1

2---

1

2---

1

2---

1

2---

30 cm

30 cm

60 cm

32 cm

47 cm

60 cm

0.84 m

0.39 m

2.1 m

9.2 m

12.4 m

8.8 m

10 m


6 The side wall of this shed is in the

shape of a trapezium and has an area

of 4.6 m2. Find the perpendicular dis-

tance between the parallel sides if one

side of the wall is 2.6 m high and the

other 2 m high.

7

Two trapeziums have corresponding parallel sides of equal length. The height of the

first trapezium is twice as large as the height of the second. The area of the second

trapezium is:

Composite shapesBreaking the shape up into simple shapes and using

the appropriate formulas can assist in finding the

areas of composite shapes. For example, the shape

at right can be found using the formulas for two cir-

cles and a square.

A twice the area of the first trapezium

B half the area of the first trapezium

C quarter of the area of the first trapezium

D four times the area of the first trapezium

E impossible to say

2 m2.6 m

h

2 m

2.6 m

h

multiple choice

MA

TH

S Q

UEST

CHALL

EN

GE

CHALL

EN

GE

MA

TH

S Q

UEST

1 A region at right consists of sevenidentical squares enclosing a totalarea of 252 m2. What is the perimeterof the region?

2 A 20-cm pizza sells for $7.50. At this rate, what will be the price of a28-cm pizza?

3 A Ferris wheel in London called the London Eye has a radius of 65 m.It takes 30 minutes for one complete revolution of the wheel. If youwere riding on this Ferris wheel, what distance would you have trav-elled in 10 minutes?


Note: The sign ≈, used in worked example 18, means ‘approximately equals’.

Find the shaded area for this shape.

THINK WRITE

The shaded area can be found by subtracting

the area of the circle from the area of the

square.

Shaded area = Asquare − Acircle

Write the formula for the area of a square. Asquare = l2

Identify the value of l. l = 2.5

Substitute 2.5 for l and evaluate. Asquare = 2.52

= 6.25 m2

Write the formula for the area of a circle. Acircle = πr2

Calculate the radius from the diameter given

in the question.

r = D ÷ 2

= 2.5 ÷ 2

= 1.25

Substitute 1.25 for r and evaluate. (Use 3.14

as an approximate value for π.)

Acircle = 3.14 × 1.252

= 4.906 25 m2

Subtract the area of the circle from the area

of the square to find the shaded area.

Shaded area = 6.25 − 4.906 25

= 1.343 75

Round the answer to 2 decimal places and

include the units.

≈ 1.34 m2

2.5 m

1

2

3

4

5

6

7

8

9

18WORKEDExample

Find the area of this shape.

THINK WRITE

The dotted line divides the figure into a

rectangle and a trapezium. To find the total

area, find the area of the rectangle and the

area of the trapezium and add them together.

Total area = Arectangle + Atrapezium

1.8 cm 4.2 cm

3 cm

1.7 cm

1

19WORKEDExample


Composite shapes

1 Find the shaded area for each of the following shapes.

a b c

d e f

THINK WRITE


rectangle.

Arectangle = lw

Identify the values of l and w. l = 3, w = 1.8

Substitute 3 for l and 1.8 for w into the

formula and evaluate.

Arectangle = 3 × 1.8

= 5.4 cm2


trapezium.

Atrapezium = (a + b) × h

Identify the values of the

pronumerals.

a = 1.8, b = 4.2, h = 1.7

Substitute 1.8 for a, 4.2 for b and

1.7 for h into the formula and evaluate.

Atrapezium = (1.8 + 4.2) × 1.7

= × 6 × 1.7

= 3 × 1.7

= 5.1 cm2

Add the two areas together to get the

total area. Include the correct area

units.

Total area = 5.4 + 5.1

= 10.5 cm2

2

3

4

51

2---

6

7

1

2---

1

2---

8

1. Many problems, such as finding a shaded area, can be solved by subtracting

one area from another.

2. Many composite shapes can be divided into two or more simpler shapes with

simple area formulas.

remember

9IEX

CEL Spreadsheet

Area

GC program

– TI

Measurement

WORKED

Example

18

GC

program–

Casio

Measurement

4 cm

28 m

m

40 mm

52 m

13 m

3.8 m

6.0 m

50 mm

16 mm88 m


g h

2 Find the area of each of the following shapes.

a b c

d e f

g h i

3 Michael is paving a rectangular yard, which is 15.5 m long and 8.7 m wide. A circular

fishpond with a diameter of 3.4 m is to be placed in the centre of the yard. What will be

the cost of paving Michael’s yard if the paving material costs $17.50 per square metre?

12 cm18 cm

30 cm

15 cm

16 cm

WORKED

Example

19

60 cm

36 cm

20 cm

6 cm

15 cm15 cm

9 cm 40 m

60 m

13 m

14 m

30 m

22 cm 28 cm

25 cm

10.5 cm

10 cm

7.5 cm15 cm

17.5 cm

30 cm

50 cm30 m

24 m

60 m


4 A farmer wishes to sow the paddocks with three different types of seed with different

costs as shown. What will be the total cost (to the nearest dollar) of the seed for the

three paddocks with the dimensions shown below?

5 Find the area of the theatre stage shown at right.

6 What will be the cost of carpeting all three rooms

shown, if the carpet costs $28.00 per square metre?

(Assume that there is no wastage or overlap.)

At the start of the chapter, did you try drawing

three different floor plans for a

one-bedroom unit that is to cover an area of

60 m2? If not, try it now.

Let’s design this one-bedroom unit in more

detail. The unit should have two entrance

doors and at least four rooms as described in

the table. Any hallways you include should be

between 1.0 and 1.2 m wide.

1 For each of the three possible floor plans you drew, make a rough sketch

showing the locations of the rooms. Think creatively about a unit you would

like.

2 Select one of your rough sketches and make a detailed plan of the one-bedroom

unit using graph paper. Determine the exact dimensions and placement of the

rooms. (Remember to keep the specifications in mind.) Label each room with its

name, dimensions and area.

3 Finish your plan by showing the locations of windows and doors.

4 Take measurements of the rooms where you live or at school. How do they

compare to those in your plan?

108 m

400 m

120 m

152

m

160 mBarley0.4 cents per m2

Corn $1.05 per m2

Oats0.65 cents

per m2

6 m

8 m

6 m

22 m

2 m

8.0 m 7.4 m5.0 m

5.0 m

6.0 m

19.0 m

DESIGN Designing a one-bedroom unit

Minimum room areas

Living room 30 m2

Bedroom 9 m2

Kitchen 6 m2

Bathroom 5 m2


Use π = 3.14 in the following questions. Where appropriate, state your answer correct to 2

decimal places.

1 True or false? The conversion of 5 km to metres is 5000 m.

2 Calculate the circumference of a circle with diameter of 5 cm.

3 Find the area of a rectangle with length 11 cm and width 6 cm.

4 Calculate the area of a triangle with base length of 12.5 m and height of 7 m.

5 Calculate the area of a photograph frame constructed

from 2 identical parallelograms as shown at right.

6 Find the area of a circle with a radius of 13 mm.

7 Calculate the area of the trapezium shown below.

8 Find the total area of the following shape.

9 Find the shaded area in the figure below.

10 What is the area of icing

needed to cover the annulus

as marked on the donut at

right?

2

11 cm

8 cm

4 cm

17 cm

13 cm

5 m

2 m 2.5 m

5 m

1 m

7 cm

3 cm


Total surface area The area formulas we have discussed in this chapter can also be used for finding the

total surface area of 3-dimensional objects.

Total surface area is the area of all outside faces of a 3-dimensional object.

To find the total surface area of an object, we need to identify the shapes of the outside

faces of the object first. We then need to find the area of each face separately (using

formulas discussed earlier in the chapter) and add them all together.

Consider, for example, a cube. It is a 3-dimensional object. Its surface area can be

thought of as the area to be painted, if we were to paint the cube all over.

The total surface area of the cube consists of six square faces. The area of each face

is given by the formula A = l2, so the total surface area (or TSA) of the cube can be

found by adding the areas of the six faces, or using the formula: TSAcube = 6l2.

Find the total surface area (TSA) of the solid shapes below.

a b

Continued over page

THINK WRITE

a Write the formula for the TSA of a

cube.

a TSAcube = 6l2

Identify the value of l. l = 5

Substitute 5 for l. TSA = 6 × 52

Evaluate (square 5 first, and then

multiply by 6) and include the

correct units.

= 6 × 25

= 150 cm2

5 cm

5 cm 5 cm

6 cm

4 cm4 cm

1

2

3

4

20WORKEDExample


Total surface area

1 Find the total surface area of the solid shapes below.

a b c

THINK WRITE

b The total surface area of a square

pyramid consists of four identical

triangles and a square (at the base).

Write a brief statement about what is

to be found.

b TSApyramid = 4 × Atriangle + Asquare


triangle.

Atriangle = bh

Identify the values of the

pronumerals.

b = 4, h = 6

Substitute 4 for b and 6 for h. Atriangle = × 4 × 6

Evaluate. = 12 cm2


square.

Asquare = l2

Identify the value of l. l = 4

Substitute 4 for l. Asquare = 42

Evaluate. = 16 cm2

Since there are 4 identical triangular

faces, multiply the area of one

triangle by 4 and then add the area of

the square to get the total surface

area of the pyramid.

TSA = 4 × 12 + 16

Evaluate and include the correct

units.

= 48 + 16

= 64 cm2

1

21

2---

3

41

2---

5

6

7

8

9

10

11

1. The total surface area of a 3-dimensional object is the area on the outside of the

object. It is the sum of the areas of each of the individual faces.

2. The total surface area of a cube is given by the formula TSAcube = 6l2.

remember

9J

SkillSH

EET 9.10

Total surface area of cubes and rectangular prisms

WORKED

Example

20a

40 cm

4.5 m

87 m


2 Find the total surface area of the solid figures below. The base of each shape is square.

a b c

3 A baby stacks up two building blocks as shown at right.

What is the surface area of the stack?

4 A rectangular piece of cheese is cut in half diagonally. What is

the surface area of a piece of cheese shown in the diagram below?

5 The roof of a typical beach hut is constructed from two rectangular sections. Its walls

are made from four rectangular sections and two triangular sections at the ends of the

roof as shown below. What would be the minimum cost of painting the external walls

and roofs of 3 typical beach huts with the same paint which costs $49.95 for a 4 litre

can? (One 4 litre can covers 55 m2.)

6 An open, cylindrical water tank is made from

two pieces of corrugated iron as shown at right.

Calculate the area of iron required to make the

tank.

GC

program–

Casio

Measurement

SkillSHEET

9.11

Totalsurfacearea of

triangularprisms

WORKED

Example

20b

GC program

– TI

Measurement

8 cm

5 cm

1.5 cm

3.7 cm82 m

80 m

8 cm17 cm

12 cm Hint: Use the formula for the area

of a triangle to find the area of

each triangular face.

2 m

2.5 m

2.5 m

1.5 m

4.5 m

2 m

Base6.28 m

Walls2 m

3 m


Volume of prisms and other shapes

Volume is the amount of space inside a

3-dimensional object.

Volume is measured in units such as cubic

centimetres (cm3) and cubic metres (m3).

Dividing a 3-dimensional shape into small cubes

with sides of length 1 cm allows us to calculate its

volume. For example, the cube at right has a

volume of 27 cm3.

Often a formula can be used to calculate the

volume of a 3-dimensional object.

Prisms

Prisms are solid shapes with identical opposite ends joined by

straight edges. They are 3-dimensional objects that can be cut

into identical ‘slices’, called cross-sections.

For example, the cube consists of layers or slices that are all squares of equal size.

The figures below are all prisms. They are named according to the shape of their

base (or cross-section).

These figures are not prisms. Can you see why?

The volume of a rectangular prism (cuboid) can be found using the formula V = lwh

where l, w and h are the length, width and height of the rectangular prism respectively.

Another way of stating this formula is:

V = area of base × height (where area of the base = lw).

Triangular prism

Rectangular prism

Hexagonal prism

Sphere ConeSquare pyramid


The following rule applies to any prism.

The volume of a prism is given by the formula: V = A × H, where A is the

cross-sectional area of a prism and H is the height of a prism.

Note: A prism might be positioned in

different ways; that is, it may stand on

the face that represents the shape of

the identical layers, or it may not. In

other words, the base of the prism (the

face on which it stands) may or may

not be its cross-section. This is why we

use the expression ‘cross-sectional

area’, rather than ‘area of the base’.

Also note that the dimension to

which we refer as the ‘height of a

prism, H’, is not necessarily the height

of the prism in the true sense of this

word. It is just a dimension that is per-

pendicular to the cross-section. If, for

example, a prism does not stand on its

cross-section, this dimension would

physically represent the depth (width

or length) rather than the height of the

prism.

The formula V = AH works whatever the shape of each end of the object is, as long

as the object has a uniform cross-sectional area.

The following figures have a uniform cross-sectional area. However, their ends are

not joined by straight edges and therefore cannot be classed as prisms.

A

H

A

HV = A × H

The base of this prism is its cross-section; H is the height of the prism.

The base of this prism is not its cross-section; H represents the depth (or length) of the prism.


Find the volume of each of the following.

a b c

THINK WRITE

a Write the formula for the volume of the

given shape.

a V = A × H

Identify the shape of the cross-section and,

hence, write the formula to find its area

Acircle = πr2

State the value of r. r = 3

Substitute the value of r into the formula

and evaluate.

A = 3.14 × 32

= 28.26 cm2

State the value of H. H = 5

To find the volume, multiply the cross-

sectional area by the height and include the

correct units.

V = 28.26 × 5

= 141.3 cm3

b Write the formula for the volume of a

prism.

b V = A × H

Identify the shape of the cross-section and,

hence, write the formula to find its area.Atriangle = bh

State the values of the pronumerals.

(Note: h is the height of the triangle, not of

the prism.)

b = 7, h = 8

Substitute the values of b and h into the

formula and evaluate.

A = × 7 × 8

= 28 cm2

State the value of H, the height of the

prism.

H = 12

To find the volume of the prism, multiply

the cross-sectional area by the height and


V = 28 × 12

= 336 cm3

c Write the formula for the volume of the

given shape.

c V = A × H

State the values of the cross-sectional area

and the height of the shape.

A = 13, H = 7

Multiply the cross-sectional area by the

height and include the correct units.

V = 13 × 7

= 91 cm3

5 cm

3 cm 8 cm

12 cm

7 cm

7 cm

A = 13 cm2

1

2

3

4

5

6

1

2 1

2---

3

41

2---

5

6

1

2

3

21WORKEDExample


Volume of prisms and other shapes

1 Which of the 3-dimensional shapes below are prisms?

a b c

d e

2 Find the volume of each of the following.

a b c

d e f

1. Volume is a measure of the amount of space inside a 3-dimensional object.

2. Volume is measured in cubic units, such as cubic centimetres (cm3) and cubic

metres (m3).

3. Prisms are solid shapes with identical opposite ends joined by straight edges.

They are 3-dimensional figures with identical layers or cross-sections.

4. The volume of a prism is given by the formula V = A × H, where A is the

cross-sectional area and H is the height of the prism (a dimension,

perpendicular to the cross-section).

5. Use an area formula appropriate to each object to find the cross-sectional area.

remember

9K

Mathcad

Volumeof a

prism

GC program

– TI

Measurement

GC pro

gram– Casio

Measurement

EXCEL Spreadsheet

Volumeof a

prism

WORKED

Example

21

6 cm

A = 14 cm2

4.5 m

A = 18 m2

SkillSHEET

9.12

Volumeof

cylinders

20 cm

15 cm

40 cm

25 cm10.5 m

9 m13 cm

9 cm


g h i

j k l

3 What volume of water will a rectangular swimming pool with dimensions shown in the

photograph below hold if it is completely filled? The pool has no shallow or deep end.

It is all the same depth.

4 How many cubic metres of cement will be needed to make

the cylindrical foundation shown in the figure at right?

5 What are the volumes of these pieces of cheese?

a b

6 cm

7 cm

4 cm

8 cm

5 cm

6 cm

SkillSH

EET 9.13

Volume of triangular prisms

8 m

8 m

10 m

7 cm

6 cm

6 cm

SkillSH

EET 9.14

Volume of cubes and rectangular prisms

2.5 cm

1.5 cm

2.0 cm

1.25 m

1.0 m

1.5 m

2.4 m

20 m

25 m

8 m

1.2 m

6 cm

5 cm

4 cm

8 cm

MOO�

CHEESE

5.0 cm

8.0 cm

4.2 cm


6 What is the volume of the bread 7 How much water will this pig trough,

bin shown below? with dimensions shown in the figure

below, hold if it is completely filled?

A radio station offers a choice of prizes for winning a competition. You are able to

win the value of a vertical stack of coins with a maximum volume of 500 cm3. The

decision to be made is whether to take the value of a stack of 5-cent, 10-cent or

20-cent coins.

Which coin would you go for to win the most prize money? Show all your

working.

GAME time

Measurement

— 002

WorkS

HEET 9.3

BREAD

18 cm

29 cm

30 cm

15 cm

15 cm55 cm

THINKING Prize money

Dimensions of the coins

Coin Diameter Approximate

height

5-cent coin 19 mm 1 mm

10-cent coin 23 mm 1.5 mm

20-cent coin 28 mm 2 mm

MA

TH

S Q

UEST

CHALL

EN

GE

CHALL

EN

GE

MA

TH

S Q

UEST

1 The areas of the three sides of a rectangular box are as

shown in the figure. What is the volume of the box?

2 A vase is shaped like a rectangular prism

with a square base of length 11 cm. It has

2 litres of water poured into it. To what

height does the water reach in the vase?

(Hint: 1 litre = 1000 cm3.)

150 cm2

180 cm2

270 cm2


Copy the sentences below. Fill in the gaps by choosing the correct word or

expression from the word list that follows.

1 The accuracy of measurements in practical situations depends upon using

an appropriate instrument, the and the .

2 The difference between a measurement and the actual measurement is

called the .

3 Absolute relative error is .

4 Absolute percentage error is .

5 The of a shape is the distance around the outside boundaries

of the shape.

6 The units of length are millimetres (mm), centimetres (cm),

metres (m) and kilometres (km).

7 The of a circle is given by the formulae C = πD or C = 2πr.

8 The of a 2-dimensional object is measured in square units,

such as square millimetres (mm2), square centimetres (cm2), square

metres (m2) and square kilometres (km2).

9 The area of a rectangle is given by the formula: .

10 The area of a triangle is given by the formula: .

11 The area of a parallelogram is given by the formula:

.

12 The area of a trapezium is given by the formula: .

summary

l

w

h

b

b

h

b

a

h


13 The area of a square is given by the formula: .

14 The area of a circle is given by the formula: .

15 The of a 3-dimensional object is the area on the outside of

the object. It is the sum of the areas of each of the individual faces.

16 The total surface area of a is given by the formula TSA = 6l2.

17 Surface area is measured in the same as other areas, for

example, square centimetres (cm2).

18 Volume is a measure of the amount of inside a 3-dimensional

object.

19 Volume is measured in such as cubic centimetres (cm3) and

cubic metres (m3).

20 Prisms are solid shapes with identical opposite ends joined by

edges. They are 3-dimensional figures with identical or cross-

sections.

21 The volume of a is given by the formula V = A × H.

l

r

W O R D L I S T

cubic units

A = (a + b) × h

metric

error

prism

cube

A = bh

layers

perimeter

area

accuracy of the

reading

A = l2

total surface area

(TSA)

A = πr2

accuracy of the

instrument

circumference

A = lw

space

units

A = bh

straight

1

2---

1

2---

¥ 100% estimated value actual value–

actual value-----------------------------------------------------------------------------------


actual value-----------------------------------------------------------------------------------


1 Estimate the angle labelled x in each of the following figures. Measure the angle and discuss

the accuracy of your estimation.

a b c

2 Estimate and draw (without measuring) lines of lengths:

a 20 cm b 4 cm.

Now measure the lengths with the ruler to check your estimations.

3 Estimate the length and the width of a desk in your classroom. Measure the dimensions and

discuss the accuracy of your estimate.

4 Irene estimated her height to be 154 cm. If her actual height is 156 cm, calculate the:

a error b absolute relative error c absolute percentage error.

5 A measurement is given as 25 ± 1 mm.

a What is the lower limit? b What is the upper limit?

6 Michael sells 500 g bags of green beans. He is allowed a tolerance of 5 g.

a What are the accepted limits of mass for the bean packets?

b Lana buys a bag of beans from Michael and discovers that it has a mass of only 496 g.

She lodges a complaint. Will the bag of beans be accepted as a legitimate sale?

7 A rectangle has its length and width given as 30 cm ± 2 cm and 20 cm ± 2 cm.

a What is the largest possible perimeter for this rectangle?

b What is the smallest possible perimeter for this rectangle?

8 Convert each of the following to the units shown in brackets.

a 5.3 mm (cm) b 7.6 cm (mm) c 15 cm (m) d 4.6 m (cm)

e 250 m (km) f 6.5 km (m) g 1.5 m (mm) h 12 500 cm (km)

9 Find the perimeter of the shapes below. Where necessary, change to the smaller unit.

a b c

10 Find the circumference of each of these circles.

a b c c

9A

CHAPTERreview

x

xx

9A

9A

9B

9B

9B

9B

9C

9C

2.2 m

3.6 m

18 cm

20 cm

10 cm2.4 cm 35 mm

5.2 cm

9D

11 cm44 mm

18 m


0.5 m

58 cm

11 Find the perimeter of these shapes.

a b c

12 A give-way sign is in the shape of

a triangle with a base of 0.5 m. If

the sign is 58 cm high, find the

amount (in m2) of aluminum

needed to make 20 such signs.

Assume no waste.

13 Find the area of the following triangles.

a b c

14 Restaurant owners want a dome like the one below over their new kitchen. Red glass is more

expensive and they want to estimate how much red glass is needed for the dome. The small

sides of the red triangles are 40 cm, the longer sides are 54 cm and their heights are 50 cm.

The trapeziums around the central light are also 50 cm high. The lengths of their parallel

sides are 30 cm and 20 cm. Calculate the area of:

a the red triangles b the red trapeziums c the total area of red glass in m2.

9D

94 m

63 m

42 m

8 cm

9E

9E

22 cm

57 cm

12 m

16 m

64 cm

42 cm

9E,H

20 cm30 cm

40 cm

54 cm

50 cm

50 cm


15 Find the area of this parallelogram.

16 What area of cardboard would be required to make

the poster shown at right?

17 Find the area of each of the circles in question 10.

18 Find the area that the 12-mm-long minute hand of

a watch sweeps out in one revolution.

19 Find the area of these trapeziums.

a b

20 Of the two parallel sides of a trapezium, one is 5 cm longer than the other. Find the height

of the trapezium if the longer side is 12 cm and its area is 57 cm2.

21 The diagram shows a design for a brooch.

a What is the total area of the brooch in square millimetres?

b If the brooch were to be edged with gold, what length of

gold strip would be needed for the edge?

22 Find the total surface area (TSA) of the following 3-dimensional objects.

a b c

23 A rectangular toy box with no lid is to be painted all over (inside and outside). If the box is

1.2 m long, 60 cm wide and 80 cm tall, find the total area that needs to be painted.

24 Find the volume of each of the following.

a b c

25 A narrow cylindrical vase is 33 cm tall and has the volume of 2592 cm2. Find (to the nearest

cm) the radius of the base of the vase.

20 m

50 m

9F

9FSquash all

squares...

80 cm

88 cm

9G9G

9H37 cm

54 cm

27 cm19 m

38 m

65 m

9H

28 mm

25 mm

9I

9J

2.5 cm

5.6 cm

3 cm8.5 cm

9J

9K64 cm

35 cm

26 cm

28 cm

22 cm

2.8 cm

A =

3 cm2

testtest

CHAPTER

yourselfyourself

9

9K

Chap 09

Documents

accurate instrument

accurate reading

accurate result

following measurements

measuring tape

measuring cylinder

liquid level

measuring angles