Measurement and Geometry Surface area and volumeweb2.hunterspt-h.schools.nsw.edu.au/studentshared... · The surface area of a solid is the total area of all the faces of the solid

3Measurement andGeometry

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 movesfrom one end of the pool to the other. The excess water inthe pool is recycled so that it can be used to produce morewaves.

n Chapter outlineProficiency strands

3-01 Areas of compositeshapes U F PS R

3-02 Surface area of aprism U F PS R C

3-03 Surface area of acylinder* U F PS R

3-04 Surface areas ofcomposite solids* U F PS R C

3-05 Volumes of prismsand cylinders U F PS R C

*STAGE 5.2

nWordbankcapacity The amount of fluid (liquid or gas) in a container

composite shape A shape made up of two or more basicshapes

cross-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 of acylinder is a rectangle when flattened.

cylinder A can-shaped solid with identical cross-sectionsthat are circles

prism A solid with identical cross-sections that arepolygons

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

surface area The total area of all faces of a solid shape

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NEW CENTURY MATHSfor the A u s t r a l i a n C u r r i c u l u m 10

n In this chapter you will:• solve problems involving the surface areas and volumes of right prisms• (STAGE 5.2) calculate the surface area and volume of cylinders and solve related problems• calculate areas of composite shapes, including circular shapes involving sectors• (STAGE 5.2) calculate the surface areas and volumes of composite solids• calculate volumes and capacities of right prisms and cylinders

SkillCheck

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

cba

d

ig

41

25

66

48

52

95

50

47

35

85

e48

75

f

14

26

20

28

35

h 14

2818

30

2 Use Pythagoras’ theorem to find, correct to one decimal place, the value of each pronumeral.

cba

25 mm

14 mm

y mm

20 mm

35 mmk mm

32 mm

25 mmd mm

Worksheet

StartUp assignment 3

MAT10MGWK10015

Skillsheet

Solid shapes

MAT10MGSS10007

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Surface area and volume

3 For each circle, find correct to two decimal places:

i its circumference ii its area.

a b c d

5.2 cm

28 cm63 cm

185 cm

4 Calculate the volume of each solid. All measurements are in metres.

cba

fed

3

7

53

47

11

516

12

8

1610

814

10

Skillsheet

What is volume?

MAT10MGSS10008

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3-01 Areas of composite shapes

Example 1

Find the area of each composite shape, correct to one decimal place where appropriate.

7 m

ba c

22 m30 m

12 m

17 mm

50 mm

14 m

m

20 mm

Solutiona Area ¼ 50 3 20� 1

23 17 3 14

¼ 881 mm2

Area of rectangle � area of triangle

b The shape is made up of a rectangle and a quadrant.

Radius of quadrant ¼ 7 mLength of rectangle ¼ 22 � 7 ¼ 15 mArea of shape ¼ area of rectangleþ quadrant

¼ 15 3 7þ 14

3 p 3 72

¼ 143:4845 . . .

� 143:5 m2

c This ring shape is an annulus, its area is enclosed bytwo circles with the same centre.

Radius of large circle ¼ 12

3 30 m

¼ 15 m

Radius of small circle ¼ 12

3 12 m

¼ 6 mArea of annulus ¼ p 3 152 � p 3 62

¼ 593:7610 . . .

� 593:8 m2

Large circle � small circle

Example 2

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

A B

5 m120°a b

80° 4.2 m

Puzzle sheet

Area

MAT10MGPS00010

A sector is a fraction of a circle‘cut’ along two radii, like apizza slice.

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Solutiona Area ¼ 80

3603 p 3 4:22

¼ 12:31504 . . .

� 12:32 m2

80360

3 area of circle

b Sector angle ¼ 360� � 120� ¼ 240�

Area of sector ¼ 240360

3 p 3 52

¼ 52:35987 . . .

� 52:36 m2

Summary

θr

Area of a sector ¼ u

3603 pr2

Exercise 3-01 Area of composite shapes1 Find the area of each composite shape.

12 m

10 m8 m

6 m 5 m

12 m

18 m

10 m

3 m

10 m

3 m

10 m

8 mm

8 mm

6 mm

16 m

10 m

10 m

16 m

10 cm

4 cm

2 cm

2 cm

8 cm

f

cb

e

a

d

There are 360� in a circle, buta sector is a fraction of a circle

See Example 1

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9 cm

15 cm4

cm

40 cm

20 cm

10 cm 10 cm

8 m

18 m

15 m20 m

14.1

m

14.1 m

20 m

300 cmkji

hg

80 cm

80 cm

80 cm

80 cm

300 cm

200

cm

100

cm10

0 cm

2 Calculate, correct to one decimal place, the area of each shape. All measurements are in metres.

40

40

15

1222

26

a b c

10 10

45

20

16

86

14

30

30

2040

90

150

g h i

j k l

d e f

22

22

7 7

6

8 6 3 3

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

10°8 m 8 m

b ca1.2 m

110°

2 m

120°

4 A circular playing field has a radius of 80 m. A rectangular cricket pitch measuring 25 m by2 m is placed in the middle. The field, excluding the pitch, is to be fertilised.a Calculate the area to be fertilised.

b How much will this cost if the fertiliser is $19.95 per 100 square metres? Give your answercorrect to the nearest dollar.

5 A circular plate of diameter 2 m has 250 holes of diameter 10 cm drilled in it. What is theremaining area of the plate? Answer correct to the nearest 0.1 m2.

6 A circular pond of diameter 10 m is surrounded by a path one metre wide.a Calculate the area of the path correct to two decimal places.

b If pavers are $75 per square metre laid, what is the cost of the path?

7 A circular sports ground ofdiameter 120 m has arectangular soccer pitchmeasuring 100 m by 50 minside it. The area outside thesoccer pitch is to be paintedin the team colour of red.

a Calculate the area that is tobe painted red, correct tothe nearest m2.

b If the cost of paint is$29.50 per 10 m2, calculatethe cost of painting thisarea.

8 A new tractor tyre has a diameter of 120 cm, while a worn tyre has a diameter of 115 cm.a Calculate the difference in circumference between a new and a worn tyre, correct to three

decimal places.

b Over 1000 revolutions, how much further (to the nearest metre) will a new tyre travelcompared to a worn tyre?

9 A square courtyard measuring 5 m by 5 m has a semi-circular area added to each side.a Calculate the total area of the semi-circular additions, correct to two decimal places.

b By what percentage (to the nearest whole number) has the area of the courtyard increased?

(This can be calculated as increase in areaoriginal area

3 100%).

See Example 2

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3-02 Surface area of a prismA cross-section of a solid is a ‘slice’ of the solid cutacross it, parallel to its end faces, rather than along it.A prism has the same (uniform) cross-section along itslength, and each cross-section is a polygon (withstraight sides).

A right prism

cross section

The trapezoidal prism shown here has cross-sections that are trapeziums.

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.

It is often useful to draw the net of a solid when finding its surface area. A net may be used toform an open solid or a closed solid. A sealed cardboard box is an example of a closed solid. Acardboard box with the lid removed is an example of an open solid.

Example 3

Find the surface area of each prism.

a

3 m

6 m7 m

Open rectangular prism

b

15 cm

12 cm8 cm

Closed triangular prism

Solutiona This open prism has five faces.

Surface area ¼ 2 endsþ 2 sidesþ base

¼ 2 3 ð3 3 6Þ þ 2 3 ð3 3 7Þ þ ð6 3 7Þ¼ 120 m2

7

3

6end end

side

side

base

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b This open prism has five faces: two identical triangles(front and back) and three different rectangles.Using Pythagoras’ theorem to find m, the hypotenuseof the 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 4

Calculate the surface area of this trapezoidal prism.

18 cm

13 cm

10 cm

12 cm

24 cm

15 cm

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

10

10

24

18

1215 13

Area of each trapezium¼12

3ð10þ24Þ312

¼204 cm2

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

Stage 5.2

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Exercise 3-02 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 Identify the prism that each net represents, then calculate the surface area of the prism. Alllengths are in metres.

21

15

12

6

13

30

26 25

66

24

72

45

24

51

a b

c d

See Example 3

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3 This classroom is being renovated. Find:a the area of the floor to be carpeted and the

total cost, 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.

3 m

10 m

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 15 m long and 10 m wide. The depth of the water ranges from 1 m to3 m. Calculate, correct to two decimal places:

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

3 m

10 m

15 m1 m

Stage 5.2

See Example 4

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Investigation: A surface area short cut

1 Consider this L-shaped prism and its net. We will find its surface area.

35

30

15

24

12

yx

15 30

30

35

15

x

x

y

y

12

12

24

a Find x and y.b This prism has eight faces: 2 ‘L-shaped’ ends and 6 rectangles. Instead of calculating the

areas of the six rectangles separately, we can combine them into one long rectangle, asshaded in the net above. The length of the rectangle is the same as the perimeter of theL-shape. What is the length of this long rectangle?

c What is the area of this long rectangle?d Copy and complete: Length of shaded rectangle ¼ p of the L-shape.e Find the surface area of the prism by copying and completing the following:

Surface area ¼ 2 ‘L-shaped’ endsþ shaded rectangle

¼ 2 3 ð15 3 30þ 20 3 12Þ þ¼

2 From question 1, it can be seen that the surface area of any prism with end faces of area A

and perpendicular height (distance between end faces) h can be calculated using theformula:

SA ¼ 2A þ Ph

where P ¼ perimeter of end face.Use this method to calculate the surface area of each prism. All measurements are incentimetres.

cba

18

20

10

8

1410

6

12

5

13

15 17 24

24

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3-03 Surface area of a cylinderA closed cylinder has three faces: two circles (the ends) and a rectangle (the curved surface). Thelength of the rectangle is the circumference of the circular end, while the width of the rectangle isthe height of the cylinder.

r

r

circumference= 2πr

height, hh

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ah2

r

Surface area of a cylinder ¼ area of two circlesþ area of rectangle

¼ 2 3 pr2 þ 2pr 3 h

¼ 2pr2 þ 2prh

Summary

Surface area of a closed cylinderA 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.

Stage 5.2

Worksheet

Surface area

MAT10MGWK10016

Puzzle sheet

Surface area

MAT10MGPS00009

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Example 5

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

40 mm

15 mm

SolutionSurface area ¼ area of 2 endsþ area of the curved surface

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 6

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

0.5 m

3 m

end

0.5 m

3 mcurvedsurface

Surface area ¼ 2 semicircle endsþ 12

3 curved surface

¼ 2 312

3 p 3 0:52� �

þ 12

3 ð2 3 p 3 0:5 3 3Þ

¼ 5:49778 . . .

� 5:5 cm2

Exercise 3-03 Surface area of a cylinder1 Calculate, correct to two decimal places, the surface area of a cylinder with:

a radius 7 m, height 10 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, correct to the nearest square metre, the surface area of each solid. 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 halfcylinder

29.316.2

e half cylinder with opentop

1.2

2.85

f half cylinder with open top,one end open

5.75

1.5

Stage 5.2

See Example 5

See Example 6

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g cylinder openboth ends

1230

h half cylinder, openboth ends

6.754.5

4.5

4 A swimming pool is in the shape of a cylinder1.5 m deep and 6 m in diameter. The inside ofthe pool is to be repainted, including the floor.Find:

6 m

1.5 m

a the area to be repainted, correct to onedecimal place

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

5 Which tent has the greater surface area?

2 m

2 m

(Note: the floor is included for both tents)

5 m

2.24 ma b

2 m 5 m

3-04 Surface area of composite solids

Example 7

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

4016

20

15

10

12

56

2536

cba

20 15

Stage 5.2

Worksheet

A page of prisms andcylinders

MAT10MGWK10017

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Stage 5.2Solutiona 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

16 – 10 = 6

15

10

12

20 – 12 = 8

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 The 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

¼ 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 . . .

¼ 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 . . . cm2

Outside curved surface area ¼ 2 3 p 3 20 3 36¼ 4523:8334 . . . cm2

Inside curved surface area ¼ 2 3 p 3 15 3 36¼ 3392:9200 . . . cm2

Total surface area ¼ 1099:5574 . . .þ 4523:8334 . . .þ 3392:9200 . . .

¼ 9016:3108 . . .

¼ 9016:3 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 semi-circle

¼ 12

3 56 ¼ 28

Do not round this partialanswer, the final answer will beinaccurate.

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Exercise 3-04 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 Three cubes of length 2 cm, 4 cm and 8 cm are glued on top ofeach other. Calculate the surface area of the new solid. 2 cm

4 cm

8 cm

3 Circular cracker biscuits of diameter 4 cm are packed in a cardboard box of length 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?

C R I S P I E S4 cm

20 cm

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

ba

15

14

20

65

c25 17

48

38 40

30

Stage 5.2

See Example 7

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Stage 5.2d16

10

10

30

e

21.2

15

35

f

282

5 A cylindrical loaf of bread that is 30 cm long with a diameter of 8 cm is cut into slices 15 mm thick.

8 cm30 cm

15 mm

8 cm

30 cm

15 mm

a Calculate the surface area of the loaf of bread before it is sliced, correct to two decimalplaces.

b Find the number of slices in a loaf.

c Calculate the surface area of each slice, correct to the nearest cm2.

6 A wedding cake with three tiers rests on a table. Eachtier is 6 cm high. The layers have radii of 20 cm,15 cm and 10 cm respectively. Find the total visiblesurface area, correct to the nearest cm2.

620

615

610

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

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

1.5 m

3 m

4 m

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8 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°

9 The curved roof of a greenhouse is to be covered inshade cloth.a Calculate, correct to one decimal place, the area of

shade cloth needed if there are no overlaps.

b Shade cloth is sold in 1.5 m wide rolls. How manylinear metres of shade cloth are needed to cover thecurved roof? Answer to the nearest 0.1 metre.

3 m

4 m 12 m

Mental skills 3 Maths without calculators

Time differences1 Study each example.

a What is the time difference between 11:40 a.m. and 6:15 p.m.?From 11:40 a.m. to 5:40 p.m. ¼ 6 hoursCount: ‘11:40, 12:40, 1:40, 2:40, 3:40, 4:40, 5:40’From 5:40 a.m. to 6:00 p.m. ¼ 20 minFrom 6:00 p.m. to 6:15 p.m. ¼ 15 min5 hours þ 20 min þ 15 min ¼ 6 hours 35 minOR:

12:00 noon11:40 a.m.

20 minutes 6 hours 15 minutes = 6 hours 35 minutes

6:00 p.m.12:00 noon 6:15 p.m.

b What is the time difference between 2030 and 0120?From 2030 to 0030 ¼ 4 hours (24 � 20 ¼ 4)From 0030 to 0100 ¼ 30 minFrom 0100 to 0120 ¼ 20 min4 hours þ 30 minutes þ 20 minutes ¼ 4 hours 50 minutesOR:

2030 2100

30 minutes 4 hours 20 minutes = 4 hours 50 minutes

0100 0120

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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.

2 Now find the time difference between:

a 11:10 a.m. and 7:40 p.m. b 6:20 pm. and 12:00 midnightc 4:45 p.m. and 8:10 p.m. d 2:35 a.m and 10:50 a.m.e 1:05 p.m. and 12:30 a.m. f 9:35 a.m. and 11:15 a.m.g 0425 and 0935 h 1440 and 2025i 7:55 a.m. and 3:50 p.m. j 2:40 p.m. and 10:20 p.m.

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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7 Start with a rectangular prism and cut out2 rectangles to create a seat. Click Windowand Materials to change the appearanceof the seat.

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

a swimming pool b a bin c a bench

3-05 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).As a prism is made up of identical cross-sections, its volume can be calculated by multiplying thearea of its base by its perpendicular height (the length or depth of the prism).

Summary

Volume of a prismV ¼ Ah

where A ¼ area of base andh ¼ perpendicular height A h

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

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A cylinder is like a ‘circular prism’ because its cross-sections are identical circles. Because of this,we can also use V ¼ Ah to find the volume of a cylinder. But for a circle, A ¼ pr2, so:

Summary

Volume of a cylinderV ¼ pr2h

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

r

h

Example 8

Find the volume of each prism.

30 cm

42 cm

15 cma b

5 m10 m

3 m6 cm

3 cm

4 cm

4 cm

c

Solutiona V ¼ 42 3 30 3 15

¼ 18 900 cm3

For a rectangular prism,volume ¼ length 3 width 3 height (V ¼ lwh)

b A ¼ 12

3 5 3 3

¼ 7:5

Area of a triangle

V ¼ 7:5 3 10

¼ 75 m3

V ¼ Ah where height h ¼ 10

c A ¼ 12

3 ð4þ 6Þ3 3

¼ 15 cm2

Area of a trapezium

V ¼ 15 3 4

¼ 60 cm3

V ¼ Ah where height h ¼ 4

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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 9

For this cylinder, calculate:

241 cm

128 cm

a its volume correct to the nearest cm3

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

Solutiona Radius ¼ 1

23 128 cm

¼ 64 cm

12

of diameter

V ¼ p 3 642 3 241

¼ 3 101 179:206 . . .

� 3 101 179 cm3

V ¼ pr2h

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

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Example 10

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

Solutiona A ¼ 40 3 12þ 20 3 12

¼ 720 cm2

Area of T cross-section

V ¼ Ah

¼ 720 3 15

¼ 10 800 cm3

b 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

4.5

4.5

26

10 10

y

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

Area of triangle � area of circle

Do not round this partial answer.

V ¼ Ah

¼ 176:3827 . . . 3 60

¼ 10 582:9649 . . .

� 10 583 cm3

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.

Stage 5.2

1019780170194655


Exercise 3-05 Volumes of prisms and cylinders1 Find the volume of each solid, given the shaded area and height.

cba

8 m

A = 63.1 m2

38 cmA = 27.5 cm2

64 cm

A = 312 cm2

2 Calculate, correct to one decimal place, the volume of each solid. All lengths are in metres.

cba

fed

ihg

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

7.2

5.6

3.5

12.83.5 2.4

2.85.5

11.3

7.7

3 For each cylinder with the given measurements, calculate:i its volume, correct to the nearest whole number

ii its capacity

a radius 7 m, height 10 m b diameter 35 cm, height 15 cmc diameter 6.2 m, height 7.5 m d radius 0.8 cm, height 2.35 cm

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

CRACKERS

20 cm

4 cm

5 This swimming pool is 15 m long and10 m wide. The depth of the waterranges from 1 m to 3 m. Calculate thecapacity of this pool in litres.

3 m

10 m

15 m1 m

See Example 8

See Example 9

102 9780170194655

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


6 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

7 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.

8 Find, correct to two decimal places, the volume of each solid. All lengths shown are incentimetres.

cba

fed

1648

8

12

20

40

10 10

radius of circle = 4 cm

50

35

15

5

15

5 5

1012

ihg11.3

7.2

19.6

12.73.2

14

10

25

45

510

5

3.6

4.8 6.4

8.3

Shut

ters

tock

.com

/Joh

nW

ollw

erth

Stage 5.2

See Example 10

1039780170194655


lkj

36

8 625

15

8

560°

5 14

100°

9 a Find, correct to two decimal places, the volume of thisgreenhouse.

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 Biggest volumeA rectangular sheet of metal measures 20 cm 3 14 cm. Square corners are to be cut from it so thatthe remaining piece can be folded and welded to form an open tray.

14 cm

20 cm

What size must the cut-out squares be for the tray to have the largest possible volume? We willuse a spreadsheet to solve this problem.

1 Create this spreadsheet.

A B C D E1 Side of square Length Width Height Volume2 0 ¼20�2*A2 ¼14�2*A2 ¼A2 ¼B2*C2*D2345

2 In cell A3, enter the formula ¼A2þ1. Use Fill Down to copy corresponding formulas intocells A4 to A9.

3 Enter appropriate formulas for cells B3, C3, D3 and E3. Hint: Look at the formulas in row 2.

4 Use Fill Down to copy corresponding formulas into rows 4 to 9.

5 The length of the cut-out square cannot be more than 7 cm. Explain why this is so.

6 The spreadsheet suggests that a cut-out square length of 3 cm will give the biggest volume.Test values above and below 3 cm (correct to one decimal place) to see whether you canfind a bigger volume.

7 What changes would we need to make to the spreadsheet if the starting dimensions weredifferent?

Stage 5.2

Worksheet

Biggest volume

MAT10MGWK10019

104 9780170194655

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


Power plus

1 The total surface area of a cube is 864 cm2. Find its volume.

2 A cylinder has a volume of 3619.11 cm3. Its height is 18 cm. Calculate the radius of itsbase.

3 Find a formula for the surface area, SA, of each solid.a A square prism of base length p and height r.b A cylinder of diameter and height x.

ba

r

p

x

x

4 The surface area of the curved surface of a can is 27 143.4 mm2. If its height is 120 mm,find the radius of the can.

5 Water flows from the top tank to the bottomtank at a constant rate. The level of water inthe top tank falls at a rate of 15 cm/h. Atwhat rate is the level of water rising in thebottom tank?

5 m

2 m

8 m

3 m4 m

6 m

6 A 10 m flat square roof drains into a cylindricalrainwater tank with a diameter of 4 m. If 5 mm ofrain falls on the roof, by how much (to the nearestmillimetre) does the level of the water in the tankrise?

10 m

10 m

4 m

2 m

1059780170194655


Chapter 3 review

n Language of maths

annulus area base capacity

circle circumference cross-section cubic

curved surface cylinder diameter external

kilolitre litre net open

perpendicular height prism quadrant radius

sector solid surface area volume

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

2 What is the formula for the curved surface area of a cylinder?

3 What is the formula V ¼ pr2h used for?

4 What is the difference between volume and capacity?

5 What is an annulus?

6 What type of measurement has units of cubic metres?

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

MAT10MGPS10023

Quiz

Area and volume

MAT10MGQZ00004

106 9780170194655

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.

Surface area of acylinder

Surface area ofcomposite shapes

Volumes of prismsand cylinders

Surface areaof prism

Area of compositeshapes

Surface area andvolume

1079780170194655

Chapter 3 review

1 Find the area of each shape. Give your answers correct to one decimal place wherenecessary.

120°14 m

d

c

15 cm

11 cm

45 m

35 m

e

a

7 cm

13 cm

7 cm

13 c

m

80 mm

34 mm

75 mm

18 m

m

b

f

6 cm

60° 60°

60°

2 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

3 Calculate, correct to one decimal place, the surface area of each solid. All lengths shown arein metres.

cba

21

35

23

15

4.8Cylinder,open atone end

2.7

See Exercise 3-01

See Exercise 3-02

Stage 5.2

See Exercise 3-03

108 9780170194655

Chapter 3 revision

4 Calculate, correct to nearest square centimetre, the surface area of each solid. All lengthsshown are in centimetres.

fed

cba

50

50

20 5 5

15

30

3030

18

18

16

34

25

282

45

12 4

20

18

127

5 Calculate, correct to nearest cubic metre, the volume of each solid. All lengths shown are inmetres.

a

45

15

b

18

14 c 1.6 2.5

5.4

d5025

25

e

24

42

28

18

f

20

23

15

6 A rectangular fish tank measures 75 cm long by 55 cm wide by 35 cm deep. Find thecapacity of the tank in litres if it is filled to 15 cm from the top.

7 A cylindrical rain water tank has a radius of 2.8 m and a height of 2.4 m.a Calculate, correct to two decimal places, the capacity of the tank in kilolitres.b If the tank is 60% full, what is the height of the water in the tank? Answer correct to two

decimal places.

Stage 5.2

See Exercise 3-04

Stage 5.2

See Exercise 3-05

See Exercise 3-05

See Exercise 3-05

1099780170194655

Chapter 3 revision

Documents

Measurement and Geometry Surface area and volumeweb2.hunterspt-h.schools.nsw.edu.au/studentshared... · The surface area of a solid is the total area of all the faces of the solid