Measurement and geometrySurface area and volumeweb2.hunterspt-h.schools.nsw.edu.au/studentshared...† (STAGE 5.3) solve problems involving surface area and volume of right pyramids,

4Measurement and geometry

Surface areaand volumeSome theme parks have wave pools, which are bigswimming pools that simulate the movement of the water ata beach. A large volume of water is quickly released into oneend of the pool, which produces a large wave that movesacross the pool to the other end. The excess water in thepool is recycled so that it can be used to produce morewaves.

n Chapter outlineProficiency strands

4-01 Surface area of a prism U F PS R C4-02 Surface area of a cylinder U F PS R4-03 Surface area of a pyramid* U F PS R C4-04 Surface areas of cones and

spheres*U F PS R C

4-05 Surface areas of compositesolids

U F PS R C

4-06 Volumes of prisms andcylinders

U F PS R

4-07 Volumes of pyramids,cones and spheres*

U F PS R

4-08 Volumes of compositesolids*

U F PS R

4-09 Areas of similar figures* U F PS R C4-10 Surface areas and volumes

of similar solids*U F PS R C

*STAGE 5.3

nWordbankcross-section A ‘slice’ of a solid, taken across the solidrather than along it

curved surface area The area of the curved surface of asolid such as a cylinder or sphere. The curved surface ofa cylinder is a rectangle when flattened.

hemisphere Half a sphere

pyramid A solid with a polygon for a base and triangularfaces that meet at a point called the apex

sector A region of a circle cut off by two radii, shaped likea piece of pizza

slant height The height of a pyramid or cone from itsapex (top) to its base along a side face rather than itsperpendicular height

Shut

ters

tock

.com

/CJP

hoto

9780170194662

NEW CENTURY MATHS ADVANCEDfor the A u s t r a l i a n C u r r i c u l u m10þ10A

n In this chapter you will:• solve problems involving the surface areas and volumes of right prisms• calculate the surface area and volume of cylinders and solve related problems• (STAGE 5.3) solve problems involving surface area and volume of right pyramids, right cones,

spheres and related composite solids• calculate the surface areas and volumes of composite solids• (STAGE 5.3) investigate ratios of areas of similar figures• (STAGE 5.3) investigate ratios of surface areas and volumes of similar solids

SkillCheck

1 Calculate the area of each shape. All measurements are in centimetres.

ca

14

26

20

28

35

b 14

2818

30

2 Find, correct to two decimal places, the area of each sector.

10°8 m 8 m

b ca1.2 m120°

2 m

110°

4-01 Surface area of a prismA cross-section of a solid is a ‘slice’ of the solid cut across it,parallel to its end faces, rather than along it. A prism has thesame (uniform) cross-section along its length, and eachcross-section is a polygon (with straight sides).

This trapezoidal prism has identical cross-sections that are trapeziums.

A right prism

cross section

Worksheet

StartUp assignment 3

MAT10MGWK10015

Skillsheet

Solid shapes

MAT10MGSS10007

Skillsheet

What is volume?

MAT10MGSS10008

Puzzle sheet

Area

MAT10MGPS00010

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

Surface area and volume

Summary

The surface area of a solid is the total area of all the faces of the solid. To calculate thesurface area of a solid, find the area of each face and add the areas together.

Example 1

Find the surface area of the prism.

15 cm

12 cm8 cm

Closed triangular prism

SolutionThe open prism has five faces: two identical triangles(front and back) and three different rectangles.Using Pythagoras’ theorem to find m, the hypotenuse ofthe triangle:

m2 ¼ 82 þ 152

¼ 289

m ¼ffiffiffiffiffiffiffiffi

289p

¼ 17Surface area ¼ 2 trianglesþ 3 rectangles

¼ 2 312

3 8 3 15� �

þ ð17 3 12Þ þ ð8 3 12Þ þ ð15 3 12Þ

¼ 600 cm2

base 12

815m

Example 2

Calculate the surface area of this trapezoidal prism.

18 cm

13 cm

10 cm

12 cm

24 cm

15 cm

1039780170194662


SolutionThis trapezoidal prism has 6 faces:two identical trapeziums (front andback) and four different rectangles.

10

10

24

18

1215 13

Area of each trapezium ¼ 12

3 ð10þ 24Þ3 12

¼ 204 cm2

Surface area ¼ ð2 3 204Þ þ ð18 3 10Þ þ ð18 3 15Þ þ ð18 3 24Þ þ ð18 3 13Þ¼ 1524 cm2

Exercise 4-01 Surface area of a prism1 Find the surface area of each prism.

cba

fed

3 m

12 m7 m

2 cm

15 cm

7 cm

41 mm

20 mm18 mm

40 mm

3 m

8 m

5 m

10 m24 mm

7 mm20 mm 6 m

2.5 m

10 m

2 Name the prism that each net represents, then calculate the surface area of the prism. Alllengths are in metres.

16

9

18 9

72

45

24

51

66

3024 13

26

25

ba dc

See Example 1

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3 This classroom is being renovated. Find:

3 m

10 m

8 m

a the area of the floor to be carpeted and thecost, at $105 per square metre

b the ceiling and wall area to be painted if theroom contains four windows, each 2.5 m by1.5 m, and a doorway 2 m by 0.8 m.

4 Calculate the surface area of each prism.

cba

fed

10 cm8.4 cm

20 cm

15 cm

8 cm

13 mm

15 mm

24 mm10 mm

6 m

3 m 2 m 10 m

10 cm

9 cm5 cm

12 cm

18 cm 12 cm

9 cm8 cm

x 14 mm48 mm

50 mm

x

5 This swimming pool is 7.6 m long and4.3 m wide. The depth of the waterranges from 1.3 m to 2.2 m. Calculate,correct to two decimal places:

2.2 m

4.3 m

7.6 m1.3 m

a the area of the floor of the poolb the total surface area of the pool.

4-02 Surface area of a cylinder

Surface area ¼ area of two circlesþ area of rectangle

¼ 2 3 pr2 þ 2pr 3 h

¼ 2pr2 þ 2prh

r

r

circumference= 2πr height, h h

See Example 2

Worksheet

Surface area

MAT10MGWK10016

Puzzle sheet

Surface area

MAT10MGPS00009

Shut

ters

tock

.com

/Sar

ah2

1059780170194662


Summary

Surface area of a cylinderSA ¼ 2pr2 þ 2prh

where r ¼ radius of circular base and h ¼ perpendicular height

The area of the two circular ends ¼ 2pr2 and the area of the curved surface ¼ 2prh.

Example 3

Find, correct to the nearest mm2, the surface area of this cylinder.

40 mm

15 mm

SolutionSurface area ¼ 2pr2 þ 2prh

¼ 2 3 p 3 152 þ 2 3 p 3 15 3 40

¼ 5183:627 . . .

� 5184 mm2

r ¼ 15, h ¼ 40

Example 4

Find, correct to two significant figures, the surface area of:

a a cylindrical tube, open at both ends, with radius 3 cm and length 55 cmb an open half-cylinder with radius 0.5 m and height 3 m.

Solutiona

55 cm

circumference

curved surface

55 cm

3 cm

Surface area ¼ curved surface

¼ 2prh

¼ 2 3 p 3 3 3 55

¼ 1036:725 . . .

� 1000 cm2

r ¼ 3 and h ¼ 55

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b Surface area ¼ 2 semicircle endsþ 12

3 curved surface

¼ 2 312

3 p 3 0:52� �

þ 12

3ð2 3 p 3 0:5 3 3Þ

¼ 5:49778 . . .

� 5:5 cm2

0.5 m

3 m

0.5 m

3 m

curvedsurface end

Exercise 4-02 Surface area of a cylinder1 Calculate, correct to two decimal places, the surface area of a cylinder with:

a radius 3.4 m, height 6 m b diameter 35 mm, height 15 mmc diameter 6.2 cm, height 7.5 cm d radius 0.8 m, height 2.35 m

2 Find, correct to the nearest whole number, the curved surface area of a cylinder with:

a radius 1.5 m, height 3.75 m b diameter 27 cm, height 41 cm

3 Calculate the surface area of each solid, correct to the nearest square metre. All lengths shownare in metres.

a closed cylinder7.2

15.1

b closed cylinder

25

15

c cylinder with one open end

1.5

0.37

d closed half cylinder

29.316.2

e half cylinder withopen top

1.2

2.85

f half cylinder with open top,one end open

5.75

1.5

g cylinder openboth ends

1230

h half cylinder, openboth ends

6.754.5

4.5

See Example 3

See Example 4

1079780170194662


4 A swimming pool is in the shape of a cylinder1.4 m deep and 5 m in diameter. The inside ofthe pool is to be repainted, including the floor.

5 m

1.4 ma Find the area to be repainted, correct to one

decimal place.

b Find the number of whole litres of paintneeded if coverage is 9 m2 per litre.

5 Which tent has the greater surface area?

1.5 m

2 m

(Note: the floor is included for both tents)

3 m

2.24 m

2 m 3 m

Technology Surface areas and volumes of

solids

In this activity, we will use Google Sketchup to construct and measure solid shapes.1 Use the arc tool and the line tool to create a semicircle.

2 To make a solid, select Push/Pull.

3 Use the Orbit tool to reorientate your solid.

4 Use the Dimension tool to obtain the dimensions of your half-cylinder. Calculate its surfacearea and volume.

5 Draw a rectangular prism using the Rectangle tool and the Push/Pull tool.

6 The Push/Pull tool can be used to cut away parts of a solid. Use the Rectangle tool tocreate rectangles on the top of the prism. Then use the Push/Pull tool to remove it.An example is shown below.

Technology worksheet

Excel worksheet:Volume calculator

MAT10MGCT00006


Excel spreadsheet:Volume calculator

MAT10MGCT00036


Excel worksheet:Volume of a box

MAT10MGCT00007


Excel spreadsheet:Volume of a box

MAT10MGCT00037

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


7 Start with a rectangular prism and cut out2 rectangles to create a seat. ClickWindow and Materials to change theappearance of the seat.

8 Draw each solid shown below and find its surface area and volume.

a swimming pool b a bin c a bench

4-03 Surface area of a pyramidA pyramid is a solid shape with a polygon for its base and triangular faces that meet at a point orvertex called its apex. Like a prism, a pyramid is named by the shape of its base.

Square pyramid Triangular pyramid Rectangular pyramid

A cone is a solid shape with a circular base and a curved surface that also has an apex. However,a cone is not a pyramid because its base is not a polygon (a circle does not have straight sides).The slant height of a pyramid or cone is the height from its apex to the base, along a side face.It is different from the perpendicular height of a pyramid or cone, which is the perpendiculardistance from the apex to the base.The surface area of a pyramid is calculated by adding the area of the base and the areas of thetriangular faces.

slant height

perpendicular height

apex

Stage 5.3


Measuring pyramids

MAT10MGCT10002


Drawing pyramids andcones

MAT10MGCT10006

1099780170194662


Stage 5.3 Example 5

Find the surface area of each square pyramid.

14 cm

14 cm

20 cm 20 cm

30 cm

a b

Solutiona Surface area ¼ area of square baseþ area of 4 triangular faces

¼ 14 3 14þ 4 312

3 14 3 20

¼ 756 cm2

b First find the slant height, s, using Pythagoras’theorem.

s2 ¼ 202 þ 152

¼ 625BC ¼ 1

23 30

AP ¼ffiffiffiffiffiffiffiffi

625p

¼ 25 cm

20 cm

A

W

X

B

Z

C

Y

s

30 cmSurface area ¼ 30 3 30þ 4 312

3 30 3 25

¼ 2400 cm2

Example 6

A rectangular pyramid with a base measuring 10 cm by 8 cm has a perpendicular height of 15 cm.Find its surface area correct to one decimal place.

SolutionFirst find the slant heights AP and AQ.

8 cm

10 cm

15 cm

E

BPC

OQ

D

A

AP2 ¼ AO2 þ OP2

¼ 152 þ 42

¼ 241

OP ¼ 12

3 8

Video tutorial

Surface area ofa pyramid

MAT10MGVT10008

This pyramid has a slant heightof 20 cm

This pyramid has aperpendicular height of 20 cm

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


AP ¼ffiffiffiffiffiffiffiffi

241p

cm

AQ2 ¼ AO2 þ OQ2

¼ 152 þ 52

¼ 250

OQ ¼ 12

3 10

AQ ¼ffiffiffiffiffiffiffiffi

250p

cm

Surface area ¼ area of rectangle baseþ 2 3 Area 4ABC þ 2 3 Area 4ADC

¼ 10 3 8þ 2 312

3 10 3ffiffiffiffiffiffiffiffi

241p

þ 2 312

3 8 3ffiffiffiffiffiffiffiffi

250p

¼ 80þ 10ffiffiffiffiffiffiffiffi

241p

þ 8ffiffiffiffiffiffiffiffi

250p

¼ 361:7328:::

� 361:7 cm2

Exercise 4-03 Surface area of a pyramid1 Find the surface area of each pyramid. Write the answer to part c correct to one decimal place.

cba

5 m

25 m

18 mm

10 mm

13 mm15 mm

8 cm

4 cm

24 c

m

20 cm

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

cba

5 m

8 m

16 mm24 mm

60 mm

25 cm

8 cm

16 cm

Stage 5.3

It is better to leave the lengthsof AP and AQ in surd formrather than round them todecimals so that the finalanswer is accurate.

See Example 5

See Example 6

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3 Calculate, correct to the nearest square centimetre, the surface area of each pyramid. Allmeasurements are in centimetres.

cba

20

16

16

32

24

40

25

37

4 Find the area of each net and hence the surface area of the corresponding pyramid. Allmeasurements are in centimetres. Write the answer to part c correct to the nearest wholenumber.

cba

1212

28.3

24

20

36

10

5 The great pyramid of Khufu (or Cheops) in Egypt has a height of 147 m, and each side of itssquare base measures 230 m. Find its surface area (excluding the base), correct to the nearestsquare metre.

6 Calculate, correct to one decimal place, the surface area of each pyramid.

cba

7 mm36 mm

25 mm24 mm

12 cm 9 cm

5 cm

12 m

20 m

10 m

7 A square pyramid has a surface area of 4704 m2 and a base area of 1764 m2. Find:a the length of its base

b the area of each triangular face

c the slant height of each triangular face

d the perpendicular height of the pyramid

Stage 5.3

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


Just for the record The Platonic solidsThe cube is an example of a regular polyhedron or Platonic solid because it has six identicalfaces. The more formal name for a cube is regular hexahedron, hex meaning ‘six’. There areonly six possible regular polyhedrons: the other five are shown below. Each face on a regularpolyhedron is a regular polygon.

Regular tetrahedron4 faces

Regular octahedron8 faces

Regular dodecahedron12 faces

Regular icosahedron20 faces

The tetrahedron, cube and octahedron occur in nature in the form of certain crystals.What are the shapes of the faces on each Platonic solid shown?

Investigation: The surface area of a cone

The net of a cone is made up of a circle (for the base) and a sector of a circle (for thecurved surface). The second diagram below shows the curved surface of a cone.

cone

O

AB

l

r

Base radius r circumference, AB,

of base = 2πr

sectorO

A

Bl

arc AB = 2πr

Net of the curved surface ofthe cone

We can use this fact to find a formula for the surface area of a cone. Suppose the cone hasa base radius of r and a slant height of l.

Looking at the second diagram, the major arc length AB is a fraction of the circumferenceof the circle and the area of the sector is a fraction of the area of the circle. They should bethe same fraction, so:

Arc lengthCircumference

¼ Area of sectorArea of circle

Stage 5.3

1139780170194662


4-04 Surface areas of cones and spheres

Surface area of a cone

Summary

Surface area of a coneSA ¼ area of curved surfaceþ area of circular base

¼ prl þ pr2

where l ¼ slant height and r ¼ radius of the base

l l

r r

h

1 The major arc length AB is equal to the circumference of the base of the cone in the firstdiagram. Write an algebraic expression for the circumference of the circle in the firstdiagram.

2 Write an algebraic expression for the circumference of the complete circle in the seconddiagram.

3 Write an algebraic expression for the area of the complete circle in the second diagram.

4 Arc lengthCircumference

¼ Area of sectorArea of circle

becomes 2pr2pl¼ Area of sector

pl2

Complete:

) Area of sector ¼ 2pr

2pl3 pl2

¼

5 But the area of the sector is equal to the curved surface area of the cone.Complete the formula for the surface area of a cone.

Surface area ¼ area of curved surfaceþ area of circular base

¼ þ

Stage 5.3



MAT10MGCT10006

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Example 7

For this cone, find correct to one decimal place:

9 cm

18 cma the curved surface area

b the total surface area

Solutionr ¼ 9 cm and l ¼ 18 cma Curved surface area ¼ prl

¼ p 3 9 3 18

¼ 508:9380 . . .

� 508:9 cm2

b Total surface area ¼ prl þ pr2

¼ p 3 9 3 18þ p 3 92

¼ 763:4070 . . .

� 763:4 cm2

Example 8

Find, correct to two decimal places, thesurface area of this cone.

10.4 cm

7.8 cmSolutionFirst calculate the slant height, l:

l2 ¼ 7:82 þ 10:42

¼ 169

l ¼ffiffiffiffiffiffiffiffi

169p

¼ 13

Surface area ¼ prl þ pr2

¼ p 3 7:8 3 13þ p 3 7:82

¼ 509:6919 . . .

� 509:69 cm2

Surface area of a sphereA sphere is a ball shape, a solid that is completely round. All points on its surface lie the samedistance (radius) from its centre. A hemisphere is half a sphere.

Stage 5.3

Video tutorial

Surface area of a coneand sphere

MAT10MGVT10009

Video tutorial

Area and volume

MAT10MGVT00004

1159780170194662


Summary

Surface area of a sphereSA ¼ 4pr2

r

where r ¼ radius of the sphere

Note that the surface area of a sphere is four times the area of the circle that slices through thecentre of a sphere.

Example 9

Find, correct to two decimal places, the surface areaof this sphere.

17 cm

SolutionSurface area ¼ 4pr2

¼ 4 3 p 3 172

¼ 3631:6811 . . .

� 3631:7 cm2

Example 10

Find the surface area of this hemisphere in exact form,in terms of p. 5 m

SolutionSurface area ¼ Area of circular baseþ curved surface area

¼ pr2 þ 12

3 4pr2

¼ pr2 þ 2pr2

¼ 3pr2

¼ 3 3 p 3 52

¼ 75p m2

Stage 5.3

Video tutorial

Surface area of a coneand sphere

MAT10MGVT10009

Answers written as surds or interms of p are exact becausethey are not decimalapproximations.

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


Exercise 4-04 Surface areas of cones and spheres1 Calculate, correct to the nearest cm2, the curved surface area of each cone. All measurements

are in centimetres.

cba

4

8

10

20

44

35

2 Find, correct to one decimal place, the total surface area of each cone.

cba

5 mm

20 mm

8 m

4 m

14 cm

7 cm

3 Calculate in exact form (in terms of p) the total surface area of each cone.

cba

12 m

5 m

14 mm24 mm

18 cm

40 cm

4 Calculate, correct to two decimal places, the surface area of each sphere.

cba

15 mm 11 m 10.8 cm

5 Find in exact form the surface area of each hemisphere.

cba 24 m

8 cm 16 mm

Stage 5.3

See Example 7

See Example 8

See Example 9

See Example 10

1179780170194662


6 Find, correct to the nearest square metre, the surface area of each solid.a a sphere with diameter ¼ 10 m

b an open cone with base radius ¼ 10 m, slant height ¼ 20 m

c an open hemisphere with radius ¼ 10 m

d a cone with base diameter ¼ 10 m, perpendicular height ¼ 20 m

7 The Earth has a radius of approximately 6400 km. Calculate the surface area of the Earth inscientific notation, correct to two significant figures.

8 Find the amount of sheet metal needed to form a conical funnel of base radius 30 cm andvertical height 50 cm, allowing for a 0.5 cm overlap at the join.

9 The curved surface of a cone is made from a sector of a circle with radius8 cm and central angle 216�. Find, correct to two decimal places:

8 cm

216°

a the length of the arc of the circle that forms thecircumference of the cone’s base

b the radius of the cone’s base

c the slant height of the cone

d the total surface area of the cone, including the base

10 Find the radius of each solid if it has a surface area of 6000 mm2. Give your answer correct tothree significant figures.

a a sphere b a closed hemisphere c an open hemisphere

11 A cone has a surface area of 2000 cm2. If the area of its base is 150 cm2, find, correct to twodecimal places:

a the radius of its base b its slant height c its perpendicular height

Mental skills 4 Maths without calculators

Estimating answersA quick way of estimating an answer is to round each number in the calculation.

1 Study each example.a 55þ 132� 34þ 17� 78 � 60þ 130� 30þ 20� 80

¼ ð60þ 20� 80Þ þ ð130� 30Þ¼ 0þ 100

¼ 100 ðActual answer ¼ 92Þb 78 3 7 � 80 3 7

¼ 560 ðActual answer ¼ 546Þc 510 4 24 � 500 4 20

¼ 50 4 2

¼ 25 ðActual answer ¼ 21:25Þ

Stage 5.3

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4-05 Surface area of composite solids

Example 11

Find, correct to one decimal place, the surface area of each solid. All measurements are incentimetres.

4016

20

15

10

12

56

2536

b ca20 15

2 Now estimate each answer.

a 27 þ 11 þ 87 þ 142 þ 64 b 55 þ 34 � 22 � 46 þ 136c 684 þ 903 d 35 þ 81 þ 110 þ 22 þ 7e 517 � 96 f 210 � 38 � 71 þ 151 � 49g 766 � 353 h 367 3 2i 83 3 81 j 984 3 16k 828 4 3 l 507 4 7

3 Study each example involving decimals.

a 20:91� 11:3þ 2:5 � 21� 11þ 3

¼ 13 ðExact answer ¼ 12:11Þb 4:78 3 19:2 � 5 3 20

¼ 100 ðExact answer ¼ 91:776Þc 37:6þ 9:3

41:2� 12:7� 38þ 9

40� 13

¼ 4727

� 5030

� 1:6 ðExact answer ¼ 1:645 . . .Þ

4 Now estimate each answer.

a 3.75 þ 9.381 þ 4.6 þ 10.5 b 14.807 þ 6.6 � 7.22c 18.47 3 9.61 d 4.27 3 97.6

e 11:07þ 18:412:2

f 38:1817:2� 9:6

g 54.75 � 18.6 � 14.4 h 18:46 3 4:939:72� 15:2

i 62.13 4 10.7 j (4.89)2

Worksheet

A page of prisms andcylinders

MAT10MGWK10017

Worksheet

A page of solid shapes

MAT10MGWK10205

1199780170194662


Solutiona This prism has 8 faces: 2 identical L-shapes

(front and back) and 6 different rectangles.

Area of L-shape ¼ 16 3 20� 10 3 12

¼ 200 cm2

Surface area ¼ Front and back L-facesþ 1st topþ 1st rightþ 2nd top

þ 2nd rightþ bottomþ left

¼ ð2 3 200Þ þ ð6 3 15Þ þ ð12 3 15Þþ ð10 3 15Þ þ ð8 3 15Þ þ ð16 3 15Þþ ð20 3 15Þ

¼ 1480 cm2

16

20

15

10

12

Note that the six rectangles can also be thoughtof as one long rectangle of width 15 cm:

Surface area ¼ ð2 3 200Þ þ ð72 3 15Þ¼ 1480 cm2

b This solid is made up of a half-cylinder(3 faces) and a rectangular prism (5 faces).

Surface area of half-cylinder ¼ 2 semi-circular endsþ curved surface area

¼ 2 312

3 p 3 282 þ 12

3 2 3 p 3 28 3 40

� 5981:5924 . . . cm2

Surface area of rectangular prism ¼ Front and back facesþ 2 side facesþ bottom face

¼ ð2 3 40 3 25Þ þ ð2 3 56 3 25Þ þ ð40 3 56Þ¼ 7040 cm2

Total surface area ¼ 5981:5924 . . .þ 7040

¼ 13 021:5924 . . .

� 13 021:6 cm2

c The hollow cylinder is made up of 2 annulus (ring) faces, anoutside curved surface area and an inside curved surface area.

Surface area of annulus faces ¼ 2 3 ðp 3 202 � p 3 152Þ¼ 1099:5574 . . .

Outside curved surface area ¼ 2 3 p 3 20 3 36� 4523:8934 . . .

Inside curved surface area ¼ 2 3 p 3 15 3 36� 3392:9200 . . .

Total surface area ¼ 1099:5574 . . .þ 4523:8934 . . .þ 3392:9200 . . .

¼ 9016:3708 . . .

� 9016:4 cm2

2 3 area between two circles

Length of long rectangle

¼ perimeter of L

¼ 6þ 12þ 10þ 8þ 16þ 20

¼ 72

Radius of semicircle

¼ 12

3 56 ¼ 28

Do not round this partialanswer, else the final answerwill be inaccurate.

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Example 12

Find, correct to the nearest square centimetre, the surface area of each solid. Allmeasurements are in centimetres.

ba

60

11

50

50

Solutiona Surface area ¼ curved surface area of cone

þ curved surface area of hemisphere

¼ prl þ 12

3 4pr2

r ¼ 11, h ¼ 60 and l ¼ ?

l 2 ¼ 112 þ 602

¼ 3721

l ¼ffiffiffiffiffiffiffiffiffiffi

3721p

¼ 61 cm

Surface area ¼ p 3 11 3 61þ 12

3 4 3 p 3 112

¼ 2868:2740 . . .

� 2868 cm2

b Surface area ¼ curved surface of cylinderþ circular base

þ curved surface of hemisphere

¼ 2prhþ pr2 þ 12

3 4pr2

¼ 2prhþ 3pr2

¼ 2 3 p 3 25 3 50þ 3 3 p 3 252

¼ 13 744:4678 . . .

� 13 744 cm2

pr2 þ 2pr2 ¼ 3pr2

r ¼ 12

3 50 ¼ 25

Stage 5.3

1219780170194662


See Example 11

Exercise 4-05 Surface areas of composite solids1 Find the surface area of each prism. All measurements are in centimetres.

9.4

8.5

10.2

3.3

2.7

a

125

67

96

53

50

b

12

12

12

6

6

6

c

2 Circular cracker biscuits of diameter 4 cm are packed in a cardboard box of length 20 cm.

C R I S P I E S4 cm

20 cm

a Calculate the surface area of the box.

b How much cardboard would be saved if the biscuits were packed into a cylindrical box?

3 Find, correct to one decimal place, the surface area of each solid. All measurements are incentimetres.

a b

15

14

20

65

c25 17

48

38 40

30

d16

10

10

30

e

21.2

15

35

f

282

4 a Find, correct to two decimal places, the totalexternal wall area of this above-groundswimming pool.

1.5 m

3 m

4 m

b Calculate the area of the water surface, correctto the nearest m2.

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


5 A wedding cake with three tiers rests on a table. Each tier is6 cm high. The layers have radii of 20 cm, 15 cm and 10 cmrespectively. Find the total visible surface area, correct to thenearest cm2.

620

615

610

6 A wedge of cheese is cut from a cylindrical blockof height 10 cm and diameter 40 cm. Find thetotal surface area of the wedge, correct to twodecimal places. wedge

40 cm

10 cm60°

60°

7 Find, correct to one decimal place, the surface area of each solid. All measurements are incentimetres.

cba

fed

7

24

8

88

6

12

12

12

40

25

15

10

10

10

10

6

Shut

ters

tock

.com

/Joh

nW

ollw

erth

Stage 5.3

See Example 12

1239780170194662


ihg5

5

12

6

4

10

19

24

14

lkj

onm

44

24

12

24

18

60

30

20

4530

24 18

24

5

8

10

6

4-06 Volumes of prisms and cylindersThe volume of a solid is the amount of space it occupies. Volume is measured in cubic units, forexample, cubic metres (m3) or cubic centimetres (cm3).

Summary

Volume of a prismV ¼ Ah

where A ¼ area of base andh ¼ perpendicular height

Volume of a cylinderV ¼ pr2h

where r ¼ radius of circular base and h ¼ perpendicular height

A h

r

h

Stage 5.3

Worksheet

A page of prisms andcylinders

MAT10MGWK10017

Puzzle sheet

Formula matchinggame

MAT10MGPS10018

Worksheet

Volumes of solids

MAT10MGWK10020

Worksheet

Back-to-front problems

MAT10MGWK10021

Worksheet

Volume and capacity

MAT10MGWK10022

Animated example

Volumes of shapes

MAT10MGAE00004

124 9780170194662

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


The capacity of a container is the amount of fluid (liquid or gas) it holds, measured in millilitres(mL), litres (L), kilolitres (kL) and megalitres (ML).

Summary

1 cm3 contains 1 mL.1 m3 contains 1000 L or 1 kL.

1 m3 = 1 kL

1 mL

1 cm3 × 1 000 000 =

Example 13

For this cylinder, calculate: 128 cm

241 cma its volume, correct to the nearest cm3

b its capacity in kL, correct to 1 decimal place.

Solutiona V ¼ p 3 642 3 241

¼ 3 101 179:206 . . .

� 3 101 179 cm3

r ¼ 12

3 128 ¼ 64

b Capacity ¼ 3 101 179 mL

¼ ð3 101 179 4 1000 4 1000Þ kL

¼ 3:101 179 kL

� 3:1 kL

1 cm3 ¼ 1 mL

mLkL L÷ 1000÷ 1000

Example 14

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

cba 40 cm

20 cm

12 cm

15 cm

12 cm

20 cm

9 cm

60 cm

26 cm

y

120°25 mm40 mm

Worksheet

Biggest volume

MAT10MGWK10019

1259780170194662


Solutiona A ¼ 40 3 12þ 20 3 12

¼ 720 cm2

V ¼ Ah

¼ 720 3 15

¼ 10 800 cm3

Area of T cross-section

b The cross-section is a triangle minus a circle.Use Pythagoras’ theorem to find y.

262 ¼ y2 þ 102

y2 ¼ 262 � 102

¼ 576

y ¼ffiffiffiffiffiffiffiffi

576p

¼ 24 cm

Radius of circle ¼ 12

3 9 ¼ 4:5

A ¼ 12

3 20 3 24� p 3 4:52

¼ 176:3827 . . . cm2

V ¼ Ah

¼ 176:3827 . . . 3 60

¼ 10 582:9649 . . .

� 10 583 cm3

4.5

4.5

26

10 10

y

Area of triangle � area of circle

Do not round this partial answer

c A ¼ 120360

3 p 3 252

¼ 654:498 . . . mm2

V ¼ Ah

¼ 654:498 . . . 3 40

¼ 26 179:938 . . .

� 26 180 mm3

Area of sector

Do not round this partial answer

Exercise 4-06 Volumes of prisms and cylinders1 Calculate, correct to one decimal place, the volume of each solid. All lengths are in metres.

cba

fed

4.53.0

1.8 2.4 25

48 0.8

2.5

3.7

4.210.1

6.4

3220

5.2

3.6

7.9

4.59.2

See Example 13

126 9780170194662

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


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 box

b the volume of the box

c 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

1279780170194662


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 ¼ 8 be the length of the prism, W ¼ 6 be the width and H ¼ 10 be the height.The thickness, T, of each layer is given by the formula T ¼ 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 F12 Number of

layers ¼3 H L W Thickness of

layersVolume oflayer

Sum ofvolumes

4 10 8 6 ¼$A$4/$D$2 ¼B4*C4*D4 ¼E45 ¼B4-$B$4/$D$2 ¼C4-$C$4/$D$2 ¼E5þF4...

13

Stage 5.3

128 9780170194662

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


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 fromquestion 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 ¼ 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 ¼ 13

Ah

where A ¼ area of the base and h ¼ 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



MAT10MGCT10006


Measuring pyramids

MAT10MGCT10002

Worksheet

Back-to-front problems(Advanced)

MAT10MGWK10206

1299780170194662


Solutiona A ¼ 27 3 32

¼ 864

b A ¼ 12

3 8 3 14

¼ 56

V ¼ 13

Ah

¼ 13

3 864 3 36

¼ 10 368 mm3

V ¼ 13

Ah

¼ 13

3 56 3 10

¼ 18623

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 ¼ 512 � 242

¼ 2025

h ¼ffiffiffiffiffiffiffiffiffiffi

2025p

¼ 45 mm

A ¼ 48 3 48

¼ 2304

V ¼ 13

3 2304 3 45

¼ 34 560 mm3

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

V ¼ 13

Ah ¼ 13

3 pr2 3 h ¼ 13

pr2h

Summary

Volume of a cone

r

h

V ¼ 13

pr2h

where r ¼ radius of the base and h ¼ 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 ¼ 13

pr2h

¼ 13

3 p 3 12:52 3 28

¼ 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 ¼ 502 � 142

¼ 2304

h ¼ffiffiffiffiffiffiffiffiffiffi

2304p

¼ 48 cm

V ¼ 13

3 p 3 142 3 48

¼ 9852:0345 . . .

� 9900 cm3

Volume of a sphere

Summary

Volume of a sphererV ¼ 4

3pr3

where r ¼ 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 ¼ 43

pr3

¼ 43

3 p 3 93 r ¼ 12

3 18 ¼ 9

¼ 3053:6280 . . .

� 3100 cm3

b V ¼ 12

343

pr3

¼ 23

pr3

¼ 23

3 p 3 1:33

¼ 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

132 9780170194662

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


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

1339780170194662


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 ofgrain 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 ¼ Ah A

h

CylinderSA ¼ 2pr2 þ 2prh

V ¼ pr2hh

r

PyramidV ¼ 1

3Ah

h

A

ConeSA ¼ prl þ pr2

V ¼ 13

pr2h

lh

r

SphereSA ¼ 4pr2

V ¼ 43

pr3r

Stage 5.3

See Example 19

Worksheet

A page of solid shapes

MAT10MGWK10205

Worksheet

Volume and capacity

MAT10MGPS00046

134 9780170194662

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 ¼ volume of cylinderþ volume of hemisphere

¼ pr2hþ 12

343

pr3

¼ pr2hþ 23

pr3

¼ p 3 102 3 35þ 23

3 p 3 103

¼ 13 089:9693 . . .

� 13 090 cm3

r ¼ 12

3 20 ¼ 10

b Capacity ¼ 13 090 mL

¼ 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

1359780170194662


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 bottomlayer 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

136 9780170194662

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 ¼ m2 : n2 orA1

A2¼ m

n2

2

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 ðA to BÞ ¼ 35 : 14

¼ 5 : 2

Ratio of areas ¼ 52 : 22

¼ 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 ¼ m2 : n2 ¼ 144 : 169

) Ratio of sides ¼ m : n ¼ffiffiffiffiffiffiffiffi

144p

:ffiffiffiffiffiffiffiffi

169p

¼ 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 ¼ 3 : 5Ratio of areas ¼ 32 : 52 ¼ 9 : 25

)A

225¼ 9

25

A ¼ 925

3 225

¼ 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

138 9780170194662

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 : 81

3 Find x if these triangles are similar.

12

x

A1 = 144π

A1 = 108 A2 = x

A = xA = 3

A2 = 324π

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.

SA1

SA2¼ m2

n2 andV1

V2¼ m3

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

140 9780170194662

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 ðX to Y Þ ¼ 3 : 4:5 ðor 2:2 : 3:3 or 2:4 : 3:6Þ

¼ 6 : 9

¼ 2 : 3

Ratio of surface areas ¼ 22 : 32

¼ 4 : 9

b Ratio of volumes ¼ 23 : 33

¼ 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 ¼ 25 : 36

) Ratio of matching sides ¼ffiffiffiffiffi

25p

:ffiffiffiffiffi

36p

¼ 5 : 6

) Ratio of volumes ¼ 53 : 63

¼ 125 : 216

Let the volume of the larger cylinder be V.

V

250¼ 216

125

V ¼ 216125

3 250

¼ 432

[ The volume of the larger cylinder is 432 cm3.

Stage 5.3

1419780170194662


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 area

ii 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

142 9780170194662

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. Show

that the slant height, l, of the pyramid is l ¼ 52

s 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 ¼ffiffiffiffiffiffi

32

r

r

.

R

2r

2r

3 A sphere and a cone have the same radius and volume. Show that the cone’s 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 cylinder’b Show that ‘Volume of cone þ Volume of sphere ¼ 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

1439780170194662


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 ¼ 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

148 9780170194662

Chapter 4 revision

Measurement and geometrySurface area and volumeweb2.hunterspt-h.schools.nsw.edu.au/studentshared...† (STAGE 5.3) solve problems involving surface area and volume of right pyramids,

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