Perimeter, area and volume 6 - WordPress.com · n 0.0051 km = m 3 Particle board ... Note: For the...

6Majid has the job of maintaining the pool at an aquatic centre. This includes replacing water and keeping the pool walls clean. How many litres of water are in anOlympic pool and what is thearea of wall that needs to be kept clean? Record an estimate of each of these.

In this chapter, you will further develop your skills in calculating length, perimeter, area and volume and apply them to many everyday situations.

Perimeter, area and volume

MQ 9 NSW 5.2P - 06 99 Sunday, July 25, 2004 8:29 AM

200

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

READY?are youAre 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 9

CD-ROM or ask your teacher for a copy.

Converting units of length

1

Convert each of the following measurements.

a

54 m to cm

b

8.7 cm to mm

c

25 mm to m

Substitution into perimeter formulas

2

Substitute the values of the pronumerals into each of the following perimeter formulas and hencefind the value of

P

(or

C

).

a

P

=

4

l

where

l

=

5

b

P

=

2(

l

+

b

) where

l

=

7 and

b

=

2

c

C

=

π

d

where

d

=

10 and

π

≈

3.142

d

C

=

2

π

r

where

r

=

4.1 and

π

≈

3.142

Perimeter of squares, rectangles, triangles and circles

3

Find the perimeter of each of the following shapes correct to 2 decimal places where necessary.

a b c d

Substitution into area formulas

4

Substitute the values of the pronumerals into each of the following area formulas and hence findthe value of

A

.

a

A

=

l

2

where

l

= 6

b

A

=

lb

where

l

= 11 and

b

= 4

c

A =

bh

where

b

= 8 and

h

= 13

d

A

=

π

r

2

where

r

= 15 and

π

≈

3.142

Area of squares, rectangles, triangles and circles

5

Find the area of each of the shapes shown in question

3

.

Surface area of cubes and rectangular prisms

6

Find the surface area of each

a b

of the prisms at right.

Volume of cubes and rectangular prisms

7

Find the volume of each of the prisms in question

6

.

6.1

6.3

6.5

5 cm

8 mm

3 mm

25 m

15 m20 m12 m

3 m

6.7

12---

6.9

6.10

4 cm

3 m

7 m5 m

6.12


C h a p t e r 6 P e r i m e t e r , a r e a a n d v o l u m e

201

Metric units of length

The metric system of units for measuring length is based on the metre (symbol m). Themetric system was devised by the French in 1791 and is now used all over the world.The metric system of units is convenient to use because it is based on powers of 10.

The metre was originally defined as one ten-millionthof the Earth’s quadrant (a quarter of the circumference), but it is now defined more accurately in terms of the wavelength of light. This allows very accurate measurement of small distances.

Four units of length, based on the metre, are commonly used:millimetre (mm) about the width of this letter ‘i’centimetre (cm) about the thickness of an exercise

book or your little fingermetre (m) about the length of one adult’s stridekilometre (km) 2 laps of an athletics track.These units are related as follows:1 cm

=

10 mm 1 m

=

100 cm 1 km

=

1000 m.The chart below can be used to make conversions from one unit to another easier.

Conversion of metric units of length

Note:

When converting to a smaller unit, we multiply and when converting to a largerunit, we divide.

q

q = 1 metre1071–—

12---

(mm)(cm)(m)(km)

× 1000 × 100 × 10

÷ 10÷ 100÷ 1000

Fill in the gaps: a 30 mm = _____ cm b 2.54 km = _____m.

THINK WRITE

a To convert from mm to cm, we divide by 10. a 30 mm = 3.0 cmb To convert from km to m, we multiply by 1000. b 2.54 km = 2540 m

1WORKEDExample

remember1. The metre is the standard unit of length in the metric system.2. (a) When converting to a larger unit, divide.

(b) When converting to a smaller unit, multiply.3. 1 cm = 10 mm 1 m = 100 cm 1 km = 1000 m

remember

MQ 9 NSW 5.2P - 06 Page 201 Sunday, July 25, 2004 8:29 AM

202

M a t h s Q u e s t 9 f o r N e w S o u t h W a l e s 5 . 2 P a t h w a y


1

Measure and record the length of 10 different objects in the classroom. Choose an appropriate unit for each length measurement.

2

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

3

Particle board sheets are sold in three sizes. Converteach of the measurements below into centimetres andthen into metres:

a

1800 mm

×

900 mm

b

2400 mm

×

900 mm

c

2700 mm

×

1200 mm.

4

A particular type of chain is sold for $2.25 per metre.What is the cost of 240 cm of this chain?

5

The standard marathon distance is 42.2 km. If amarathon race starts and finishes with one lap ofStadium Australia, which is 400 m in circumference,what distance is run on the road outside the stadium?

6

Fabric is sold for $7.95 per metre. How much will4800 mm of this fabric cost?

7

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?

8

Timber is sold in standard lengths, which increase in 300 mm intervals from thesmallest available length of 900 mm. (The next two standard lengths available aretherefore 1200 mm and 1500 mm.)

a

Write down the next 4 standard lengths (after 1500 mm) in mm, cm and m.

b

How many pieces of length 600 mm could be cut from a 2.4 m length of timber?

c

If I need to cut 8 pieces of timber each 41 cm long, what is the smallest standardlength I should buy?

6A

SkillSH

EET 6.1

Convertng units of length

WORKEDExample

1

Mathca

d


Cabri

Geometry

Changingunits

EXCEL

Spreadsheet

Lengthconverter

MQ 9 NSW 5.2P - 06 Page 202 Tuesday, July 27, 2004 11:04 AM

C h a p t e r 6 P e r i m e t e r , a r e a a n d v o l u m e 203

9 An interior designer wishes to cover a window which is 3.6 m wide with curtainmaterial, which is 900 mm wide. How many widths of material will be needed if thecurtains are to be 180 cm wider than the window, to allow for seams and gathering?

10 In 1991, the world 10 000 m record of 27:08.23 (27 min, 8.23 s) was held by ArturoBarrios of Mexico.a How long did Barrios take to cover 100 m at this pace, assuming that his speed was

constant? b How many 400 m laps do the runners complete in this event?c How long did Barrios take to cover each kilometre at this pace, assuming that his

speed was constant?d What was his average speed in kilometres per hour?

11 The world’s longest bridge is the Akashi-Kaikyo Bridge which links the Honshu andShikoku islands of Japan. Its central span covers 1.990 km.a How long is the central span, in metres?b How much longer is the span of the Akashi-Kaikyo Bridge than the Sydney

Harbour Bridge, which spans 1149 m?

12 The orange roughy is a valuable fish found in Australian waters at depths of up to1.2 km. The Southern Bluefin Tuna inhabits the top 30 metres of the ocean. a What is the difference between the depths at which these different fish live?b How long will it take to retrieve a fishing line from a depth of 1.2 km if it is drawn

into a boat at the rate of 60 m per minute?

MA

TH

SQUEST

C H A L L

EN

GE

MA

TH

SQUEST

C H A L L

EN

GE

1 Which would you rather have: a kilometre of 20-cent pieces lined upside-by-side or a kilometre of 10-cent pieces?

2 How long is the night when the sun sets at 6.39 pm and rises at 7.16am?

3 A special camera takes 10 000 000 pictures each second. A bullet travel-ling at a speed of 500 m/s was photographed for a distance of 1 cm.How many seconds of normal-speed projection (24 pictures per second)would be required to show just 1 cm of the bullet’s travel?


204 M a t h s Q u e s t 9 f o r N e w S o u t h W a l e s 5 . 2 P a t h w a y

PerimeterThe perimeter of a shape is the total distance around the outside of the shape.

For example, the perimeter of the triangle at right is 25 + 53 + 36 = 114 mm.

In many cases, it is necessary to measure each side and add the side lengths together to find the perimeter of a shape.

Some basic shapes have a formula for perimeter. These are shown in the table below.

Note: For the circle we use the term circumference instead of perimeter. When cal-culating π without a calculator, an approximate value such as 3.142 can be used. Whenusing a formula we begin by writing the formula, next we substitute the known valuesinto the formula and then evaluate the expression.

Square Rectangle Circle

Perimeter = 4l Perimeter = 2(l + b) Circumference = 2πr = πd

25 mm36 mm

53 mm

Find the perimeter of this figure, in millimetres.

THINK WRITE

Measure each side accurately and write the lengths in mm, as required in the problem.

To find the perimeter, add the side lengths together.

P = 50 + 43 + 25 = 118 mm

1

43 mm

50 mm25 mm

2

2WORKEDExample

l

l

ll

l

b

r

d



There are examples of other figures for which we use a combination of formula andaddition of extra sides.

Find the circumference of the circle at right. Give your answer correct to 2 decimal places.

THINK WRITE

Since radius is known, choose the formula for the circumference that contains radius.

C = 2πr

Identify the value of the pronumeral. r = 2.5Substitute the value of the pronumeral into the formula.

C = 2 × π × 2.5

Use a calculator to multiply and include the appropriate unit.

C ≈ 15.71 cm

2.5 cm

1

23

4

3WORKEDExample

Find the perimeter of the shape shown at right. Give your answer correct to 2 decimal places.

THINK WRITE

The perimeter of the given shape consists of two straight sections and 2 semicircles. The 2 semicircles together make up a full circle, whose diameter is known. Write the formula for the circumference that contains diameter.

C = πd

State the value of the pronumeral. d = 25Substitute the value of the pronumeral into the formula and evaluate using a calculator.

C = π × 25C ≈ 78.54

To find the perimeter, add the lengths of the 2 straight sections and the length of the circumference together. Remember to include the correct unit.

P = 78.54 + 23 + 23 = 124.54 mm

23 mm

25 mm

1

23

4

4WORKEDExample



Perimeter

1 Find the perimeter of the following figures, in millimetres after measuring the sides.a b

c d

e f

2 Find the perimeter of each of the figures below. Give your answers in centimetres (cm).a b c

d e f

remember1. The perimeter of a shape is the total distance around that shape.2. For circular figures the term circumference is used instead of perimeter.3. A formula can be used to find the perimeter (or circumference) of each of the

following shapes.(a) Square P = 4l (l = side length)(b) Rectangle P = 2(l + b) (l = length, b = breadth or width)(c) Circle C = 2πr (r = radius) or C = πd (d = diameter)

4. When finding the perimeter of a shape, make sure that all measurements have the same units.

remember

6B

SkillSH

EET 6.2

Measuring the length of a line

WORKEDExample

2

EXCEL

Spreadsheet

Perimeterandarea

Mathca

d

Perimeter

GCpro

gram CasioMeasurement

GCpro

gram TI

Measurement 4.2 m

4.7 m170 cm

25 cm

35.4 cm

30 cm

0.80 m

48 cm

98 mm

350 mm 32 cm

460 mm

18 mm

18 mm 27 mm

38 mm


C h a p t e r 6 P e r i m e t e r , a r e a a n d v o l u m e 2073 Find the perimeter of each of the squares below.

a b c

4 Find the circumference of the circles below. Give your answer correct to 2 decimal places.a b c

d e f

5 Find the perimeter of the rectangles below.a b c

d e f

6 Find the perimeter of the shapes shown below. Give your answer correct to 2 decimalplaces.a b c

d e f

g h i

SkillSHEET

6.3

Substitutioninto

perimeterformulas

SkillSHEET

6.4

Roundingto a givennumber of

decimalplaces

2.4 cm

11.5 mm

7.75 km

WORKEDExample

3

8 cm4 m

22 mm

Cabri Geometry

Circumferenceand area of

a circle

7.1 cm3142 km

1055 mm

60 m

36 m

500 mm

110 mm 50 cm

0.8 m

2.8 cm

9 mm

1.8 km

3 km

100 cm

3 m

SkillSHEET

6.5

Perimeter ofsquares,

rectangles,triangles and

circles

WORKEDExample

4

25 cm

30 cm

21 cm

20 cm

12 cm

160 cm

0.6 m

80 mm

66 cm

99 cm

20 cm

60 m

40 m

8.5 cm



j

7 Find the perimeter of the racetrack from the plan at right.

8 Yacht races are often run over a triangular course as shown at right.What distance would the yachts cover if they completed 3 laps of the shown course?

9 Use Pythagoras’ theorem to find the length of the missing side and, hence, find the perimeter of the triangular frame shown.

10 The Russian Radioastron satellite, which forms part of a huge radio telescope network, orbits the Earth at a height ofabout 70 000 km. If the radius of the Earth itself is about 6378 km, find the distance that the satellite covers in one orbit of the Earth.

11.5 mm

50 m

39 m

5.5 km 5.1 km

1900 m

SkillSH

EET 6.6

Using Pythagoras’ theorem

4 m

3 m

Earth

70 000 km

r = 6378 km


II’’vve ge got a sprained ankleot a sprained ankle..WhatWhat’’s that?s that?

AD

G H

E

I

N

RO

VTS

C

4.89 m

3.74 m

6 m

3 m

4.5 m

3 m

7.1 m 7 m

7.92 m

5.3 m

7.4 m

6.37 m

4.98 m

5 m

4 m

7 m

4.73 m

1.9 m

3.62 m

8.2 m

7.08 m7.47 m

5.95 m 4.09 m

6.38 m

1.9 m

M

4.39 m

6.43 m

5.16 m

L3 m

2 m

4 m

17.26 m

16.90 m

15.43 m

23.89 m

25.92 m

25.13 m

11.37 m

22.36 m

25.13 m

25.92 m

22.36 m

18.85 m

18.00 m

25.92 m

12.91 m 18.20 m 17.72 m 29.62 m 17.26 m 20.37 m 25.92 m 16.90 m 22.36 m




AreaThe area of a shape is a measure of the amount of surface enclosed by that shape.

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

The conversion chart for units of area

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

The area of a shape can be found by dividing it into centimetre squares and counting them. For example, a rectangle shown in the diagram at right can be divided into 12 centimetre squares and, thus, its area is 12 cm2.

The formula A = lb can also be used to find the area of a rectangle. In this formula, A represents the area of a rectangle of length l and breadth b.

The area of many simple shapes can be found using an appropriate formula.

Some frequently used formulas for calculating area are shown in this table.

Square Rectangle Triangle

A = l2 A = lb A = bh

Circle Parallelogram Trapezium

A = πr2 A = bh A = h(a + b)

squaremillimetres

(mm2)

squarecentimetres

(cm2)

squaremetres(m2)

squarekilometres

(km2)

× 10002

10002 = 1 000 000 1002 = 1 0 000 102 = 1 00

× 1002 × 102

÷ 102÷ 1002÷ 10002

EXCEL

Spreadsheet

Areaconverter(DIY)

Cabri

Geometry

Changingunits

3 cm

4 cm

Cabri

Geometry

Perimeter and area of arectangle

Cabri

Geometry

Area ofa triangle

Cabri

Geometry

Circumferenceand area ofa circle

Cabri

Geometry

Area of aparallelogram

Cabri

Geometry

Area of atrapezium

l l

b b

h

12---

r h

b

h

a

b

12---



The areas of composite shapes can be found by breaking the shape up into simple shapes and using the appropriate formulas from the table on page 210.

For example, the shape at right can be divided into 2 semicircles and a square as shown by the dotted lines. Its area can then be found using the formulas for a circle and a square.

Find the area of this shape.

THINK WRITE

Write the formula for the area of a triangle.

A = bh

Identify the values of the pronumerals. b = 88, h = 40Substitute the values of the pronumerals into the formula.

A = × 88 × 40

Evaluate. Remember to include the correct unit in your answer.

A = 1760 cm2

40 cm

88 cm

112---

23

12---

4

5WORKEDExample

12 cm

Find the area of the composite shape shown at right, giving your answer correct to 1 decimal place.

Continued over page

THINK WRITE

Divide the shape into parts using a dotted line.

Identify the shapes that constitute the total area.

Total area = Asemicircle + Asquare

The area of a semicircle is half the area of a complete circle.

Asemicircle = πr2

State the value of the pronumeral: the diameter of the circle is 50 mm; hence, its radius is 25 mm.

r = 25

50 mm1

50 mm

2

312---

4

6WORKEDExample



Area

1 Find the area of each of the following shapes. (Where appropriate, give your answercorrect to 2 decimal places.)a b c

d e f

g h i

THINK WRITE

Substitute the value of the pronumeral into the formula and evaluate.

Asemicircle = × π × 252 ≈ 981.7 mm2

Write the formula for the area of a square. Asquare = l2 State the value of the pronumeral. l = 50Substitute the value of the pronumeral into the formula and evaluate.

Asquare = 502 = 2500 mm2

Add the two areas together to find the total area of the given shape.

Total area = 981.7 + 2500 = 3481.7 mm2

512---

6

78

9

remember1. The area of a shape is a measure of the amount of surface enclosed by that shape.2. Area is measured in units based on the square metre, such as the square

centimetre (cm2). The conversion between square units is the square of the conversion between linear units.

3. A formula can be used to calculate the area of simple shapes.(a) Square A = l2 (l = length)(b) Rectangle A = lb (l = length, b = breadth or width)(c) Triangle A = bh (b = base, h = height)(d) Circle A = πr2 (r = radius)(e) Parallelogram A = bh (b = base, h = height)(f) Trapezium A = h(a + b) (a, b = parallel sides, h = height)

4. The area of a composite shape can be found by dividing the shape up into simpler shapes and using the appropriate formula.

12---

12---

remember

6C

SkillSH

EET 6.7

Substitution into area formulas

WORKEDExample

5

EXCEL

Spreadsheet

Perimeterandarea

6 cm

4 cm

4 mm 43 cm

25 cm

Mathca

d

Area

GCpro

gram Casio

Measurement

13 cm

23 cm

4.8 m

6.8 m

2 cm

5.5 cm

3 cm

GCpro

gram TI

Measurement

30 cm

15 cm

13 cm1 cm

2.5 cm

25 m58 m

50 m



213

j

k l

2

To convert an area measurement from square kilometres to square metres:

A

divide by 1000

B

multiply by 1000

C

divide by 1 000 000

D

multiply by 1 000 000

3

Fill in the gaps:

a

13 400 m

2

=

km

2

b

0.04 cm

2

=

mm

2

c

3 500 000 cm

2

=

m

2

d

0.005 m

2

=

cm

2

e

0.043 km

2

=

m

2

f

200 mm

2

=

cm

2

4

Find the area of the composite shapes shown. Where appropriate, express youranswers correct to 1 decimal place.

a b c

d e f

g h i

j

Cabri Geometry

Circumferenceand area of

a circle

4 m

2 mm

3.4 m

SkillSHEET

6.8

Convertingunits of area

mmultiple choiceultiple choice

SkillSHEET

6.9

Area ofsquares,

rectangles,triangles

andcircles

6

50 mm

2 m

3 m

21 cm

20 cm

18 cm

120 m

80 m3.0 m

1.5 m

1.2 m

60 m

40 m

8 cm

26 cm

18 cm

29 cm

23 cm

11.5 mm

MQ 09 NSW 5.2P - 06 Page 213 Thursday, March 23, 2006 3:47 PM


5 Find the area of glass in the window shown, correct to 2 decimal places. (Ignore the supports.)

6 What would be the cost of covering the sportsground shown in the figure at right with turf, if the turf costs $7.50 per square metre?

7 The Murray–Darling river basin is Australia’s largest catchment. Irrigation of farms inthe Murray–Darling Basin has caused soil degradation due to rising salt levels.Studies indicate that about 500 000 hectares of the basin could be affected in the next50 years.

a Convert the possible affected area to square kilometres. (1 km2 = 100 hectares)b The total area of the Murray–Darling Basin is about 1 million square kilometres,

about one seventh of the continent. What percentage of this total area may beaffected by salinity?

1.8 m

0.8 m

0.9 m

43 m

58 m


C h a p t e r 6 P e r i m e t e r , a r e a a n d v o l u m e 2158 The plan at right shows two rooms,

which are to be covered with carpet. Find the cost of carpeting the two rooms if the carpet costs $45 per square metre.

9 Find the area of the regular hexagon shown at right by dividing it into two trapeziums.

10 Find the area of the regular octagon by dividing it into two trapeziums and a rectangle, as shown in the figure.

11 An annulus is a shape formed by two concentric circles (two circles with a commoncentre). Find the area of the annulus shown below by subtracting the area of the smallercircle from the area of the larger circle.a b

c d

9 m7 m

13 m

7.5 m

8.5 m

24 cm21 cm

12 cm

5 cm

2 cm

1.45 cm

6 cm

2 cm

18 m

20 m

WorkS

HEET 6.1

10 cm22 cm

4 mm

3 mm



Career profileB R E N DA G A L E Y —

F o r e s t a s s e s s m e n t m a n a g e m e n t o f f i c e r

Qualifications: Bachelor of Science(Geography)

Employer: Department of Natural Resource and Environment, Forest Assessment Section

I love geography and through my degree learned how to use aerial photographs to accumulate and map a wide variety of information about forests. Things like the height of the trees, how old they are, the species and if there has been any disturbance such as fire or logging. I use mathematical formulas in many tasks. For example:1. The scale of an aerial photograph is very

important when making accurate measurements of distance, area or height from the photograph. To calculate the photograph scale factor, the following formula is used.

Photograph scale factor = where,

GD = ground distance between two pointsPD = photo distance between two points.

2. When you know the scale you can measure the distance to a section of forest that will be evaluated and compare it to the mapping on the ground. We call this field verification. We use things like road or stream intersections to find our location on the ground that corresponds to a location on the photographs. Then we measure on the photograph and convert that to ground distance.GD = PD × photograph scale factorFor example, if the photograph distance is 5 cm and the scale factor is 10 000 then,

GD = 5 cm × 10 000 GD = 50 000 cm.So the ground distance is 500 m or 0.5 km.

3. We use a special tool to measure the area of trees. It is called a dot grid. It is a transparent sheet with uniformly spaced dots and it is placed over an aerial photo. The dots represent the centres of small squares. This is a good way to find out, for example, how many hectares (ha) have been burned by a fire in a remote country area.

Ha/dot =

If you have 4 dots/cm2 on a 10 000 scale photograph, each dot represents 0.25 ha.If you have 200 dots on the photograph of a forest, there would be 50 hectares of forest.

A good background in mathematics is important, as it is used continually in my job as well as in many other interesting career areas such as oceanography, geology, medicine, computer graphics, to name a few.

Questions1. If Brenda were given an aerial photograph of

an area of forest that contained distinguishing features such as a road and a river, and she knew the ground distance between two points on these features from a map, explain how she could work out the photograph scale factor.

2. Calculate the photograph scale factor if the ground distance between these features is 642 m and it corresponds to a photo distance of 2.4 cm.

3. Calculate the ground distance between two water holes on the same photograph if the photograph distance between them is 7.6 cm.

4. A fire has burnt out part of a forest. Using a dot grid, Brenda is able to calculate the area of the forest affected using the following information:10 000 scale aerial photographdot grid with 4 dots/cm2 345 dots in the burnt section of forest. Can you work out the area of fire-affected forest?

GDPD--------

Number of ha/cm2

Number of dots/cm2-------------------------------------------------



Area and perimeter of a sectorA sector is a portion of a circle formed by two radii and the arc between them.

We have seen that the area of a semicircle can be found by halving the area of a circle with the same radius. That is, the area of a semicircle can be calculated using the formula A = π r2.

Another special type of a sector is a quadrant (a quarter circle). Again, the area formula for the circle can be adapted to find the area of a quadrant, as shown in the next worked example.

So far we have considered special sectors with the angle between the radii being either180° (for a semicircle), or 90° (for a quarter circle). The area of any sector with the

angle between the radii being θ can be obtained by finding the fraction of thearea of a full circle.

Area of a sector is given by the formula A = πr2, where θ is

the angle included between the radii.

The perimeter of any sector can be found by finding the length of the curved side andthen adding all the sides together. The curved side is part of the circumference and can

be obtained by finding the fraction of the full circumference.

12---

Find the area enclosed by the figure at right, correct to 1 decimal place.

THINK WRITE

The given shape is a quadrant and thus its area can be found by dividing the area of a circle by 4. Write the appropriate formula.

Aquadrant = πr2

State the value of the pronumeral. r = 11Substitute the value of the pronumeral into the formula. A = × π × 112

Evaluate and round your answer to 1 decimal place. ≈ 95.0 cm2

11 cm1

14---

23

14---

4

7WORKEDExample

θ360°-----------

q360--------- r

θ

Find the area of the sector shown, correct to 1 decimal place.THINK WRITE

Write the formula for the area of the sector. A = πr 2

Identify the values of the pronumerals. θ = 30o, r = 5

Substitute the values of the pronumerals into the formula. A = × π × 52

Evaluate and round your answer correct to 1 decimal place. Include the appropriate unit.

≈ 6.5 m2

30°5 m

1θ

360---------

2

330360---------

4

8WORKEDExample

θ360---------



The length of the curved side is given by the formula l = × πd or

l = × 2πr, where θ is the angle included between the radii.

The perimeter is found by adding all the sides together.

Area and perimeter of a sector

1 Find the area of the semicircles below, correct to 2 decimal places.a b c d

θ360---------

θ360---------

Find the perimeter of the sector shown, correct to 1 decimal place.

THINK WRITE

Write the formula for the length of the curved side.

l = × 2πr

Identify the values of the pronumerals. θ = 80°, r = 3

Substitute the values of the pronumerals into the formula.

l = × 2 × π × 3

Evaluate and round your answer correct to 1 decimal place.

≈ 4.2 cm

Add all the sides together to calculate the perimeter. Include the appropriate unit.

P = 4.2 + 3 + 3=10.2 cm

3 cm80°

1θ

360---------

2

380

360---------

4

5

9WORKEDExample

remember1. A sector is a portion of a circle formed by two radii and the arc between them.

2. To find the area of a sector, use the formula A = πr2, where θ is the angle

included between the radii.

3. To find the perimeter of a sector, use the formula l = × 2πr, where θ is the

angle included between the radii, to find the curved side. Then add the lengths of all sides to find the total perimeter.

θ360---------

θ360---------

remember

6D

Mathca

d

Area ofa sector

Cabri

Geometry

Area ofa sector

6 cm

20 cm

rr = 4.2 cm D

D = 24 mm


C h a p t e r 6 P e r i m e t e r , a r e a a n d v o l u m e 2192 Find the area enclosed by each of the figures below, correct to 1 decimal place.

a b c d

3

Which is the correct formula for finding the area of this sector?

A A = πr2 B A = πr2 C A = πr2 D A = πr2

4 Find the area of each of the sectors shown, correct to 1 decimal place. a b c

d e f

5 A searchlight lights up the ground to a distance of 240 m. What area does the searchlight illuminate if it can swing through an angle of 120°, as shown in the diagram at right? (Give your answer correct to 1 decimal place.)

6 Find the perimeter of the quadrants in question 2, correct to 1 decimal place.

7 Find the perimeter of the sectors shown in question 4, correct to 1 decimal place.

8 Find the perimeter, correct to 1 decimal place, of lime-green cardboard in the cartoon figure at right.

WORKEDExample

7

4 cm

12.2 cm

a = 11.4 m

a

1.5 m

WORKEDExample

8


36°

r

r

14--- 3

4--- 1

2--- 1

10------

238°

30 cm

45°24 m

60°

9 cm

140°

77 m r

r = 74 cm

200°

48 cm

10°

Illuminatedarea

240 m

120°

SearchlightWORKEDExample

9

40°

80 cm



9 A goat is tethered by an 8.5 m rope, to the outside of a corner post in a paddock, as shown in the diagram at right. Find the area of grass (shaded) on which thegoat is able to graze. (Give your answer correct to 1 decimal place.)

10 A beam of light is projected onto a theatre stage as shown in the diagram at right.a Find the illuminated area (correct to

1 decimal place) by finding the area of the sector.

b Find the percentage of the total stage area that is illuminated by the light beam.

Tethered donkey1 A donkey inside a square enclosure is tethered to

a post at one of the corners. Determine the length of the rope required so that the donkey eats only half of the grass in the enclosure.

2 Suppose 2 donkeys are tethered at opposite corners of the square region shown at right. How long should the rope be so that the donkeys together can graze half of the area?

3 This time four donkeys are tethered, one at each corner of the square region. How long should the rope be so that all the donkeys can graze only half of the area?

4 Another donkey is tethered to a post inside an enclosure in the shape of an equilateral triangle. The post is at one of the vertices. Calculate how long the rope should be so that the donkey eats only half of the grass in the enclosure.

5 This time the donkey is tethered halfway along one side of the equilateral triangular region shown at right. How long should the rope be so that the donkey can graze half of the area?

8.5 m

Fence

Illuminatedarea

Light

20 m

17 m 68°

5 m

GAME

time

Perimeter,area and volume — 001

150 m

100 m

100 m 100 m

100 m



1 Convert 45.45 m to cm.

2 The dimensions of a room are given as 556 cm by 423 cm. Convert these dimensionsto m.

3 Find the total perimeter of the object in the figure at right.

4 Find the circumference of a circle with a diameter of 45.2 cm.

5 A yacht race consists of 12 laps around a triangular course. The triangle is equilateral with each side 810 m. Find the total length of the race, in kilometres.

6 A box is built to contain 12 tennis balls, as shown in the figure at right.

If the radius of each tennis ball is 4.6 cm, find the perimeter of the smallest possible rectangle that will contain the balls.

7 Find the area of the trapezium shown at right.

8 Find the area of the parallelogram shown at right.

9 A skating rink consists of 4 quarter circles at each corner and 4 straight lines connecting them, as shown in the figure at right.

Find the total perimeter (to 1 decimal place) ifthe radius of the circles is 1.2 m and each straight line is 10 m.

10 Find the area of the skating rink in question 9.

1

10 cm

6 cm

15 cm

9 cm

5 cm

12 cm

20 cm

12 cm

35 cm

1.2 m

10 m



Surface area of rectangular and triangular prisms

A prisms is a 3-dimensional figure that has a uniform cross-section that is a polygon.

Here are examples of prisms.

Imagine ‘slicing’ these shapes up into pieces like slicing a loaf of bread. Each ‘slice’of a prism has the same size and shape. The cross-sections formed by slicing the prismare congruent.

‘Slicing’ a prism into pieces produces congruent cross-sections.

The shapes below are not prisms, because they do not have congruent cross-sections.

Surface area of prismsImagine calculating the area that needs to be painted to cover the outside of a prism.This area is the surface area (SA) of the prism.

To find the surface area of a prism, simply find the area of each face using the appropriate area formula, and add them together. Often, several faces will have the same area, making the task easier.

Consider a rectangular prism (or a cuboid) shown in the figure at right.

The surface of the prism is represented by 6 rectangular faces.

Triangular prism

Rectangular prism (cuboid)

Hexagonal prism

Cone

Square pyramid

Sphere

Triangular pyramid

l

h

b


C h a p t e r 6 P e r i m e t e r , a r e a a n d v o l u m e 223The surface area can be found using the formula generated below.

SA = Afront + Aback + Atop + Abottom + Aleft side + Aright sideSA = (l × h) + (l × h) + (l × b) + (l × b) + (b × h) + (b × h)

= lh + lh + lb + lb + bh + bh = 2lh + 2lb + 2bh = 2(lh + lb + bh)

The surface area of a rectangular prism (or a cuboid) of length l, breadth b and height h is given by the formula SA = 2(lh + lb + bh).

Find the surface area (SA) of this fish tank which is a rectangular prism (cuboid).

THINK WRITE

The aquarium is a rectangular prism. Write the formula for the surface area of a rectangular prism.

SA = 2(lh + lb + bh)

Identify the value of the pronumerals. l = 50, b = 20, h = 24


SA = 2 × (50 × 24 + 50 × 20 + 20 × 24)

Evaluate. = 5360 cm2

24 cm

20 cm 50 cm

1

2

3

4

10WORKEDExample



Find the surface area (SA) of this triangular prism.

THINK WRITE

The ends of the triangular prism (the front and back) are congruent triangles. Write the formula for the area of a triangle.

Afront = Aback = bh

Identify the value of the pronumerals. b = 4, h = 3.5

Substitute the values of the pronumerals and calculate the area of each end.

Afront = Aback = × 4 × 3.5 = 7 cm2

The other faces are congruent rectangles. Write the formula for the area of a rectangle.

Aleft side = Aright side = Abottom = lb

Identify the values of the pronumerals. l = 18, b = 4

Substitute the values of the pronumerals into the formula and so calculate the area of each rectangular face.

Aleft side = Aright side = Abottom = 18 × 4

= 72 cm2

To find the surface area, add areas of all faces together. Remember to include the appropriate unit.

SA = (2 × 7) + (3 × 72) = 230 cm2

3.5 cm4 cm

4 cm

4 cm

18 cm

112---

2

312---

4

5

6

7

11WORKEDExample

remember1. Prisms are 3-dimensional figures, which have uniform cross-sections (that are

polygons).

2. The surface area of a prism is the area of its outside surface.

3. To find the surface area of a prism, find the area of each face using the correct area formula, and add them together.

4. Look for identical faces, which will have the same area.

5. The surface area (SA) of a rectangular prism (or a cuboid) of length l, breadth b and height h is given by the formula SA = 2(lh + lb + bh).

remember



Surface area of rectangular and triangular prisms

1 Find the surface area (SA) of the rectangular prisms (cuboids) below.

a b c

d e f

2 Find the surface area (SA) of each of the triangular prisms below.

a b c

3 Find the surface area (SA) of each of the triangular prisms below.

a b c

4 What area of cardboard would be needed to construct a box to pack this prism assuming that no overlap occurs?

6ESkillSHEET

6.10

Surfacearea of

cubes andrectangular

prisms

EXCEL Spreadsheet

Surfacearea andvolume

Mathcad

Surface areaof rectangularand triangular

prisms

GC program

TI

Measurement

WORKEDExample

10

GC

program Casio

Measurement

4 cm3 cm

3 cm3 m

2 m 5 m1.3 m0.8 m

1.1 m

140 cm0.8 m

0.5 m

70.5 cm41.2 cm

25.8 cm

0.9 m0.7 m

0.2 m

WORKEDExample

11 3.5 cm

4 cm

6 cm

8 m

6.1 m7 m

h

h = 0.87 cm

2.5 cm

1 cm

SkillSHEET

6.11

Surface areaof triangular

prisms

17 cm 15 cm

8 cm18 cm

8.8 cm

7.2 cm6.2 cm 14 mm

44 mm

36 mm25 m

m

4.3 cm

16 cm

5 cm



5 Maya is planning to buy and paint the outside surface of one of these shippingcontainers. How many cans of paint should Maya buy, if the base of the container isnot painted, and each can of paint covers about 40 m2?

6 The aim of the Rubik’s cube puzzle is to make each face of the cube one colour. Find the surface area of the Rubik’s cube if each small coloured square is 1.2 cm in length. Assume that there are no gaps between the squares.

7 A glass water tank for display purposes is a triangular prism with the dimensions shown at right. The top of the tank is open. What area of glass is required to construct the tank?

8 How many square metres of iron sheet are needed to construct the water tank shown?

6.5 m

3.2 m

2.8 m

1.2 cm

h h = 0.43 m

1.4 m

0.5 m

1.9 m 3.2 m

1.4 m


C h a p t e r 6 P e r i m e t e r , a r e a a n d v o l u m e 2279 A simple tent is constructed as shown. What area of canvas is needed to make the tent,

if the floor is included?

10 How many square centimetres of cardboard are needed to construct the shoebox at right, assuming no overlap? Draw a sketch of a net, which could be used to make the box.

11 a Find the surface area of the toy block shown.

b If two of the blocks are placed together as shown, what is the surface area of the prism which is formed?

c What is the surface area of the prism formed by three blocks?

d Use the pattern to find the surface area of a prism formed by 8 blocks arranged ina line.

12 Find the surface area of a square-based prism of height 4 cm, given that the side lengthof its base is 3 cm.

2.0 m

1.6 m

1.5 m

1.0 m

320 mm

110 mm

160 mm

6E

BLACKSUEDE

5 cm

5 cm

WorkS

HEET 6.2



Surface area of a cylinderThe top and bottom of the cylinder are identical circles while the curved surface of thecylinder if it were cut open and unfolded would form a rectangle. The net of thecylinder is shown below.

If the cylinder has a radius of r and a height of h as shown in the net above then itsSA can be calculated as follows.

Radius of circles = rLength of the rectangle = circumference of circular ends

= 2πrWidth of rectangle = hSA = Aside + Atop + Abottom

= lb + πr2 + πr2

= 2πrh + 2πr2

The surface area of a cylinder is given by the formula: SA = 2πrh + 2πr2, where r is the radius of the base and h is the height of a cylinder.

r

h

h

2 rπ

A = r 2π

a Use the formula A = 2πrh to find the area of the curved surface of the cylinder, correct to 1 decimal place.

b Use the formula SA = 2πrh + 2πr2 to find the surface area of the cylinder, correct to 1 decimal place.

THINK WRITE

a Write the formula for the area of the curved surface of a cylinder.

a A = 2πrh

Identify the values of the pronumerals.

r = 2, h = 3


A = 2 × π × 2 × 3

Evaluate and round off to 1 decimal place.

A ≈ 37.7 m2

3 m

2 m

1

2

3

4

12WORKEDExample



Surface area of a cylinder

1 Use the formula A = 2πrh to find the area of the curved surface of each of the cylindersbelow. (Express your answers correct to 1 decimal place.)a b c

d e f

2 Use the formula SA = 2πrh + 2πr2 to find the total surface area of each of the cylindersin question 1.

THINK WRITE

b Write the formula for the surface area of a cylinder.

b SA = 2πrh + 2πr2

Identify the values of the pronumerals (these are the same as in part a).

r = 2, h = 3


SA = (2 × π × 2 × 3) + (2 × π × 22)

Evaluate and include the appropriate unit.

≈ 62.8 m2

1

2

3

4

remember1. The outer surface of a cylinder is made up of two circles and a rectangle.

2. The area of the rectangular curved surface is given by the formula A = 2πrh.

3. To find the surface area of a cylinder, use the formula SA = 2πrh + 2πr2, where r is the radius of the base and h is the height of the cylinder.

remember

6F

EXCEL Spreadsheet

Surfacearea andvolume

GC program

TI

Measurement

WORKEDExample

12a

GC

program Casio

Measurement

Mathcad

Surfacearea of acylinder

4 m

3 m1.5 cm

3 cm

32 m

20 m

h

h = 21 cm

17 cm

1.4 m

1.5 m

h

r = 2.4 mh = 1.7 m

r

WORKEDExample

12b


230

M a t h s Q u e s t 9 f o r N e w S o u t h W a l e s 5 . 2 P a t h w a y

3

Michael constructed the 13 m-high storage tank at right from stainless steel (including the lid and the base).

a

What is the surface area of the tank?

b

How much did the steel cost if it comes in sheets 1 m wide that cost $60 a metre?

4

Find the surface area of the cylindrical marzipan bar shown at right.

5

Find the area of the outer curved surface of the concrete pipe shown in the diagram at right given the following measurements.

t

=

30 mm

D

=

18 cm

l

=

27 cm

6

Wooden mouldings are made by cutting cylindrical dowels in half as shown. Calculate the surface area of the moulding.

Click on the link below when using the CD for extension work on surface area.

3 m

13 m

MARZIPAN

15 cm

45 mm

l

D

t

12 cm

45 mm

extensioneextensionxtension — Surface area of spheres, cones and composite shapes

MQ 9 NSW 5.2P - 06 Page 230 Monday, July 26, 2004 8:32 AM


1 Convert 67 500 m to km.

2 Determine the dimensions of a square whose perimeter equals that of a 10 cm by15 cm rectangle.

3 Determine the area of a circle with a diameter of 4.2 m.

4 Find the area of the object in the figure below.

5 Find the perimeter of the sector below.

6 Find the surface area of a cube whose edges are 6.7 mm.

7 Convert your answer from question 6 to cm2.

8 Find the surface area of a cylinder with a radius of 4 cm and a height of 20 cm.

9 A prism has a regular hexagonal face, as shown below.

The length of each side of the hexagon is 2 cm, and its area is 10.4 cm2. Find thesurface area of the prism.

10 Determine the area of the ‘curved’ surface of a cylinder whose radius is 4.3 cm andheight is 5.9 cm.

2

3 m

2 m

5 m

9 m

70° 16 cm

2 cm

15.1 cm



Volume of prisms and cylindersVolume

The volume of an object is the amount of space it occupies.Volume is measured in cubic units, such as cubic millimetres (mm3), cubic centimetres (cm3) and cubic metres (m3).

The units of volume are related to each other as shown below.

Conversion chart for units of volume

Note: The conversion of units for volume is the cube of the linear conversions.The volume of a solid figure can often be found by simply dividing the solid up into

small cubes with 1 cm sides as shown below.

The volume of this cuboid is 12 cm3.

Volume of a prismThe volume of any prism can be found by multiplying its cross-sectional area (A) by its height (h).

The volume of a prism is given by the formula V = Ah, where A is the cross-sectional area and h is the height.

Cubicmillimetres

(mm3)

Cubiccentimetres

(cm3)

Cubicmetres(m3)

× 103× 1003

÷ 1003 ÷ 103

103 = 1000 1003 = 1 000 000

This solid contains12 cubic centimetres.

2 cm3 cm

2 cm

1 cm

A

h


C h a p t e r 6 P e r i m e t e r , a r e a a n d v o l u m e 233Note: The cross-section of the prism is often referred toas the ‘base’. In such instances the base does not alwaysreflect the true meaning of the word because the base isnot necessarily a face on which the prism stands. Thebase must be one of the identical ‘slices’; that is, cuttingthe prism parallel to the base should produce a ‘slice’ ofthe same size and shape.

Similarly, the height is not necessarily the same asheight as we generally think of it. The height of a prismis perpendicular to the base and not necessarily vertical.

In some cases a special formula can be developed from the formula V = Ah.Cube Volume = base area × height

= area of a square × height

= l2 × l

= l3

Rectangular prism Volume = base area × height = area of a rectangle × height = lbh

Volume of a cylinderCylinder Volume = base area × height

= area of a circle × height

= πr2h

The volume of a cylinder can also be found by multiplying its cross-sectional area byits height.

Height

Base

l

l

b

h

r

h

Find the volume of the hexagonal prism.

THINK WRITE

Write the formula for the volume of a prism.

V = Ah

Identify the values of the pronumerals. A = 40, h = 8Substitute the values of the pronumerals into the formula and evaluate. Remember to include the correct unit.

V = 40 × 8 = 320 cm3

A = 40 cm2

8 cm

1

23

13WORKEDExample



Capacity

The capacity of a container is the amount that it can hold.

Capacity is usually applied to the measurement of liquids.

The most commonly used units of capacity are the litre (L) and the millilitre (mL). A larger unit, the kilolitre (kL) is sometimes used.

The units of capacity are related as follows:

1000 millilitres = 1 litre1000 litres = 1 kilolitre

The measuring cylinder shown at right measures capacity in both millilitres (mL) and litres (L).

Clearly, capacity and volume are closely related concepts. The capacity of a solid figure is the same as its volume. The units of capacity and volume are connected.

Capacity Volume1 millilitre (mL) = 1 cubic centimetre (cm3)1000 litres (L) = 1 kilolitre (kL) = 1 cubic metre (m3)

Find the volume of the prism.

THINK WRITE

Write the formula for the volume of a prism.

V = Ah

The cross-section of the prism is a triangle. Write the formula for the area of the triangle.

A = × base × height of triangle

Substitute the known values into the formula and evaluate the area of the cross-section.

A = × 4 × 3

= 6 cm2

State the value of h — the height of the prism.

h = 8

Substitute the values of the cross-sectional area and the height into the formula for the volume and evaluate. Remember to include an appropriate unit.

V = 6 × 8 = 48 cm3

3 cm

4 cm

8 cm

1

212---

312---

4

5

14WORKEDExample



Find the volume of water, in litres, in a cylindrical water tank that has a diameter of 5.4 m and a height of 3 m. (Give your answer correct to 1 decimal place.)

THINK WRITE

Draw a diagram of the tank, including all given measurements.

Write the formula for the volume of a cylinder.

V = Ah

The diameter is known, so halve the diameter to find the size of the radius.

r = 2.7

State the value of h: the height of the cylinder.

h = 3

Substitute the values of A and h into the formula to find the volume of the tank, in cubic metres (m3).

V = 22.9022 × 3 ≈ 68.7066 m3

Convert this volume to litres. 1 m3 = 1000 L, so multiply by 1000 by moving the decimal point 3 places to the right.

V = 68.7066 × 1000 = 68 706.6 L

1 5.4 m

3 m

2

3

4

5

6

15WORKEDExample

remember1. The volume of a solid is the amount of space it occupies.

2. Volume is measured in cubic units, such as cubic millimetres (mm3), cubic centimetres (cm3) and cubic metres (m3).

3. The volume of any prism can be found by multiplying the area of its cross-section (or base), A, by its height, h; that is, V = Ah.

4. The volume of a cylinder is given by the formula V = πr2h.

5. Capacity is a term usually applied to the measurement of liquids.

6. The capacity of a solid figure is the same as its volume, if the thicknesses of the walls of the figure are ignored.

7. The most commonly used units of capacity are the litre (L) and the millilitre (mL).

8. The units of capacity and volume are connected as follows:(a) 1 millilitre (mL) = 1 cubic centimetre (cm3)(b) 1000 litres (L) = 1 kilolitre (kL) = 1 cubic metre (m3).

remember



Volume of prisms and cylinders

1 Which of the following figures are not prisms?

a b c d

e f g h

2 Find the volume of the cuboids below. Assume that each small cube has sides of 1 cm.

a b c

3 Find the volume of these objects.

a b c

4 Find the volume of these rectangular prisms.

a b c

6G

EXCEL

Spreadsheet

Volume

Mathca

d

Volume of a rectangular or triangular prism

GCpro

gram Casio

Measurement

GCpro

gram TI

Measurement

WORKEDExample

13

3 cm

A

A = 4 cm2

4 cm

A

A = 17 cm2

h

A

A = 3.2 m2

h = 3.0 m

SkillSH

EET 6.12

Volume of cubes and rectangular prisms

4 cm3 cm

3 cm3 m

2 m 5 m1.3 m0.8 m

1.1 m


C h a p t e r 6 P e r i m e t e r , a r e a a n d v o l u m e 2375 Find the volume of the prisms shown below.

a b

c d

6 Find the volume of the cylinders below. Give your answers correct to 1 decimal place.a b c

d e f

7 Find the volume of water, in litres (L), in a cylindrical water tank that has a diameter of 3.2 m and a height of 1.8 m. (Express your answer correct to 1 decimal place.)

8 Find the total volume, in litres, of the storage tank shown at right.

SkillSHEET

6.13

Volume oftriangular

prisms

WORKEDExample

14

26 cm

30 cm18 cm

2.4 m

15 m8 m

5 m

28 cm

17 cm

37 cm

SkillSHEET

6.14

Volume ofcylinders

32 m

20 mh

h = 21 cm

17 cm

1.4 m

1.5 m

h

r = 2.4 mh = 1.7 m

r

2.7 m

3.8 m

2 mm

6 mm

WORKEDExample

15 3.6 m

7.4 m



9 What volume of goods can be stored in the Esky shown at right if it is completely filled?

10 What is the capacity (in mL) of the cylindrical coffee plunger shown, if it is filled as shown at right?

11 The world’s biggest landfill garbage dump is Fresh Kills, on Staten Island outside NewYork City. (New Yorkers throw out about 100 000 tonnes of refuse weekly.) Find theapproximate volume (m3) of the Fresh Kills landfill (shown below) if it covers an areaof 1215 hectares and is about 240 m high. (1 hectare = 10 000 m2)

0.84 m

0.42 m0.5 m

9 cm

18 cm


C h a p t e r 6 P e r i m e t e r , a r e a a n d v o l u m e 23912 Find the volume, in litres, of the refrigerator

shown at right.

13 A cylindrical glass below is designed to hold 1.25 L. Use trial and error to find some possible measurements for the height and radius of the glass so that its volume would be 1.25 L.

14 Sudhira is installing a rectangular pond in a garden. The pond is 1.5 m wide, 2.2 m long and has a uniform depth of 1.5 m.a Find the volume of soil (m3) that

Sudhira must remove to make the hole, prior to installation of the pond.

b What is the capacity of the pond in litres? (Ignore the thickness of the walls in this calculation.)

15 Find the volume of plastic needed to make the door wedge shown at right.

16 Find the internal volume of the wooden chest shown (ignore the thickness of the walls).

17 A local water board is constructing a storage tank with the dimensions shown. Find the volume of the tank, in kilolitres (kL).

1.5 m

85 cm90 cm

3.5 cm

3.5 cm

7.5 cm

27 cm

52 cm95 cm

12.5 m

28 m



Volume of pyramids, cones and composite solids

Pyramids (including cones) are not prisms as the cross-section changes size from thebase upwards.

It has been found in the investigation above that the volume of apyramid is one-third of the volume of an equivalent prism with thesame base area and height.

Since a cone is a pyramid with a circular cross-section, the volume of a cone is one-third thevolume of a cylinder with the same base area andheight.

Volume of a pyramid = × base area × height

= Ah

Volume of a cone = × base area × height

= πr2h

Comparing volumes of pyramids and prisms

You will need a set of 3-dimensional volumetric shapes of pyramids and prisms with the same base area and height, and water (or rice).

For this activity, you need to compare the volumes of pairs of open 3-dimensional containers by considering the amount of water (or rice) each can hold.

Consider the following pairs of containers:

Square-based pyramid and cubeRectangle-based pyramid and rectangular prism Triangle-based pyramid and triangular prismCone and cylinder

1 Fill the first container with water (or rice), then pour the contents into the second container. Continue filling the first container and pouring the contents into the second until the second container is full. How many times was it necessary to do this?

2 From your results, how would you compare the volume of a pyramid with that of a prism of the same base area and height.

3 If the volume of a prism has the general formula V = Ah, suggest a general formula for the volume of a pyramid.

13---

13---

13---

13---

h

base (A)

r

h



Volume of composite solidsTo calculate the volume of a composite solid, the solid is divided into components,individual volumes of the components calculated and then added. This technique isshown in the following worked example.

Find the volume of each of the following solids.a b

THINK WRITE

a Write the formula for finding the volume of a cone.

a V = πr2h

Identify the values of r and h. r = 8, h = 10Substitute and evaluate. V = × π × 82 × 10

≈ 670.2 cm3

b Write the formula for volume of a pyramid.

b V = Ah

Find the area of the square base. A = l2 where l = 8A = 82

= 64 cm2

Identify the value of h. h = 12Substitute and evaluate. V = × 64 × 12

= 256 cm3

8 cm

10 cm 12 cm

8 cm

113---

2

313---

113---

2

3

413---

16WORKEDExample

Calculate the volume of the composite solid shown.

Continued over page

THINK WRITE

The given solid is a composite figure, made up of a cube and a square-based pyramid.

V = Volume of cube + Volume of pyramid

1.5 m

3 m

1

17WORKEDExample



Volume of pyramids, cones and composite solids

1 Find the volume of each of the following cones, correct to 1 decimal place.a b

2 Find the volume of each of the following pyramids.a b c d

THINK WRITE

Find the volume of the cube. Vcube = l3 where l = 3Vcube = (3)3

= 27 m3

Write the formula for finding the volume of a square-based pyramid.

Vsquare-based pyramid = Ah

Find the area of the square base. A = l2

= 32

= 9 m2

Identify the value of h. h = 1.5Substitute and evaluate the volume of the pyramid.

Vsquare-based pyramid = × 9 × 1.5

= 4.5 m3

Find the total volume by adding the volume of the cube and pyramid.

V = 27 + 4.5 = 31.5 m3

2

313---

4

5

613---

7

remember1. Volume of a 3-dimensional figure is the amount of space it takes up.2. Volume is measured in cubic units.3. Volume of a prism = Ah, where A is the cross-sectional (or base) area and h is

the height of the prism.4. The height of a prism is the dimension perpendicular to the prism’s cross-section.

5. Volume of a cone = πr2h.

6. Volume of a pyramid = Ah.

13---

13---

remember

6HWORKEDExample

16a

4.5 cm

8 cm 26 mm20 mm

WORKEDExample

16b 12 cm

14 cm

42 cm

20 cm

32 cm

1.7 m

2.3 m

4.1 m

12 cm

15 cm

18 cm



243

3

Calculate the volume of each of the following composite solids.

a b c

4

Calculate the volume of each of these solids.

5

Note:

There may be more than one correct answer.

The pyramid at right is cut across into two parts labelled A and B.Which of the following statements is correct?

A

The volume of A equals the volume of B.

B

The volume of A is greater than the volume of B

.

C

The volume of

B

is greater than the volume of A

.

D

The volume of A is less than the volume of B.

6

A cylindrical water tank has a diameter of 1.7 m and a height of 3 m. What is thevolume of the tank?

7

A cylindrical tank (radius, 30 cm and height, 1 m) is filled with flour. The flour fromthe tank is then transferred to conical containers of radius 50 cm and height 80 cm.How many conical containers are required?

8

An axle from a truck, shown at right, is pulled out during repair works. What is the volume of the axle?

9

A cylindrical piece is obtained by machining the rectangular piece of wood, as shown at right.What is the volume of wood machined out to obtain the cylindrical piece?

17

16 m

1.3 m

1.1 m

0.65 m 20 cm

25 cm30 cm10 cm

80 cm0.8 m

1.6 m

1.8 m 1.1 m

15 mm

8 mm 6 mm

18 mm

[Base area: 35 mm2] [Base area: 28 cm2]

15 cm

6 mm

14 mm

26.5 mm

a b c d


5 cm

5 cm

10 cm

10 cm

B

AhA =

hB =

50 cm

6 cm

10 cm

30 cm

30 cm

1.2 m 30 cm

MQ 09 NSW 5.2P - 06 Page 243 Thursday, March 23, 2006 3:47 PM


Volume of spheres and composite solids

Volume of a sphere

The volume of a sphere can be found using the formula V = πr3, where r is the radius of the sphere.

Maintaining an Olympic poolAt the beginning of the chapter we met Majid, who is responsible for the maintenance of the Olympic (50 m) pool at an Aquatic Centre. The figure below shows the dimensions of an Olympic pool.

1 What is the shape of the pool?2 Calculate the area of the ‘base’ of the prism.3 Calculate the volume of the pool, in cubic metres.4 What is the capacity of the pool, in litres?5 Majid needs to replace the water in the pool every 3 months. If the pool is drained

at 5000 L per hour and refilled at 3000 L per hour, how long will it take to:a drain?b refill (in hours and minutes)?

6 When the pool is empty, Majid decides to have the bottom and sides resurfaced. Find the area that needs to be resurfaced.

7 How many litres of sealant will be needed to cover the walls and bottom, given that it will need two coats of sealant and each litre of sealant has a coverage of 5 m2?

22 m

2 m1 m

50 m

43---

Find the volume of the sphere shown at right. Express your answer correct to 1 decimal place.

THINK WRITE

Write the formula for the volume of a sphere. V = πr3

State the value of the pronumeral. r = 6

Substitute 6 for r into the formula. V = × π × 63

Evaluate and round to 1 decimal place. ≈ 904.8 cm3

6 cm

143---

2

343---

4

18WORKEDExample



Volume of composite solidsThe volume of some objects can be found by dividing the object up into pieces whichare prisms or other regular figures. The next worked example illustrates the concept.

Find the volume of the hemisphere shown at right. Give your answer correct to 2 decimal places.

THINK WRITE

The volume of a hemisphere can be found by halving the volume of the sphere. Write the formula.

V = × πr3

= πr3

State the value of the pronumeral. r = 15Substitute the value of r into the formula. V = × π × 153

Evaluate and round, correct to 2 decimal places.

≈ 7068.58 cm3

15 cm

112--- 4

3---

23---

23

23---

4

19WORKEDExample

Find the volume of the object shown. Give your answer correct to 1 decimal place.

THINK WRITE

The object is made up of a rectangular prism and a hemisphere. Find the volume of the rectangular prism first. Write the formula. (The cross-section is a square.)

Vprism = Ah

A = l2

Write the formula for the area, A, and evaluate. Then identify the values of the pronumerals for volume.

= 42

= 16 cm2

A = 16, h = 2Substitute the values of the pronumerals into the formula and evaluate the volume of the rectangular prism.

Vprism = 16 × 2 Vcuboid = 32 cm3

Write the formula for the volume of a hemisphere.

Vhemisphere = πr3

State the value of the pronumeral, r, by halving the diameter.

r = 2

Substitute the value of r and evaluate the volume of the hemisphere. Round off to 1 decimal place.

Vhemisphere = × π × 23

≈ 16.8 cm3

Add the volumes of the rectangular prism and the hemisphere together to find the total volume of the given solid.

V = 32 + 16.8 = 48.8 cm3

2 cm

4 cm1

2

3

4

523---

6

723---

8

20WORKEDExample



Volume of spheres and composite solids

1 Find the volume of the spheres shown below. Express your answer correct to1 decimal place.a b c

d e f

2 Find the volumes of spheres with the following radii. Express your answer correct to1 decimal place.a 7 cm b 4 cm c 13 cmd 3.2 cm e 11.45 m f 120 km

3 Find the volume of the hemispheres shown below. Express your answer correct to2 decimal places.a b c

d e f

remember1. The volume of a sphere can be found using the formula V = πr3.

2. The volume of composite figures can be found by dividing the figure up into pieces which are prisms or other regular figures.

43---

remember

6IWORKEDExample

18

Mathca

d

Volumeof asphere

5 cm12 cm

8.3 cm

GCpro

gram Casio

Measurement

GCpro

gram TI

Measurement13 cm 11.5 m 0.8 m

WORKEDExample

1910 cm

3.4 m

7 cm

2.5 mm

20.5 cm11.8 cm


C h a p t e r 6 P e r i m e t e r , a r e a a n d v o l u m e 2474 Find the volume of the objects shown below. Express your answer correct to

1 decimal place.a b c

d e

f g h

5 Metal nuts are constructed from a piece of metal shown in the figure at right. The area of the hexagonal base of the nut is 2.2 cm2. Find the volume of metal needed to manufacture 2000 of these nuts. Express your answer correct to 1 decimal place.

6 A space helmet is constructed from a sphere of lightweight plastic material with a diameter of 40 cm with other materials added. Calculate the volume of air inside the spherical helmet (correct to 1 decimal place).

7 A terrarium is made from transparent plastic, as shown in the diagramat left. a Calculate the volume of soil in the terrarium, if the soil is 6.5 cm deep.b Calculate the total volume of the terrarium.c Find the volume of air in the terrarium.Express your answers correct to 1 decimal place.

WORKEDExample

20

8 cm1.7 m

3.4 m

12 cm

45 mm

15 cm

17 cm12 cm

8 cm

30 cm

1.5 m

1.4 m

1.8 m

3.1 m

3 m

2.5 m

37 cm

19 cm

14 cm

22 cm

0.5 m

1.2 m2 cm2 cm

4 cm

5 mm

35 cm

32 cm

Soil



6378 km

8 Find the capacity of one test tube used to collect blood in millilitres if it is completelyfilled. Express your answer correct to 1 decimal place.

9 The Earth’s radius is approximately 6378 km. Assuming that the Earth is a perfect sphere, find its volume in km3. Express your answer in scientific notation.

10 Find the amount of helium gas (m3) needed to fill the blimp with dimensions shown below.

12 mm

150 mm

GAME

time

Perimeter, area and volume — 002

WorkS

HEET 6.3 MATHS ISMARVELLOUS

MATHS ISMARVELLOUS

18 m28 m

MA

TH

SQUEST

C H A L L

EN

GE

MA

TH

SQUEST

C H A L L

EN

GE

1 An A4 sheet of paper measures 295 mm by 210 mm. What percentageof the paper is enclosed by the largest circle that can be drawn on thesheet?

2 How does the area of a parallelogram compare with the area of atriangle if both have the same height and base length?


Australian inventions!Australian inventions!

Volume = G m3 (2d.p.)

Area = H m2

Volume = I cm3

Area = K mm2

Area = L cm2

Volume = M m3

Area = N m2

Area = A cm2

Volume = B mm3

Area = C m2 (2d.p.)

Volume = D m3

Area = E cm2

Volume = F cm3 (2d.p.)

Volume = O m3

Area = P cm2 (2d.p.)

Volume = R m3

Area = T mm2

Volume = X cm3 (2d.p.)

Area = Y m2

13 cm3 m

700 cm 400 cm

2 m

8 cm

5 mm

11 m8 m12 m

2 mm

2 mm

10 cm

The answers to the measurement problems give the puzzle code.

15 mm

area = 3 cm2

8 cm

3 m

3 m

1 m

1 m

9 cm

8 cm5 cm

15 cm

3 cm 4 cm

20 mm

3 m

13 m5 m

12 m

6 m

5 m8 m

2 cm

16 mm

5 cm

6 cm

10 m

8 m

6 m

20 4.5 4.5 4.528.28 67.5 92 120 27 50.27 4 92 30 32056

20 4 92 28.27 1200 20 56 471.24 125.66 4 4.5 14.14 110 320

27 67.5 27 56 120 24 120 67.5 28.27 56 120 1.5 67.5 120

30




Copy the sentences below. Fill in the gaps by choosing the correct word or expression from the word list that follows.

1 The metric system of units for measuring length is based on the.

2 To convert from kilometres to metres, you by 1000.

3 When converting to a larger unit, .

4 The of a shape is the distance around the shape.

5 The formula C = is used to find the circumference (perim-eter) of a circle.

6 The of a shape is a measure of the amount of surfaceenclosed by the shape.

7 The area of a circle can be found using the formula A = .

8 Area is measured using units based on the metre.

9 A is a portion of a circle formed by two radii and the arc ofthe circle between them.

10 To find the area of a sector, multiply the formula for the area of a circle bythe of the circle formed by the sector.

11 Prisms are 3-dimensional figures, which have uniform .

12 To find the surface area of a prism, find the area of each andadd them together.

13 The surface of a cylinder is made up of two and a.

14 The volume of a solid object is the amount of it occupies.

15 is usually applied to the measurement of liquids.

16 The volume of a prism can be found by multiplying the area of its cross-section (or base) by its .

17 The formula V = πr2h is used to find the volume of a .

18 The formula V = πr3 is used to find the volume of a .

summary

13---

43---

W O R D L I S Tcapacityfacespherecirclescone

perimeterdividefractionsectorarea

squarerectangle πr2

metrespace

multiplyheight2πrcross-sections



1 Fill in the gaps.

a 26 mm = cm b 1385 mm = cm

c 1.63 cm = mm d 1.5 km = m

e 0.077 km = m f 2850 m = km

2 Find the cost of 1.785 km of cable, if the cable costs $4.20 per metre.

3 Find the circumference of circles with the following dimensions (correct to 1 decimal place).

a radius 4 cm b radius 5.6 m c diameter 12 cm

4 Find the perimeter of the following shapes (correct to 1 decimal place).

a b

c d

e f

g h

6A

CHAPTERreview

6A6B

6B11 cm

30 cm

294 mm 9 mm

6.2 mm

8.5 m

48 mm

1.9 m

3.6 m 120 m

17 cm

24 mm



5 Find the area of the following shapes.a b

c d

e f

g h

6 Find the area of the following 2-dimensional shapes by dividing them into simpler shapes.(Where necessary, express your answer correct to 1 decimal place.) a b

c d

7 Find the area of the sectors below (correct to 2 decimal places). a b c d

6C

5 m

38 cm

25 cm

31 cm

25 cm

60 m

140 m

64 m

20 mm

10 mm

9 mm70 cm

31 cm

18.5 cm

30.2 cm

94 mm

6C

30 cm

26 cm9 m

23 m23 m

14 m

25 cm

22 cm23.9 cm

7 cm

9.9 cm

6D

20 m

8.7 cm

18 cm

70°

124 m

40°


C h a p t e r 6 P e r i m e t e r , a r e a a n d v o l u m e 2538 Find the perimeter of the sectors in question 7 (correct to 2 decimal places).

9 Find the surface area of the bedroom storage box shown.

10 Find the surface area of the triangular prism at right.

11 Find the cost of painting an outer surface (including the lid) of a cylindrical water tank, which has a radius of 2.2 m and a height of 1.6 m, if the paint costs $1.90 per square metre.

12 Find the volume of each of the following.

a b c

d e f

g h i

6D6E

35 cm 80 cm

42 cm

6F22 cm

40 cm25 cm

6F

6G,H, I

7 cm

12 cm

8 cm

7 cm

35 cm

40 cm

1 m

3.7 m

10 cm

12 cm

30 cm

12 cm

10 cm

11 cm

9 cm

30 cm

20 cm

42 cm

12 mm



13 A hemisphere is mounted on top of a cube of side length 10 cm, as shown at right. If the diameter of a hemisphere is equal to the side length of a cube, find the volume of this compound solid. (Give your answer correct to 2 decimal places.)

Non-calculator questions

14 a Calculate the area of a rectangle with side lengths 10 cm and 8 cm.b Calculate the area of a parallelogram with

length 8 cm and perpendicular height 8 cm.c Calculate the volume of a pyramid with a

base area of 20 cm2 and a perpendicular height of 6 cm.

d Calculate the volume of a cone with a base area of 50 cm2 and a perpendicular height of 30 cm.

15 Calculate the area of these shapes. (Leave your answer in terms of π where necessary.)a b c

16 a Calculate the surface area of these shapes. (Leave your answer in terms of π where necessary.)i ii iii

b Calculate the volume of the shapes in part a. (Leave your answer in terms of π where necessary.)

6I

4 cm

6 cm

5 cm

3 cm 4 cm

10 cm

5 cm 20 cm10 cm

A = 10.8 cm2

10 cm

5 cm

testtest

CHAPTERyyourselfourself

testyyourselfourself

6


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