Year 10 Surface Area and Volume 5 - Dobmathsdobmaths.weebly.com/.../year_10_surface_area__volume_5.pdfmeasurements to centimetres.) 8 Calculate the surface area of each shape (correct

Year 10 Mathematics Surface Area and Volume Practice Test 5

Name__________________________

1 Find the surface area of this solid

315C H A P T E R 1 2 S U R F A C E A R E A A N D V O L U M E

!3 Find the surface area of this solid correct to 2 decimal places.

First, we need to find the height of the triangle.

Using Pythagoras’ theorem:

12.22! h 2

" 52

h 2! 12.22

# 52

! 123.84

! h ! 11.13 cm (correct to 2 decimal places)

Area of front face ! $12$ % 10 % 11.13 ! 55.65

Area of back face ! 55.65

Area of right-side face ! 24.6 % 12.2 ! 300.12

Area of left-side face ! 300.12

Area of base ! 10 % 24.6 ! 246

Total surface area ! 55.65 " 55.65 " 300.12 " 300.12 " 246 ! 957.54

The surface area is 957.54 cm2 (correct to 2 decimal places).

S U R F A C E A R E A S O F R I G H T P R I S M S

"1 Find the surface area of each shape:

"2 Find the surface area of a cube with side length:a 7 cm b 8.4 cm c 0.9 m

"3 Find the surface area of a rectangular prism with:a length ! 4.8 m, width ! 2.4 m and height ! 5.2 mb length ! 14.8 cm, width ! 3.8 cm and height ! 7.6 cm

10 cm12 cm

18 cm

27.3 cm

18.2 cm18.2 cm

4.8 m

a b c

Exerc ise 12A

12.2 cm 12.2 cm

10 cm

12.2 cm

10 cm

24.6 cm

5 cm

h h

2 Find the surface area of this trapezoidal prism

3 Calculate the surface area of tis triangular prism (correct to 1 decimal place). You will

need to use Pythagoras’ theorem to calculate an unknown length.

4 For this closed cylinder, find correct to 2 decimal places:

a) the area of the circular base

b) the area of its curved surface

c) the total surface area

5 A cylindrical can is open at one end. Find the external surface area of metal correct to 1 decimal place.

316 C O N N E C T I O N S M A T H S 9 S T A G E 5 . 2 / 5 . 1

!4 Find the surface area of each solid (correct to 1 decimal place), given its net:

!5 Find the surface area of each triangular prism:

!6 Find the surface area of each trapezoidal prism:

!7 A glazier is commissioned to build a glassdisplay case in the shape of a trapezoidalprism. The case is made from panes of glassheld together by metal edging. How manysquare centimetres of glass are needed tobuild the display case? (Hint: First change allmeasurements to centimetres.)

!8 Calculate the surface area of each shape (correct to 1 decimal place where necessary).You will need to use Pythagoras’ theorem to calculate an unknown length.

x cm

6 cm

12 cm

8 cm

x cm

5 cm

13 cm

22 cm

a b

17.0 m

31 m

3.0 m17.2 m

2 m

6.4 cm

18 cm

11 cm

7.7 cm

16 cm

a b

6 cm

a b

5.2 cm

4.8 cm

15.2 cm

4 cm

6 cm 12 cm

15 cm

10.3 cm

8.4 cm

a b 8.7 m

2.4 m

1.8 m

31.5 mm

18.2 mm

12.7 mm

2.337 m

1.14 m

76.2 cm

88.9 cm

63.5 cm63.5 cm


!9 Find the surface area of each composite shape:

Surface areas of right cylindersCylinders are like prisms in that they have uniform cross-sections.However, while the faces of a prism are all plane figures (that is, flat),a cylinder has a curved surface.

The two circular faces of a cylinder are congruent (identical).In a right cylinder, these faces are perpendicular to the curved surface.

The curved surface is a rolled-up rectangle. If you unwrap the paperlabel from a can of beans, you will see that it is a rectangle with widthequal to the height of the can, and length equal to the perimeter of itscircular end.

12 m

34 cm

20 cm

45 cm

16 cm

28 cm

36 cm

30 cm

4 m 4 m

6 m

41 m

37 m

84 m20 m

a b

c d

11 cm

18 cm

x cm

23 cm

16 cm

14 cm

x cm

24 cm

19 cm

32 cm

38 cm

c d

plane circular face

plane circular face

curvedsurface


c The surface is made up of two circular faces and a curved surface.

Total surface area ! 2 " 265.90 # 479.78! 1011.58

The total surface area is 1011.58 m2 (correct to 2 decimal places).

!2 A cylindrical can is open at one end. Find the external surface area of metalcorrect to 1 decimal place.

For the base area:

A ! !r 2

! ! " 72

! 153.938 04 (using a calculator)

For the curved surface area:

A ! 2!rh! 2 " ! " 7 " 12! 527.787 565 8 (using a calculator)

The surface area is made up of one circular face and a curved surface.

Total surface area ! 153.938 04 # 527.787 565 8! 681.725 605 8

The external surface area of metal is 681.7 cm2 (correct to 1 decimal place).

S U R F A C E A R E A S O F R I G H T C Y L I N D E R S

"1 For each cylinder, find correct to 2 decimal places:i the area of a circular baseii the area of the curved surface

a b

c d

14 cm

23 cm

10 cm

7.8 mm

34 cm

3.7 m19.6 mm

1.2 m

Exerc ise 12B

7 cm

closed

open

12 cm


!2 Find the curved surface area of each cylinder in terms of !:

!3 For each cylinder, find to 3 significant figures:i the area of the two circular endsii the area of the curved surfaceiii the total surface area

!4 Find the total surface area of the outside of a pipe 15 m long with radius 0.25 m. (A pipedoes not have any ends.) Give your answer correct to 1 decimal place.

!5 A tank is to have a base and a curved surface, but no lid. If the tank is 68 cm in diameterand 123 cm high, what is its outside surface area correct to 3 significant figures?

!6 Give the total surface area in square centimetres correct to 1 decimal place of a closedcylinder with dimensions:a radius 4 cm and height 9 cm b radius 2.4 m and height 92 cmc diameter 1.4 m and height 75 cm d diameter 128 mm and height 82 cm

1.8 cm 2.2 m

2.6 m

a b

c d

2.6 cm

7.5 m

3 m

97 mm48 mm

4 m

a b

c d

5.6 cm

2.4 cm

2.7 m

1.1 m

18 mm

12 m 39 mm

6 Find the volume of this trapezoidal prism

7 Find the volume of this prism

8 Find the volume of each cylinder to 1 decimal place

9 Find the volume of this solid to 1 decimal place

10 Find the volume of each pyramid to the nearest cubic centimetre

a)

b)

c)


!4 Use Pythagoras’ theorem to find the height of eachtriangle, then calculate the volume of each triangularprism to the nearest cubic centimetre:

!5 Find the volume of each trapezoidal prism to the nearest square unit:

!6 a What is the volume of a packet of muesli bars that is 18 cm long, 14.5 cm high and3.5 cm wide?

b If each packet holds 8 muesli bars, what is the volume of each bar correct to 1decimal place (assuming there is no space between the bars)?

!7 One of the longest hand-squared wooden girders ever cut on the north coast of NSWwas made in 1935. It measured 33 cm by 30 cm by 36 m. This blackbutt girder formedthe keel of a boat. What is the volume of this wood, in cubic metres?

!8 What are the dimensions of a rectangular prism with volume 64 cm3 that has thesmallest possible surface area?

!9 Give the dimensions of three different rectangular prisms with volume 24 m3.

!10 Find the volume of each prism (correct to 1 decimal place if neccessary):

Do not confuse the perpendicularheight of the triangle with the

height of the prism.

a b c13 cm 9 cm

5 cm15.2 cm

25.4 cm

71.6 cm

20 cm

26 cm

19 cm

20 cm

12 cm15 cm

3.5 m6.4 m

4.2 m

2.2 m

5.8 m

7.1 ma b c

2.1 m

6.5 m

8 cm

9 cm

10 cm

22 cm

3.4 m5.3 m

12.1 m

8.7 m

6.6 m

31 cm

23 cm

2 cm

2 cm

2 cm 2 cm

a b

c d

10 cm

3.7 cm

2.8 cm

3.0 cm


!4 Use Pythagoras’ theorem to find the height of eachtriangle, then calculate the volume of each triangularprism to the nearest cubic centimetre:

!5 Find the volume of each trapezoidal prism to the nearest square unit:

!6 a What is the volume of a packet of muesli bars that is 18 cm long, 14.5 cm high and3.5 cm wide?

b If each packet holds 8 muesli bars, what is the volume of each bar correct to 1decimal place (assuming there is no space between the bars)?

!7 One of the longest hand-squared wooden girders ever cut on the north coast of NSWwas made in 1935. It measured 33 cm by 30 cm by 36 m. This blackbutt girder formedthe keel of a boat. What is the volume of this wood, in cubic metres?

!8 What are the dimensions of a rectangular prism with volume 64 cm3 that has thesmallest possible surface area?

!9 Give the dimensions of three different rectangular prisms with volume 24 m3.

!10 Find the volume of each prism (correct to 1 decimal place if neccessary):

Do not confuse the perpendicularheight of the triangle with the

height of the prism.

a b c13 cm 9 cm

5 cm15.2 cm

25.4 cm

71.6 cm

20 cm

26 cm

19 cm

20 cm

12 cm15 cm

3.5 m6.4 m

4.2 m

2.2 m

5.8 m

7.1 ma b c

2.1 m

6.5 m

8 cm

9 cm

10 cm

22 cm

3.4 m5.3 m

12.1 m

8.7 m

6.6 m

31 cm

23 cm

2 cm

2 cm

2 cm 2 cm

a b

c d

10 cm

3.7 cm

2.8 cm

3.0 cm


V O L U M E S O F R I G H T C Y L I N D E R S

!1 Find the volume of each cylinder correct to 2 significant figures:a radius 4 cm and height 15 cm b radius 7.8 cm and height 6.5 cmc radius 1.4 m and height 1.7 m d radius 95 cm and height 4.7 me radius 0.5 m and height 136 cm f radius 2.5 m and height 250 cm

!2 Find the volume of each shape, correct to 2 decimal places if necessary:

!3 Find the volume in cubic centimetres correct to 1 decimal place of a soft-drink can withheight 120 mm and radius 33 mm.

!4 a Which of the following cylinders has the larger volume?

i ii

b Are the surface areas of the cylinders the same? Explain.

!5 How many times larger than the volume of cylinder i is the volume of cylinder ii?

!6 a Find the volume of each cylindrical can, leaving your answers in terms of !:

b A manufacturer has to choose one of these cans for its product. Given that the marketing department says that they all have equal chance of success, whichcontainer should the manufacturer choose? (Hint: Consider surface area. Why?)

2 cm

4 cm

10 cm 2.5 cm 1 cm

40 cm

0.625 cm

8 cm

i ii iii iv

a b i iii ii

5 cm 5 cm 5 cm

10 cm

10 cm

10 cm 10 cm 20 cm

20 cm

10 cm

10 cm

20 cm

a b c d

8.1 cm

95 cm

1.2 m

1.4 cm

3.1 cm

3.6 m24.7 cm315.7 cm

Exerc ise 12D


!7 Calculate the volume of each solid correct to 3 significant figures:

!8 The cross-sections below are for some wood mouldings available from ahardware store:

a Find the area of each cross-section to the nearest square centimetre.b Find the volume of wood in cubic centimetres for a 1 m length of each moulding.

!9 A rectangular ditch 600 cm long is dug to hold a pipe beinginstalled for a drainage system. The pipe’s diameter is 90 cm,and the width of the ditch is 150 cm. The pipe sitsat the bottom of the rectangular ditch, which is120 cm deep. What is the volume of earth (tothe nearest cubic metre) needed to fill inaround the pipe, assuming that the pipe isthe same length as the ditch?

1 cm

i ii iii

23.5 cm

10.8 cm

2 cm

48 cm

(hole cut throughcentre of cylinder) (cylinder of diameter

8.4 cm cut through cube)

25.4 cm

52 cm 2.4 m

2.4 m

a b c

d e f

4 cm

1.3 m

1.6 m

32 m

14 m

3.8 m

600 cm120 cm

90 cm

150 cm


The height given is the slant height. Draw a triangle and usePythagoras’ theorem to find the perpendicular height, LO.

LO 2! 14.52

" 6.12

! 173.04

! LO ! 13.15 cm (correct to 2 decimal places)

Now: V ! #13

# Ah

! #13

# $ 101.26 $ 13.15! 443.856 333 3 (using a calculator)

The volume of the pyramid is 443.86 cm3 (correct to 2 decimal places).

V O L U M E S O F R I G H T P Y R A M I D S

!1 Find the volume of each pyramid (correct to 1 decimal place where necessary):

!2 Find the volume of each pyramid to the nearest cubic centimetre:

!3 Show that these three pyramids have the same volume. Note that a and b arerectangular pyramids.

a b c

5 m

6 m8 m 5 m

6 m

8 m5 m

6 m

8 m

a b c

d e

38 cm

24 cm

0.4 m

31 cm

0.25 m

0.14 m 63 cm

0.5 m20.1 cm

23.7 cm 16.8 cm

A

B

C

D

P

O

AC ! 1.2 mBP ! 43 cmDO ! 68 cm

a b c

4 m

24 m2

7.3 cm

26.5 cm2

39.4 cm2

A

O

AO ! 12.5 cm

Exerc ise 12F

14.5 cm

6.1 cm PO

h

L OP ! #12

# $ HI

! #12

# $ 12.2! 6.1 cm



LO 2! 14.52

" 6.12

! 173.04


Now: V ! #13

# Ah

! #13







a b c

5 m

6 m8 m 5 m

6 m

8 m5 m

6 m

8 m

a b c

d e

38 cm

24 cm

0.4 m

31 cm

0.25 m

0.14 m 63 cm

0.5 m20.1 cm

23.7 cm 16.8 cm

A

B

C

D

P

O

AC ! 1.2 mBP ! 43 cmDO ! 68 cm

a b c

4 m

24 m2

7.3 cm

26.5 cm2

39.4 cm2

A

O

AO ! 12.5 cm

Exerc ise 12F

14.5 cm

6.1 cm PO

h

L OP ! #12

# $ HI

! #12

# $ 12.2! 6.1 cm

40 cm



LO 2! 14.52

" 6.12

! 173.04


Now: V ! #13

# Ah

! #13







a b c

5 m

6 m8 m 5 m

6 m

8 m5 m

6 m

8 m

a b c

d e

38 cm

24 cm

0.4 m

31 cm

0.25 m

0.14 m 63 cm

0.5 m20.1 cm

23.7 cm 16.8 cm

A

B

C

D

P

O

AC ! 1.2 mBP ! 43 cmDO ! 68 cm

a b c

4 m

24 m2

7.3 cm

26.5 cm2

39.4 cm2

A

O

AO ! 12.5 cm

Exerc ise 12F

14.5 cm

6.1 cm PO

h

L OP ! #12

# $ HI

! #12

# $ 12.2! 6.1 cm

11 Find the volume of these cones to 1 decimal place

12 Find the volume of this solid to 1 decimal place 13 Find the volume of these solids to 1 decimal place

14 Find the volume of these solids to 1 decimal place

a) b)

a)

b)

a) b)


!2 What is the volume of this cone correct to 2 decimal places?

The slant height is given, but we require the perpendicular height.

Using Pythagoras’ theorem:

h2! 8.52

" 2.52

! 66! h ! 8.12 cm (correct to 2 decimal places)

Now: V ! #13

# !r 2h

! #13

# $ ! $ 2.52$ 8.12

! 53.145 275 70 (using a calculator)

The volume of the cone is 53.15 cm3 (correct to 2 decimal places).

V O L U M E S O F R I G H T C O N E S

"1 Find the volume of each cone correct to 2 decimal places:

"2 A cone has base diameter 3.6 m and height 2.8 m. Find its volume correct to 1 decimal place.

"3 A cone’s base diameter is equal to its height. If its height is 6.6 m, what is its volume?Answer correct to 2 decimal places.

"4 What happens to the volume of a cone if:a its height is doubled?b its radius is doubled?c both the radius and height are doubled?

3.2 m

1.8 m

12.1 cm

6.4 cm

a b c

18.7 cm

20.4 cm

Exerc ise 12G

2.5 cm

8.5 cm

2.5 cm

8.5 cmh


!5 Paricutin is a conical volcano in Mexico. It first grew in a cornfield in 1943 and nowstands 410 m tall and is 1300 m across at its base. Calculate the volume of Paricutin.Give your answer in scientific notation correct to 4 significant figures. Why is thiscalculation very approximate?

!6 Use Pythagoras’ theorem to find r or h, then calculate the volume of each cone correctto 3 significant figures:

!7 Suppose the dimensions of a cylinder are doubled. What changes will have to be madeto the dimensions of a cone that just fits inside the original cylinder so that the ratio ofthe volumes of the new cone and the new cylinder will still be 1 : 3?

!8 Find the maximum volume of this funnel.

!9 Find the volume of each solid to the nearest cubic centimetre:

!10 A piece of circular filter paper has diameter 10.0 cm. A quadrant was cut out anddiscarded. The remaining piece was joined together along the cuts to form a cone.a What is the circumference of the

circular base of the cone (correct to2 decimal places)?

b What are the radius and height of thiscone (correct to 2 decimal places)?

c Calculate its volume correct to1 decimal place.

a b

10.2 cm

14.7 cm

16.4 cm

8 cm10 cm

16 cm20 cm

2.4 m

21.5 cm

3.9 cm4.5 cm

a b c

1.3 m

16.8 cm

6 cm

4 cm

1 cm

12 cm

discard

To help you see how toanswer this question,

construct a conefrom a circularpiece of paper.


!5 Paricutin is a conical volcano in Mexico. It first grew in a cornfield in 1943 and nowstands 410 m tall and is 1300 m across at its base. Calculate the volume of Paricutin.Give your answer in scientific notation correct to 4 significant figures. Why is thiscalculation very approximate?

!6 Use Pythagoras’ theorem to find r or h, then calculate the volume of each cone correctto 3 significant figures:

!7 Suppose the dimensions of a cylinder are doubled. What changes will have to be madeto the dimensions of a cone that just fits inside the original cylinder so that the ratio ofthe volumes of the new cone and the new cylinder will still be 1 : 3?

!8 Find the maximum volume of this funnel.

!9 Find the volume of each solid to the nearest cubic centimetre:

!10 A piece of circular filter paper has diameter 10.0 cm. A quadrant was cut out anddiscarded. The remaining piece was joined together along the cuts to form a cone.a What is the circumference of the

circular base of the cone (correct to2 decimal places)?

b What are the radius and height of thiscone (correct to 2 decimal places)?

c Calculate its volume correct to1 decimal place.

a b

10.2 cm

14.7 cm

16.4 cm

8 cm10 cm

16 cm20 cm

2.4 m

21.5 cm

3.9 cm4.5 cm

a b c

1.3 m

16.8 cm

6 cm

4 cm

1 cm

12 cm

discard

To help you see how toanswer this question,

construct a conefrom a circularpiece of paper.


!2 Find the volume of each solid correct to 1 decimal place:

!3 The Montgolfier brothers ofFrance made one of the firsthot-air balloons to carrypeople. In 1783 they sent up alarge spherical smoke-filledcloth bag 10.6 m across.Calculate the volume of gas inthis balloon to the nearestcubic metre.

!4 The circumference of Earth at the equator is about 40 000 km.a Use the formula C ! 2!r to find the radius of Earth

correct to the nearest 100 km.b Use this radius to find the volume of Earth correct

to 3 significant figures. Write your answer in scientific notation.

!5 A spherical steel shell’s outer diameter is 20 cm, and its inner diameter is 18 cm.a What are the inner and outer radii of the shell?b What is the thickness of steel in the shell?c Calculate the volume of steel in the shell to the nearest

cubic centimetre.d If the mass of 1 cm3 of steel is 7.2 g, what is the mass of

this steel shell, in kilograms correct to 2 decimal places?

!6 Find the volume of each solid correct to 3 significant figures:

a b

2.1 m

1.3 m

6.7 cm

11.8 cm

a b c

2.4 m 12.2 cm

3.5 m

r

equator

18 cm20 cm











a b

2.1 m

1.3 m

6.7 cm

11.8 cm

a b c

2.4 m 12.2 cm

3.5 m

r

equator

18 cm20 cm











a b

2.1 m

1.3 m

6.7 cm

11.8 cm

a b c

2.4 m 12.2 cm

3.5 m

r

equator

18 cm20 cm











a b

2.1 m

1.3 m

6.7 cm

11.8 cm

a b c

2.4 m 12.2 cm

3.5 m

r

equator

18 cm20 cm

Documents

Year 10 Surface Area and Volume 5 - Dobmathsdobmaths.weebly.com/.../year_10_surface_area__volume_5.pdfmeasurements to centimetres.) 8 Calculate the surface area of each shape (correct