Chapter 4 Surface Area and Volume

ihg

7.2

5.6

3.5

12.83.5 2.4

2.85.5

11.3

7.7

2 Rice crackers of diameter 4 cm are packed in acardboard box of height 20 cm. Calculate, correctto one decimal place: WAFERS

20 cm

4 cm

a the volume of the crackers in the boxb the volume of the boxc the percentage of the box that is empty space.

3 This swimming pool is 25 m long and10 m wide. The depth of the waterranges from 1 m to 3 m. Calculatethe capacity of this pool in kilolitres. 3 m

10 m

25 m1 m

4 A wedding cake with three tiers rests on a table. Eachtier is 6 cm high. The layers have radii of 20 cm, 15 cmand 10 cm respectively. Find the total volume of thecake, correct to the nearest cm3.

620

615

610

5 A fish tank that is 60 cm long, 30 cm wide and 40 cm high is filled with water to 5 cm belowthe top. Calculate the volume of the water in litres.

6 Find, correct to two decimal places, the volume of each solid. All lengths shown are in centimetres.

cba

fed

1648

8

12

20

40

10 10

radius of circle = 4 cm

50

35

15

5

5

510 45

15 5 5

1012

Shut

ters

tock

.com

/Joh

nW

ollw

erth

See Example 14

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NEW CENTURY MATHS ADVANCEDfor the A u s t r a l i a n C u r r i c u l u m1010A

lkj

36

8 625

15

8

560

5 14

100

ihg11.3

7.2

19.6

12.73.2

14

10

25

3.6

4.8 6.4

8.3

7 a Find, correct to two decimal places, the volume ofthis greenhouse.

b If this greenhouse costs 0.5c per m3 per hour to heat,how much is this per day, correct to the nearest cent?

3 m

4 m 10 m

Technology Approximating the volume of

a pyramid

In this activity, we will use a spreadsheet toapproximate the volume of a rectangularpyramid by slicing it into tiny layers ofrectangular prisms of equal thickness. 6

8

10

Let L 8 be the length of the prism, W 6 be the width and H 10 be the height.The thickness, T, of each layer is given by the formula T H

number of layers).

Starting at the bottom, the length and width of each layer are decreased by the amountsL

number of layersand W

number of layerswith each step.

1 Set up your spreadsheet as shown.A B C D E F

12 Number of

layers 3 H L W Thickness of

layersVolume oflayer

Sum ofvolumes

4 10 8 6 $A$4/$D$2 B4*C4*D4 E45 B4-$B$4/$D$2 C4-$C$4/$D$2 E5F4...

13

Stage 5.3

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Surface area and volume

2 Let the number of layers be 10. Enter 10 in cell D2.3 Copy each formula down to row 13.4 Explain the results in cells E13 and F13.5 How accurate was your result in F13? Explain.6 Print out your spreadsheet and paste it into your book.7 Enter 40 (layers) in cell D2 and copy each formula down to row 43.8 In one or two sentences compare your result in F43 with the previous result in F13 from

question 4.9 Enter each value in cell D2 and copy down the formulas as requested.

a 100 (copy down to row 103) b 200 (copy down to row 203)c 400 (copy down to row 403)

10 Use the formula V 13

Ah to calculate the exact volume of the pyramid.

11 Write a brief report about your results in questions 9 and 10.

4-07Volumes of pyramids, cones andspheres

Volume of a pyramid

Summary

Volume of a pyramid

h

A

V 13

Ah

where A area of the base and h perpendicular height.

Example 15

Find the volume of each pyramid.

ba

27 mm 32 mm

36 mm

8 m

10 m

14 m

Stage 5.3

Technology worksheet

Drawing pyramids andcones

MAT10MGCT10006


Measuring pyramids

MAT10MGCT10002

Worksheet

Back-to-front problems(Advanced)

MAT10MGWK10206

1299780170194662


Solutiona A 27 3 32

864b A 1

23 8 3 14

56

V 13

Ah

13

3 864 3 36

10 368 mm3

V 13

Ah

13

3 56 3 10

186 23

m3

Example 16

Find the volume of a square pyramid with base length 48 mm and slant height 51 mm.

SolutionFirst find h, the perpendicular height of the pyramid.

48 mm

h

51 mmh2 512 242

2025

h ffiffiffiffiffiffiffiffiffiffi

2025p

45 mm

A 48 3 48 2304

V 13

3 2304 3 45

34 560 mm3

Volume of a coneA cone is like a circular pyramid so:

V 13

Ah 13

3 pr2 3 h 13

pr2h

Summary

Volume of a cone

r

h

V 13

pr2h

where r radius of the base and h perpendicular height.

Stage 5.3


Approximating thevolume of a cone

MAT10MGCT10003

Video tutorial

Area and volume

MAT10MGVT00004

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Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16


Example 17

Find, correct to the nearest cubic millimetre, thevolume of this cone.

25 mm

28 mm

Solution

V 13

pr2h

13

3 p 3 12:52 3 28

4581:4892 . . . 4581 mm3

Example 18

A cone has a base radius of 14 cm and a slant height of 50 cm. Find its volume, correct to twosignificant figures.

Solution

First find the height, h.

h

14 cm

50 cm

h2 502 142

2304

h ffiffiffiffiffiffiffiffiffiffi

2304p

48 cm

V 13

3 p 3 142 3 48

9852:0345 . . . 9900 cm3

Volume of a sphere

Summary

Volume of a sphererV 4

3pr3

where r radius of the sphere.

Stage 5.3

1319780170194662


Example 19

Find, correct to two significant figures, the volume of each solid.ba

18 cm

1.3 m

Solution

a V 43

pr3

43

3 p 3 93 r 12

3 18 9

3053:6280 . . . 3100 cm3

b V 12

343

pr3

23

pr3

23

3 p 3 1:33

4:6013 . . . 4:6 m3

Exercise 4-07 Volumes of pyramids, cones andspheres

1 Find the volume of each pyramid.

cba

fed

8 cm

9 cm

10 cm

10 cm

6 cm

8 cm

12 cm

5 cm

14 m

18 m

8 m

20 cm

12 cm

15 cm

5 m

8 m

6 m

Stage 5.3

See Example 15

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2 For each pyramid, find correct to one decimal place:i its perpendicular height ii its volume

cba

fed

18 cm

18 cm

15 cmh

h

60 m

41 m18 m

50 m25 mm 25 mm

14 mm14

mm

68 mm 61 mm

11 mm

11 mm32 mm 32 mm

8.5 m 8.5 m

3.6 m 3.6 m3.6 m 3.6 m

160 cm

126 cm

116 cm

105 cm

3 Find, correct to the nearest whole number, the volume of each cone.

cba

9 m

4 m

10 cm

12 cm

17 mm

20 mm

fed

12 cm7 cm 10 cm

15 cm

30 mm

18 mm

4 For each cone, find correct to one decimal place:i its perpendicular height ii its volume

cba

fed

7 cm

3 cm

4.4 m

4.5 m

10 cm

8 cm

0.8 m

3.6 m

68 m

247 m83 cm

83 cm

Stage 5.3

See Example 16

See Example 17

See Example 18

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5 For each solid, find correct to the nearest whole number:i its volume ii its capacity

cba

fed

15 mm 11 m10.8 cm

24 m8 cm

16 mm

6 The Earth has a radius of approximately 6400 km. Calculate its volume in scientific notationcorrect to two significant figures.

7 A grain hopper is in the shape of a square pyramid. 4.5 m

5 m

4.5 m

a Find the volume of grain that it holds when full.b If there are 750 kg of wheat per m 3, find the mass of

grain in the hopper when it is filled to three-quarters ofcapacity. Give your answer correct to the nearest tonne.

8 A pyramid has a volume of 360 m3 and a base area of 48 m2.Calculate its perpendicular height.

9 A square pyramid has a volume of 800 cm3 and a perpendicular height of 12 cm. Calculate,correct to one decimal place, the length of its base.

10 A cone has a volume of 600 m3 and a base radius of 10 m. Calculate, correct to one decimalplace, its perpendicular height.

11 A cone has a volume of 160 cm3 and a perpendicular height of 20 cm. Calculate, correct toone decimal place, its radius.

12 Calculate, correct to one decimal place, the radius of a sphere with a volume of 81 585 mm3.

4-08 Volumes of composite solids

Summary

PrismV Ah A

h

CylinderSA 2pr2 2prh

V pr2hh

r

PyramidV 1

3Ah

h

A

ConeSA prl pr2

V 13

pr2h

lh

r

SphereSA 4pr2

V 43

pr3r

Stage 5.3

See Example 19

Worksheet

A page of solid shapes

MAT10MGWK10205

Worksheet

Volume and capacity

MAT10MGPS00046

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Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16


Note that the formulas for surface area involve two dimensions, for example, r2 or rh, while theformulas for volume involve three dimensions, for example, lwh, r2h or r3.

Example 20

a Find, correct to the nearest cubic centimetre, the volume of this solid.b Find, correct to the nearest litre, the capacity of this solid.

20 cm

35 cm

Solutiona Volume volume of cylinder volume of hemisphere

pr2h 12

343

pr3

pr2h 23

pr3

p 3 102 3 35 23

3 p 3 103

13 089:9693 . . . 13 090 cm3

r 12

3 20 10

b Capacity 13 090 mL 13:09 L 13 L

Exercise 4-08 Volumes of composite solids1 The storage tank shown is completely filled with water.

4 m

2 m

4 m

a Calculate, correct to the nearest cubic metre, the volume ofthe tank.

b Find the capacity of the tank, correct to the nearest kilolitre.

2 Find the volume of each solid. All measurements are in centimetres.

ba c

4

7

7

910

10 6

12

12

12

Stage 5.3

See Example 20

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fed

20

15

12

25

18 24

21

30

20

10

15

3 For each solid, find:i the volume (to the nearest cm3)ii the capacity (in litres, correct to three decimal places).

All measurements are in centimetres.

cba

40

15

2014

24

5

56

12

4 A conical tank (A) and a hemispherical tank (B) have measurements as shown. How muchmore does tank B hold compared to tank A? Answer correct to two decimal places.

3 m

3 m BA

3 m

3 m

5 Spherical balls of diameter 10 cm are stacked inside a box inthe shape of a rectangular prism, as shown.

30 cm40 cm

50 cm

a How many balls will fit in the bottom layer?b If the balls are stacked in the same manner as in the bottom

layer until the box is full, how many balls will fit in the box?

c Calculate, correct to the nearest cubic centimetre, the volumeof the space occupied by the balls when the box is full.

d What percentage of the box is empty space? Give your answercorrect to the nearest whole percentage.

Stage 5.3

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Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16


6 The sand in this hourglass takes up three-quarters of thevolume of the bottom cone.

20 cm

50 cm

a Calculate, correct to the nearest cubic centimetre, the volumeof sand in the hourglass.

b If the sand takes one hour to fall from the top cone to thebottom cone, at what rate is it falling? Give your answer incm3/s, correct to two significant figures.

7 a Calculate the volume of this swimming pool.

10 m

1 m

20 m

10 m

2 m

b Calculate the capacity of the pool if it isfilled to a depth of 20 cm from the top.

c If water costs $1.98/kL, find the cost offilling the pool.

4-09 Areas of similar figures

Summary

Areas of similar figuresIf the matching sides of two similar figures are in the ratio 1 : k, then their areas are in theratio 1 : k2.If the matching sides are in the ratio m : n, then their areas are in the ratio m2 : n2.

A1 : A2 m2 : n2 orA1A2 m

n22

Example 21

What is the ratio of the areas of the similar rectangles shown?

B

14 mm

8 mm

20 mm

35 mm

ASolutionRatio of matching sides A to B 35 : 14

5 : 2

Ratio of areas 52 : 22

25 : 4

Stage 5.3


Excel worksheet: Areaof similar shapes

MAT10MGCT00013


Excel spreadsheet:Area of similar shapes

MAT10MGCT00043

1379780170194662


Example 22

Two similar pentagons have areas in the ratio 144 : 169. Find the ratio of the lengths of theirmatching sides.

SolutionRatio of areas m2 : n2 144 : 169

) Ratio of sides m : n ffiffiffiffiffiffiffiffi

144p

:ffiffiffiffiffiffiffiffi

169p

12 : 13

Example 23

Two similar triangles have matching sides in the ratio 3 : 5. If the area of the larger triangle is225 cm2, find the area of the smaller triangle.

SolutionLet the area of the smaller figure be A.

A3 5

225 cm2Ratio of matching sides 3 : 5Ratio of areas 32 : 52 9 : 25

) A225 9

25

A 925

3 225

81 cm2

The area of the smaller triangle is 81 cm2.

Exercise 4-09 Areas of similar figures1 For each pair of similar figures, find the ratio of their areas.

ba

dc

1 cm3 cm

1.5 m

2.5 m

9 cm 5 cm 4 cm 6 cm

Stage 5.3

See Example 21

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Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16


2 For each ratio of the areas of two similar figures, find the ratio of the lengths of their matchingsides.

a 9 : 25 b 1 : 100 c 64 : 25 d 16 : 813 Find x if these triangles are similar.

12

x

A1 = 144

A1 = 108A2 = x

A = xA = 3

A2 = 324

7.85.2

2.80.8

Area = 12 cm2

Area = 3 cm2

7 cma b

c d

x cm

4 Two circles have radii in the ratio 3 : 5. If the larger area is 150 cm2, find the area of thesmaller circle.

5 Similar squares have sides in the ratio 7 : 4. If the area of the smaller square is 14.4 cm2, findthe area of the larger square.

6 Two similar triangles have areas in the ratio 4 : 9. If the length of the base of the smallertriangle is 5 cm, find the length of the base of the larger triangle.

7 Two similar rectangles have their areas in the ratio 36 : 121. If the width of the smallerrectangle is 84 cm, find the width of the larger rectangle.

8 If the radius of a circle is doubled, how has its area changed?

9 If the area of a square is divided by 9, how have the sides changed?

10 If the sides of a triangle are increased by 2.5, how has its area changed?

11 If the area of a trapezium is decreased by 1100

, how have the sides changed?

Investigation: Surface areas and volumes of similar solids

1 a Calculate the volume of this rectangular prism.2 cm

6 cm

8 cm

b Calculate the surface area of the rectangular prism.c If the length, width and height are all doubled, what

happens to:i the volume? ii the surface area?

d Copy and complete:If the length, width and height are all doubled, the volume is increased ______ times andthe surface area is increased ______ times.

Stage 5.3

See Example 22

See Example 23

1399780170194662


4-10Surface areas and volumes of similarsolids

Summary

Surface areas and volumes of similar solidsIf the matching sides of two similar solids are in the ratio 1 : k, then their surface areas are inthe ratio 1 : k2 and their volumes are in the ratio 1 : k3.If the matching sides are in the ratio m : n, then their surface areas are in the ratio m2 : n2

and their volumes are in the ratio m3 : n3.

SA1SA2 m

2

n2and

V1V2 m

3

n3

2 a Explain why these rectangular prisms are similar solids.

2 cm

1 cm3 cm

2 cm

6 cm

4 cmb What is the ratio of their matching sides?c What is the ratio of their surface areas?d What is the ratio of their volumes?

3 For the spheres A and B, find the ratio of:a their radiib their surface areasc their volumes

9 cm

3 cm

A

B

4 How is the ratio of the surface areas of similar solids related to the ratio of matchingsides?

5 How is the ratio of the volumes of similar solids related to the ratio of their matchingsides?

Stage 5.3

NSW

Worksheet

Areas and volumes ofsimilar figures

MAT10MGWK10207

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Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16


Example 24

For these two similar triangular prisms, find the ratio of their:

a surface areasb volumes

2.2 cm2.4 cm

3 cmX3.3 cm

3.6 cm

4.5 cmY

Solutiona Ratio of sides X to Y 3 : 4:5 or 2:2 : 3:3 or 2:4 : 3:6

6 : 9 2 : 3

Ratio of surface areas 22 : 32

4 : 9

b Ratio of volumes 23 : 33

8 : 27

Example 25

Two similar cylinders have their surface areas in the ratio 25 : 36. If the volume of the smallercylinder is 250 cm3, find the volume of the larger solid.

SolutionRatio of surface areas 25 : 36

) Ratio of matching sides ffiffiffiffiffi

25p

:ffiffiffiffiffi

36p

5 : 6

) Ratio of volumes 53 : 63

125 : 216

Let the volume of the larger cylinder be V.

V

250 216

125

V 216125

3 250

432

[ The volume of the larger cylinder is 432 cm3.

Stage 5.3

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Exercise 4-10 Surface areas and volumes of similarsolids

1 For each pair of similar solids, find the ratio of:i the smaller surface area to the larger surface areaii the smaller volume to the larger volume

3 cm

a b

c d

5 cm

3.6 m 2.4 m

12 cm15 cm

22.5 m

9

2 Two similar pyramids have surface areas of 81 cm2 and 100 cm2. Find the ratio of their:a matching side lengths b volumes.

3 Two similar prisms have volumes of 125 cm3 and 343 cm3. Find the ratio of their:a matching sides b surface areas.

4 Blocks of chocolate are sold in the shape of similar triangular prisms. The areas of thetriangular faces of two prisms are 6400 mm2 and 1600 mm2. If the volume of the smallerprism is 9600 mm3, find the volume of the larger prism.

5 There are two similar cylindrical drink cans. The larger can is 15 cm high and contains 350 mLof drink. If the smaller can is 9 cm high, how much drink does it contain?

6 A box of washing powder is 12 cm tall and contains 750 g of washing powder. A similar box is18 cm tall. How much washing powder does it contain?

7 A large fish tank has a capacity of 624 L. A smaller, similar fish tank has half the length, widthand depth of the large tank. Find the capacity of the smaller tank.

8 A cylinder has its height and radius increased 1.5 times. By what factor has its:a surface area increased? b volume increased?

9 A spherical balloon has a radius of 8 cm. By what factor is the volume decreased if the radiuschanges to 6 cm?

Stage 5.3

See Example 24

See Example 25

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Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16


Power plus

1 A square prism and square pyramid have the same base and the same surface area. Showthat the slant height, l, of the pyramid is l 5

2s where s is the length of the base.

2 A cylinder with diameter and height 2r has the same surface area as a sphere of radius R.

Show that R ffiffiffiffiffiffi

32

r

r

.

R

2r

2r

3 A sphere and a cone have the same radius and volume. Show that the cones height, h, isfour times the radius, r.

r

h

r

4 A sphere and a cone fit inside identical cylinders with the same base diameter and height.

2r

2r

2r

2r

a Find the ratio Volume of cone : Volume of sphere : Volume of cylinderb Show that Volume of cone Volume of sphere Volume of cylinder

5 A cube is divided into six identical square pyramids as shown, each with a perpendicularheight that is half the length of the base edge. Show that the volume of each pyramid isone-third the volume of a square prism with the same base edge and perpendicularheight.

2s

2s2s

s

2s

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Chapter 4 review

n Language of maths

apex base capacity circle

cone cross-section cubic curved surface

cylinder diameter hemisphere kilolitre

litre perpendicular height pyramid radius

ratio sector similar figures similar solids

slant height sphere surface area volume

1 Which word means a slice of a prism or cylinder?

2 Name three solids that have a curved surface area.

3 What is the formula for the curved surface area of a cone?

4 Explain the difference between the perpendicular height and the slant height of a pyramid.

5 What is the formula V 13pr2h used for?

6 Describe the relationship between the volumes of similar solids.

n Topic overview

Copy and complete the table below.

The best part of this chapter was

The worst part was

New work

I need help with

Puzzle sheet

Surface area andvolume crossword

(Advanced)

MAT10MGPS10208

Quiz

Area and volume

MAT10MGQZ00004

144 9780170194662

Copy and complete this mind map of the topic, adding detail to its branches and usingpictures, symbols and colour where needed. Ask your teacher to check your work.

Compositesolids Prisms

Cylinder Cone Sphere Pyramids

SURFACEAREA

Similar solids ratio of areas :

VOLUME Similar solids ratio of volumes :

1459780170194662

Chapter 4 review

1 Find the surface area of each prism.

cba

fed

0.4 m

0.5 m

0.8 m0.3 m

45 mm

15 mm7 cm

48 cm50 cm

3.6 m

12 m

3 m

8 m

6 cm

4 mm

5 mm24 mm

2 Calculate, correct to one decimal place, the surface area of each solid.

cba

21

35

23

15

4.8Cylinder,open atone end

2.7

fed

50 cm

50 cm

20 cm5 cm 5 cm

15 cm

30 cm

30 cm30 cm

18 cm 34 cm

25 cm

3 Find the surface area of each pyramid.

cba

16 cm16 cm

22 cm

54 cm

36 cm

30 cm

14 cm

25 cm

See Exercise 4-01

See Exercise 4-02

Stage 5.3

See Exercise 4-03

146 9780170194662

Chapter 4 revision

4 Find, correct to the nearest square metre, the surface area of each solid. All measurementsare in metres.

cba

fed

8

20

closed

48

40

open

11

60

closed

6 m

17 m

25 m

5 Find, correct to the nearest square centimetre, the surface area of each solid. All measurementsare in centimetres.

fed

30

16

18

12

25

25

cba

18

16

282

45

12 4

20

18

127

6 Calculate, correct to nearest cubic metre, the volume of each solid. All measurements are inmetres.

a5025

25

b

24

42

28

18

c

20

23

15

Stage 5.3

See Exercise 4-04

See Exercise 4-05

Stage 5.3

See Exercise 4-06

1479780170194662

Chapter 4 revision

7 Find, correct to two decimal places (where necessary), the volume of each solid.

b ca

11 m

11 m

8 m

15 cm 18 cm

25 m

m

14 mm 14 mm

12 cm

ed

8 cm

20 cm

28 mm

50 mm

f

6 m

8 Find, correct to the nearest whole number, the volume of each solid.

cba

fed

80 mm

45 mm

80 mm

45 mm

45 mm

45 mm

6 cm

8 cm

8 cm

8 cm

4.5 m

4.5 m

4.5 m

18 cm

24 cm

12 cm

24 m

44 m

9 a Two similar circles have radii in the ratio 4 : 5. If the smaller area is 150cm2, find the areaof the larger circle.

b The radius of a circle is increased by a factor of 2 12. By what factor has the area increased?

10 a The areas of the bases of two similar rectangular prisms are in the ratio of 25 : 64. If thevolume of the larger prism is 1024 cm2, find the volume of the smaller prism.

b Two similar pyramids have volumes of 216 cm3 and 343 cm3. Find the ratio of theirsurface areas.

Stage 5.3

See Exercise 4-07

See Exercise 4-08

See Exercise 4-09

See Exercise 4-10

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Chapter 4 revision

Documents

Chapter 4 Surface Area and Volume