maths a - chapter 4

syllabussyllabusrrefefererenceenceStrand:Applied geometry

Core topic:Elements of applied geometry

In thisIn this chachapterpter4A Changing units and

calculating perimeters4B Calculating areas4C Total surface area4D Volume and capacity

4

Length, area and volume

MQ Maths A Yr 11 - 04 01 Wednesday, July 4, 2001 4:11 PM

102

M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d

Introduction

Peta is an expert in the craft of origami (the Japanese art of paper folding). She deftlyfolds and creases paper, the end product being a crane, an elegant swan, a gift box . . .the list is endless. In fact, many two-dimensional and three-dimensional objects can befolded from a single piece of paper without any cutting. In creating these works of art,it is necessary to have an understanding of the spatial relationships of geometry. It isessential to fold the paper at the correct angles and to have the side lengths in thecorrect proportions.

We are surrounded by geometric shapes —in our daily lives, in nature, architecture, etc.Throughout this chapter you’ll discover prop-erties of geometry by conducting investi-gations. Each investigation is designed toreinforce the particular geometry concept ofthat section of study.

MQ Maths A Yr 11 - 04 Page 102 Wednesday, July 4, 2001 4:11 PM

C h a p t e r 4 L e n g t h , a r e a a n d v o l u m e

103

1

What do you understand by the word

length

?

2

Give some units you would use to measure length.

3

What do you understand by the term

perimeter

?

4

Calculate the perimeter of the following shapes:

a b c d

5

Define the term

area

.

6

Give some units you would use to measure area.

7

Calculate the areas of the figures in question

4

above.

8

Identify the shapes of the following figures:

a b

c d

e f

9

Use your calculator to determine answers to the following (correct to 2 decimalplaces, if necessary).

a

1.46

2

b c

4.3

3

d

10

Distinguish between the terms

volume

and

capacity

.

11

Give some units you could use to measure

a

volume

b

capacity.

12

Convert the following quantities to the units indicated.

a

6 mm

→

cm

b

0.25 m

→

mm

c

40 cm

2

→

m

2

d

0.45 km

→

m

e

300 mL

→

L

f

5000 cm

3

→

m

3

g

25 L

→

kL

h

5 cm

3

→

mm

3

i

100 cm

3

→

L

j

25 kL

→

m

3

Objects all around us can be classified as two-dimensional (2-D) or three-dimensional(3-D). A

two-dimensional

object can be described by two measurements (for examplea length and a width), whereas a

three-dimensional

object requires threemeasurements to describe its shape (for example a length, a width and a height).

6 cm

10 m

4 m4.33 m

5 m

3 m

SkillSH

EET 4.1EXC

EL Spreadsheet

Rounding

3.07 28.763


104


It’s important to realise that sometimes it is not possible to construct a 3-D object that has been drawn on paper in two dimensions. Look carefully and you will see that this is the case in the situation at right. Try drawing the shape and you will understand the problem!

Paper folding 1

Constructing a 3-D package from a 2-D shape

Resources: Cardboard, scissors, protractor, ruler.

For our first investigation, let’s look at the relative positions, shapes and sizes of the resulting faces when a 2-D piece of cardboard is folded into a 3-D shape to form an unusual postage box.

1

Trace the above shape onto a piece of thin cardboard. Basically, it consists of two rectangles.

2

Cut along all the blue lines.

3

The lines marked – – – – represent ‘valley’ folds. Crease firmly along these lines with the fold pointing down like a valley.

4

The line XY marked – – • – – • represents a ‘mountain’ fold. Crease firmly along this line, with the fold pointing upwards like a mountain.

5

Rotate the top rectangle through 90º at the point Y, along the crease line XY, in an anticlockwise direction, until it lies on top of the lower rectangle.

6

Fold the top and bottom flaps in along the crease lines in a 3-D manner. Fold in the side flaps along the crease lines in a 3-D manner.

7

This container, on a larger scale, would be a suitable package for posting rectangular objects through the mail.

8

Open the package and

a

measure angle ZXY

b

measure angle

a

.

9

Would it be possible to vary these angles and still form a rectangular container? Investigate the consequences.

inve

stigationinvestigatio

n

w2 w2

w1w1

a

X

Z Y


C h a p t e r 4 L e n g t h , a r e a a n d v o l u m e 105

Two-dimensional objectsAs mentioned previously, the shape of 2-D objects can be specified by two measure-ments. These objects lie in one plane. Among other properties, they possess a perimeterand an area.

PerimeterThe perimeter represents the distance around the boundary of a figure. (We areassuming that all the figures we are dealing with are closed; that is, they begin and endat the same point.) Any line inside the boundary is ignored when calculating the perim-eter. The units used to measure perimeter are those of linear measure: millimetre (mm),centimetre (cm), metre (m) and kilometre (km).

We’ll first discuss converting measurements from one unit to another. You are fami-liar with the following conversions.

10 millimetres = 1 centimetre100 centimetres = 1 metre

1000 metres = 1 kilometre

Using these conversion factors, we can now construct a ‘conversion ladder’.

The smallest unit (mm) is at the narrow top of the ladder; working down the rungswe approach the largest unit (km) which is at the wide base of the ladder. The con-version factors are placed on the rungs in between the units. In changing measurementsfrom a smaller unit to a larger unit, we divide by the relevant conversion factor(s)because we know that our answer must be a lesser amount. When converting from alarger to a smaller unit, we multiply by the relevant conversion factor(s) since we knowthat we require a greater amount as the result.

10 Consider using this container to post a book. On your flat shape, identify:a on which part the book would lieb which sections would dictate the maximum length, width and thickness of

the book.

11 Why is the width w1 greater than the width w2?

12 Taking the measurements of your text book, design and construct a package of this style which could be used to post your book.

SkillSH

EET 4.2

mm

10

cm

100

m

1000

km

Multiplywhengoingto a

smallerunit

Dividewhengoingto a

largerunit

Smallest unit

× ÷

Largest unit


106 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d

We shall now review familiar formulas used to determine the perimeter of commonshapes.

Shape Perimeter

Square P = 4S

Rectangle P = 2(L + W)

Other polygons P = sum of lengths of all sides

Circle C = 2πr orC = πD

Sector C = length of arc + 2 radii

C = × 2πr + 2r

Complete each of the following.a 30 mm = cm b 4800 m = km c 6.5 m = cm d 8400 mm = m

THINK WRITE

a Changing millimetres to centimetres: divide by 10.

a 30 mm = 30 ÷ 10 cm= 3 cm

b To change metres to kilometres: divide by 1000.

b 4800 m = 4800 ÷ 1000 km= 4.8 km

c To change metres to centimetres: multiply by 100.

c 6.5 m = 6.5 × 100 cm= 650 cm

d To change millimetres to metres: divide by 10 (to change to centimetres) then divide by 100 (to change to metres).

d 8400 mm = 8400 ÷ 10 cm= 840 ÷ 100 m= 8.4 m

1WORKEDExampleEXCEL

Spreadsheet

Length con-versions

Mathca

d

Unit con-versions

SkillSH

EET 4.3

SkillSH

EET 4.4

S

W

L

r D

Arc length

Arc

θ° r

θ°360°-----------



Perimeter of composite figuresIn many instances, figures are not of one distinct shape; they may be composed of sev-eral shapes. The perimeter of such shapes is still the distance around the boundary ofthe composite figure. Remember to ignore any lines inside the figure. It is often helpfulto start at one point in the figure, work your way around the boundary in a clockwise oranticlockwise direction, identifying shapes and adding the lengths of all sides, until youreach your starting point.

Find the perimeter around the following piece of pizza.

THINK WRITE

Identify the shape: sector.

Perimeter means ‘distance around outside’; that is, arc plus 2 radii.

Write the appropriate formula. P = × 2πr + 2r

Identify the values of the variables. θ° = 40°, r = 10 cm

Substitute values in the formula.

Calculate the answer, not forgetting units.

P = + (2 × 10)

P = 6.98 + 20P = 26.98 cm

10 cm

40°

1

2

3θ°

360°-----------

4

5

6

40360--------- 2× π× 10×

2WORKEDExample

Find the perimeter of this shape.

THINK WRITE

Start at X and travel in a clockwise direction until reaching X again. Identify the sides as 3 straight lines and one semicircle.

P = Side 1 + Side 2 + Side 3 + circumference of circle

Write the formulas. P = S1 + S2 + S3 + πD

Identify the values of the variables. S1 = 20, S2 = 10, S3 = 20, D = 10

Substitute the values of the variables in the formula.

Calculate the answer, not forgetting units.

P = 20 + 10 + 20 + ( × π × 10)

P = 50 + 15.7P = 65.7 cm

X20 cm

10 cm

112---

212---

3

4

5

12---

3WORKEDExample



Changing units and calculating perimeters

1 Copy and complete each of the following.

2 Richard is planning to have a garage built. The garage is 5.2 m long, 2.4 m wide and2.5 m high. All builders, however, work in millimetres. What are the dimensions of thegarage, in millimetres?

3 Find the perimeters of the following figures (to the nearest whole units).a b c

d e f

4 Find the perimeters of the following figures.a b c

a 70 mm = cm b 600 cm = m c 5000 m = kmd 9 cm = mm e 12 m = cm f 9 km = mg 86 mm = cm h 9.2 km = m i 2400 m = kmj 6.4 cm = mm k 11.25 m = cm l 2.2 cm = mm

remember1. Recall unit conversions for length.2. Multiply when changing to a smaller unit and divide when changing to a larger

unit.3. The perimeter is the distance around a closed, 2-dimensional figure.4. When finding the perimeter of a composite figure, start at any place on the

perimeter and continue in a clockwise or anticlockwise direction until reaching the starting point. Ignore any lines inside the perimeter.

5. Don’t forget to include units in the answer.

remember

4AWWORKEDExample

1

WORKEDExample

2

7 m

12 m

5 m4 m

23.7 cm

17.8

cm

15.4

cm

27.5 cm

13.5 mm

7.5 m

11.5 m

5 m

4 m

120

m

210

m

90 m

WORKEDExample

3

12 m

25 m

10 m14 m

20 m2 m

3.5

m

12 m

17 m


C h a p t e r 4 L e n g t h , a r e a a n d v o l u m e 109d e f

5

The perimeter of the figure shown in centimetres is:A 34B 24 + 5πC 24 + 2.5πD 29 + 5πE 29 + 2.5π

6

The perimeter of the enclosed figure shown is156.6 metres. The unknown length, x, is closest to:A 20.5 mB 35.2 mC 40.2 mD 80.4 mE Cannot be determined

Paper folding 2Resources: A4 paper, scissors.

Having looked at some 2-D shapes, let’s investigate a ‘Geometry jigsaw’.

1 Cut a square from a sheet of unlined A4 paper. Scribble a pattern on one side of the paper, so that you can distinguish the ‘top’ of the jigsaw from the ‘underneath’.

2 Fold the square sheet in half along its diagonal. Unfold it, and cut it along the crease. You should now have two triangles.

3 Take one of these triangles, fold it in half and cut it along the crease line.

4 Take your second triangle from above and lightly crease it to find the midpoint of the longest side. Fold it so that the vertex of the right angle touches that midpoint and cut it along the crease. This forms a trapezium and a small triangle.

5 Take the trapezium, fold it in half and cut along the crease line. You should now have two smaller trapeziums.

125 mm

24 mm90 mm48 mm

21 cm

16 cm

8 cm

10 c

m

12 cm

10 cm

20 m

22 m

7 m

13 m

11 m

34 m

44 m

mmultiple choiceultiple choice

12 cm

7 cm

2 cm

3 cm


35.2

m

20.5 mx

inve


n



6 Fold the acute base angle of one of the trapeziums to the adjacent right base angle and cut along the crease. This should result in a square and a small triangle.

7 Take your other trapezium and fold its right base angle to the opposite obtuse angle. Cut along the crease. This should result in a parallelogram and a small triangle.

8 You should now have 7 shapes: 5 triangles, 1 square and 1 parallelogram.

9 See if you can now assemble this Geometry jigsaw to form the original square from which it was cut.

10 See how many different shapes you can form using some or all of the 7 shapes.

This investigation should enhance your understanding of geometric shapes.

Task 1Take your 7 shapes. Use the smallest triangle as the basic unit of area. Arrange your pieces in order of increasing area. Give the area of each piece in terms of ‘small triangular units’. Some shapes may have the same area. Copy and complete the following table:

Task 2Your pieces can be fitted together in different combinations to form squares of vari-ous sizes. Experiment with your pieces to see whether you can form a square usingthe following number of pieces. Copy and complete the table below.

Shape Sketch of shape Area (e.g. 3 ‘triangular units’)

1

2

3

4

5

6

7

Pieces Sketch Pieces Sketch

1 5

2 6

3 7

4



AreaArea represents the amount of space within the boundary of a closed figure. The unitsused to measure area are those of square measure: mm2, cm2, m2 and km2. There aretwo units of area which are not square units — the hectare (ha) in the metric systemand the acre in the imperial system.

By constructing a conversion ladder, as we did before for linear measure, we find:

To obtain the conversion factors for square measures, it is necessary to square the linear measure conversion factors. As previously, we multiply when converting to a smaller unit and divide when converting to a larger unit.

One hectare is equivalent to the area of a square of side length 100 metres.

So 1 ha = 100 m × 100 m

That is 1 ha = 10 000 m2

Square measureResources: Paper, pencil, ruler.

How is it possible to determine unit conversions for square measure knowing the relevant conversion for linear measure?

1 Draw a square of side 1 cm.

2 You are aware that each centimetre can be divided into 10 millimetres. Mark these millimetre divisions on each side of the square, then join opposite side markings.

3 This creates a grid of smaller squares. How many of these smaller squares are there? Each smaller square is 1 mm2 while the large square is 1 cm2. So how many mm2 are there in 1 cm2?

4 Use the reasoning applied above (it is not really practicable to draw a square of side length 1 metre) to determine the number of cm2 in 1 m2.

Converting units in square measure is simply a matter of applying the above technique.

inve


n

mm2

10 × 10 = 100

cm2

100 × 100 = 10 000

m2

1000 × 1000 = 1 000 000

km2

× ÷

Mathcad

Unitconversions

100 m

100 m1 ha

MQ Maths A Yr 11 - 04 1 Wednesday, July 4, 2001 4:11 PM


We’ll now review familiar formulas used to find the areas of common shapes.

Shape Area

Square A = S2

Rectangle A = L × W

Parallelogram A = b × hwhere the height measurement must be at a right angle to the base measurement.

Trapezium A = (a + b)hwhere the height measurement must be at a right angle to the base measurement.

Triangle A = bhwhere the height measurement must be at a right angle to the base measurement.

Complete each of the following:a 250 mm2 = _____ cm2 b 5 km2 = _____ ha

THINK WRITE

a Change mm2 to cm2, so divide by 100. a 250 mm2 = 250 ÷ 100 cm2

250 mm2 = 2.5 cm2

b Change km2 to m2 (multiply by 1 000 000).Change m2 to ha (divide by 10 000).

b 5 km2 = 5 × 1 000 000 m2

5 km2 = 5 000 000 m2

5 km2 = 5 000 000 ÷ 10 000 ha5 km2 = 500 ha

4WORKEDExample

EXCEL

Spreadsheet

Area of a square

S

EXCEL

Spreadsheet

Area of a rectangle

L

W

Area of a parallelogram

b

h

b

h

a12---

Area of a triangle

Vic Gen fig ch 6. 57a

b

h

G

h

b

h

b

12---



Triangle(Heron’sformula)

A = where s = (a + b + c)(Use when height measurement is unknown.)

Circle A = π r2

SectorA = × πr2

Shape Area

a

bcs s a–( ) s b–( ) s c–( )

12---

EXCEL Spreadsheet

Area ofa circler

D

Cabri Geometry

Area ofa circler

θ°

θ°360°-----------

Find the area of the triangle at right.

THINK WRITE

Write the formula. A = × b × h

Substitute for the base and the height. = × 12.8 × 9.4

Calculate the area. = 60.16 cm2

9.4 cm

12.8 cm112---

212---

3

5WORKEDExample

Find the area of each of the following shapes.a b

THINK WRITE

a Write the formula. a A = b × h

Substitute the base and height. A = 14 × 9

Calculate the area. A = 126 m2

b Write the formula. b A = × (a + b) × h

Substitute the sides and height. A = × (5.9 + 11.4) × 7.2

Calculate the area. A = 62.28 cm2

9 m

14 m7.2 cm

5.9 cm

11.4 cm

1

2

3

112---

212---

3

6WORKEDExample

EXCEL Spreadsheet

Area of atriangle

Mathcad

Substitu-tion 2

Cabri Geometry

Area of atrapezium



Mathca

d

Area of a triangle

Find the area of this shape.

THINK WRITE

Identify the shape (in this case it is a triangle with no height measurement) and write down the appropriate formula for the area (Heron’s formula).

A =

where s = (a + b + c)

Identify the values of the pronumerals. a = 7, b = 8, c = 10To find s, substitute a, b and c values into the formula and simplify.

s = (7 + 8 + 10)

= × 25

= 12.5Substitute the values of a, b, c, and s into the formula for the area.

A =

Simplify.(a) Evaluate the brackets first.(b) Multiply the values together.(c) Take the square root.

A = = = 27.810 744 33

(Round off the answer to 1 decimal place and include the units.)

A = 27.8 cm2

7 m 8 m

10 m1 s s a–( ) s b–( ) s c–( )12---

23 1

2---

12---

4 12.5 12.5 7–( ) 12.5 8–( ) 12.5 10–( )

512.5 5.5 4.5 2.5×××773.4375

7WORKEDExample

SkillSH

EET 4.5 A minute hand moving from the number 12 to the number 4 position sweeps out a sector. If the hand is 10 cm long, what is the area of this sector?THINK WRITE

Write down the formula for the area of the sector.

Area of sector =

Identify the value of the radius. r = 10 cmCalculate the angle of the sector:The angle between the numbers on a clock = 360° ÷ 12

= 30°.From 12 to 4 there are 4 intervals between the numbers. So to find the angle of a sector, multiply 30° by 4.

θ° = 30° × 4= 120°

Substitute the values of r and θ into the formula and evaluate.

Area of sector = × π × 102

= 104.719 755 1 cm2

Write an answer sentence with the number rounded off appropriately and units given.

The minute hand as it rotates through an angle of 120° sweeps an area of 104.7 cm2.

4

12 12

3

10 cm1θ°

360°----------- πr2×

23

4 120360---------

5

8WORKEDExample



Area of composite figuresThe term composite means made up of distinct parts. Composite figures in geometryare figures comprising a number of distinct shapes. Depending upon the compositefigure, to find the overall area you may need to add these individual shapes or subtractone from another.

For example, the composite figure in the diagram at right has beenformed using a semicircle and a square.The area of this shape can be found as follows:Area of total figure = Area of a semicircle (A1) + Area of a square (A2)When finding the area of a composite figure, follow the steps givenbelow.

1. Identify the basic shapes that make up the total figure and number them.

2. Write the expression for the total area in terms of individual shapes.

3. Calculate the area of each individual shape.

4. Add or subtract areas to find the total area of the given shape.

In the diagram above, we added the areas of the square and the semicircle. If the diagram had been shown as at right, we would have subtracted the area of the semicircle from that of the square.

A1

A2

A1

A2

A clock has a minute hand that is 6 cm long and an hour hand that is 3 cm long. In one full revolution of each hand, the minute hand would sweep out a larger circle than the hour hand. What is the difference in the area they cover (to the nearest square centimetre)?

THINK WRITE

The area required is the area between two circles. Write down the appropriate formula.

A = outer area − inner areaA = πR2 − πr2

Identify the value of R (radius of larger circle) and the value of r (radius of smaller circle).

R = 6, r = 3

Substitute the values of the pronumerals into the formula and evaluate.

A = π × 62 − π × 32

= 113.097 − 28.274= 84.823 cm2

Write an answer sentence with the value rounded to the nearest square centimetre.

The difference in area covered by the two hands is approximately 85 cm2.

3 cm6 cm

Rr

1

2

3

4

9WORKEDExample



Calculating areas

1 Copy and complete each of the following.a 70 mm2 = _____ cm2 b 6000 cm2 = _____ m2

c 3 m2 = _____ cm2 d 2.5 km2 = _____ m2

e 4.5 ha = _____ m2 f 3 km2 = _____ ha

2 Find the area of each of the figures below.

a b c

d e f

g h i

remember1. The area is the space enclosed by the boundaries of a two-dimensional shape.2. Area is measured in square units. To change from one unit to another, square

the appropriate linear measure conversion factor. Multiply by the conversion factor when changing to a smaller unit and divide when converting to a larger unit.

3. Remember these area formulas:Square A = S2

Rectangle A = L × WParallelogram A = b × hTrapezium A = (a + b)h

Triangle A = bh (if perpendicular height is known).If three sides are known, use Heron’s formulaA = where s = (a + b + c)

Circle A = πr2

Sector A = × πr2

4. For composite figures, identify shapes of parts of the figure. After calculating these individual areas, add or subtract them to give the total area.

5. Remember to provide units in the answer.

12---

12---

s s a–( ) s b–( ) s c–( ) 12---

θ°360°-----------

remember

4B

WorkS

HEET 4.1WORKEDExample

4

Area of triangle

8 cm29 mm

3.6 km

3 m

9 m27 mm

38 mm

47 cm

62 cmWORKEDExample

5 4.2 m

9.7 m

8.4 km

6.3 km3.7 m



3 Look at the figure at right.a Find the area of the outer rectangle.

b Find the area of the inner rectangle.

c Find the shaded area by subtracting the area of the inner rectangle from the area of the outer rectangle.

4 Find the shaded area in each of the following.a b

c d

5

The area of the triangle at right is:

j k l

m n o

p q r

s t u

A 36 cm2 B 54 cm2

C 108 cm2 D 1620 cm2

WORKEDExample

6a12.8 km

16.9 km

38 mm

87 mm8 m

80 cm

WORKEDExample

6b

1 m

12 m

9 m

2.8 m

3.65 m

0.4 m

3.6 cm

9.5 cm5.4 cm

WORKEDExample

738 cm

27 cm15 cm

12 mm

8 mm7 mm65 m

16 m58 m

WORKEDExample

845°

6 cm 120° 2 m

300°

5 mm

WORKEDExample

9 3 m 12 m

20 m

7 m

10 m

8 m14 m

16 m

5 cm

5 cm

9 cm

3 m

9 m10 m

8 m5 m

8 m12 m

12 m

8 m


15 cm12 cm

9 cm



6

Which of the two statements is correct for the two shapes at right?

Statement 1. The rectangle and parallelogram have equal areas.Statement 2. The rectangle and parallelogram have equal perimeters.

7

The area of the figure at right is:

8 Len is having his lounge room carpeted. Carpet costs $27.80/m2. The lounge isrectangular with a length of 7.2 m and a width of 4.8 m.a Calculate the area of the lounge room.b Calculate the cost of carpeting the room.

9 A rectangular garden in a park is 15 m long and 12 m wide. A concrete path 1.5 mwide is to be laid around the garden.a Draw a diagram of the garden and the path.b Find the area of the garden.c What are the dimensions of the rectangle formed by the path?d Find the area of concrete needed for the path.

10

Examine the diagram at right.a The circles cover an area of approximately:

b The shaded area is approximately:

11 A family-size pizza is cut into 8 equal slices. If the diameter of the pizza is 33 cm,find (to the nearest square centimetre) the area of the top part of each slice.

12 The collectable plate shown at right is 22 cm in diameterand has a golden ring that is 0.5 cm wide.Find (to 1 decimal place) the area of the golden ring if itsouter edge is 1 cm from the edge of the plate.

A Statement 1 B Statement 2 C Both statements D Neither statement

A 54 m2 B 165 m2

C 225 m2 D 255 m2

A 402 cm2 B 201 cm2 C 804 cm2 D 805 cm2 E 603 cm2



19 cm

38 cm 38 cm

19 cm


17 m

7 m

15 m

15 m


32 cm

22 cm

1 cm

0.5 cm

WORKEDExample

9


C h a p t e r 4 L e n g t h , a r e a a n d v o l u m e 11913 Early-model vehicles

had a single wind-screen-wiper bladeto remove waterfrom the windscreen.(The bus at left has two single blades of this type.)Using the dimensions given in the diagram:a what area (to the nearest whole number)

did the blade cover?b what percentage (to 1 decimal place) of

the windscreen was cleared?

Calculate the area (or shaded area) of each of the figures drawn below. Where necessary, give your answer correct to 1 decimal place.

1 2 3

4 5

6 7 8

9 10

140°

45 cm

120 cm

60 cm

1

12 cm

6.3 m

8 cm30°

62 mm

91 mm

10 cm

4 cm24 cm

25 cm

30 cm

25 cm

20 cm40 m

20 m

40 cm

12 cm

76 mm

32 m

m

15 cm

12 cm

6 cm



Maximising an area of landFarmer Brown needs to build a paddock for her sheep to graze. She has 1000 m of fencing with which to build this paddock.

1 If farmer Brown builds the paddock 100 m long and 400 m wide, the area will be 40 000 m2. If she builds it 200 m long and 300 m wide, the area will be 60 000 m2. What dimensions should farmer Brown choose for her paddock so it has the maximum possible area?a Set up a spreadsheet with the headings:

LENGTH WIDTH AREA

b Enter an initial value of 50 m for the length of the paddock, then provide a formula and copy it down to generate length measurements in increments of 50 m; for example:

50100150•

••

450c Since the length of the fencing is 1000 m,

length + width = 500 mProvide a formula in the width column incorporating the values in the length column, then copy this formula down the column.

d Enter a formula under the AREA heading to calculate the area of the figure. Copy this formula down the column.

e What length and width provide the greatest area? What shape is the paddock?

2 If one side of the paddock is a river, only three sides need to be fenced. If farmer Brown still uses 1000 m of fencing, what dimensions should she now choose for her paddock to maximise its area?

EXCEL

Spreadsheet

Maximising area of land

Area of a rectangle

Maximum area

inve


n



a Set up a spreadsheet similar to the one above. In this case:width + length + width = 1000 m

Provide formulas in the columns, then copy them down.b What length and width provide the greatest area in this case? Is the result the

same as for the previous case?

3 Write a paragraph outlining recommendations for farmer Brown. Justify your conclusions with mathematical evidence.

4 Investigate further to see whether you could write your conclusions in a general form for any given length of fencing.

Effect of scale factors on perimeter and area

Resources: Paper, scissors.

A variety of shapes can result when paper is folded.Applying a scale factor to a figure affects its perimeter and area differently. Let’s investigate.1. Cut two strips of paper: one should be 1 cm

wide, the other should be 2 cm wide.2. Taking the ends of the strips, fold them as you

would in knotting a piece of string.3. Gently flatten the knot and sharply crease the edges.

Trim off the ends of the paper strips beyond the knot.

1 What shape does each knot form?

2 Measure the interior angles of each knot. What do you conclude?

3 Measure the side lengths of each knot. What do you find?

4 The two knots are similar in shape. What is the scale factor S?

5 Using your measurements, calculate the perimeter of each figure. What is the relationship between your two answers?

6 Look at the flat surface of the knot. Because we don’t know a formula for the area of this shape, we can break it up into familiar shapes whose areas we can identify. Locate a trapezium and a triangle on each of the figures. Take the required measurements and calculate the areas of the smaller knot and the larger knot. What is the relationship between the two areas?

7 From your two previous answers, what would you expect the perimeter and area to be if you were to construct a knot from a 3-cm strip of paper?

8 Deduce a general statement indicating what happens to the perimeter and area of a figure when we apply a scale factor S.

9 Open each knot out flat.a What is the overall shape of the paper?b What shapes do you find within the creases?c Compare the perimeters and areas of your two flat shapes. Do your answers

support your conclusion in question 8?

inve


n



Patchwork designsResources: Graph (or unlined) paper, pencil, ruler.

In this investigation we look at two common patchwork designs for quilts, and consider the shapes within blocks and the requirements for completing the quilts.

Patchwork quilts are frequently constructed from a set of ‘blocks’ of the same pattern. The eight-pointed star block is shown at right.

Each block is constructed from a set of smaller shapes sewn together to create the pattern. A border of a complementary colour usually frames the blocks.

1 Identify the shapes in the block above. How many of each shape are required to form the pattern?

2 Each block has dimensions 50 cm square. Sketch a design for a single-bed quilt 173 cm by 218 cm, including the border. How many blocks and how many pieces of each shape would be required?

3 Trace four blocks. Place them together at a point and shade or colour them to illustrate the overall effect of your quilt.

4 Experiment with other patterns incorporating the eight-pointed star block. The quilt displayed provides some suggestions.

inve


n



Three-dimensional objectsAs mentioned previously, the shape of 3-D objects can be described by three measure-ments. These objects can be classified as prisms or non-prisms.

PrismsPrisms are 3-D figures that have a constant cross-section in one direction. This con-stant cross-section is parallel to the face which is called the base of the prism. Thename of the prism comes from the shape of its base.

Common examples of prisms are:

Non-prismsNon-prisms do not have a constant cross-section in any direction. The two types of 3-Dobject that we will study in this category are pyramids and spheres.

PyramidsIn pyramids, the cross-section parallel to the base reduces in size as the cross-sectionprogresses from the base to the apex.

Common examples of pyramids are:

Another interesting block is the cube lattice shown at right.

5 Trace a set of four of these. Place them together at a point and experiment with different shading or colouring of the faces to produce different 3-D effects. Note that dark colours recede and light colours come to the fore in terms of vision.

6 Identify the number and types of different shapes in each block. Draw a design for a single-bed quilt 173 cm by 218 cm (including borders) using blocks of this type. Determine the number of pieces of each shape required.

7 Design a patchwork block of your own. Draw a completed quilt incorporating your block.

CubeRectangular prism

Cylinder

Square-basedpyramid

Triangle-basedpyramid Cone



Spheres

A sphere has no flat faces. When spheres are sliced, the flat surface exposed is alwayscircular.

Common examples are:

Two properties which three-dimensional figures possess are surface area and volume.

Surface areaThe surface area of a 3-D object represents the total area of all its exposed faces. Tofind the surface area, we must calculate the area of each face of the object, as identifiedby its net, then add all of these areas to find the total. The units used for total surfacearea (TSA) are the same as those used for area.

Object Net TSA

Cube TSA = 6S2

Rectangular prism

TSA = area of TSA = 6 rectanglesTSA = 2(WH + LW + LH)

Cylinder TSA = area of 2 circles + curved surface

= 2πr2 + 2πrH

Sphere

Hemisphere

S

1 3

2

6

S

S

4 5

WL

H W

HH

W

L

L

H

Hr H

r

r

2πr



Square-based pyramid

TSA= area of square + area of 4triangles

= b2 + 4 × ( bh)

Cone Curved surface of a cone is formed by removing a sector out of a circle.

TSA = area of base (circle) + area of curved surface= πr2 + πrS

Sphere Not shown TSA = 4πr2

HemisphereOpen

Not shown TSA = 2πr2

Closed Not shown TSA = 2πr2 + area flat circle

TSA = 2πr2 + πr2

TSA = 3πr2

Object Net TSA

b

h b

h12---

S = SlantheightH

r

Arc length Slant height= radius of

sector

Circumference of base = arc length

of sector

r = S

Minorsector

Majorsector

Mathcad

TSA

r

r



Find the total surface area of the object shown.THINK WRITE

Identify the shape. Shape: rectangular prism

Write down the formula for the TSA of a rectangular prism.

TSA = area 6 rectanglesTSA = 2(WH + LW + LH)

Allocate a value to the pronumerals. W = 9, H = 17, L = 19

Substitute the values of the pronumerals into the formula.

TSA = 2(9 × 17 + 19 × 9 + 19 × 17)

Evaluate (brackets first, then multiply by 2).

= 2(153 + 171 + 323)= 2 × 647= 1294

Write the answer, including units. TSA = 1294 cm2

17 cm

19 cm9 cm1

2

3

4

5

6

10WORKEDExample

Find the surface area of an open cylindrical can that is 12 cm high and 8 cm in diameter (to 1 decimal place).

THINK WRITE

Form a net of the open cylinder, transferring all the dimensions to each of the surfaces.(Note that the cylinder has no top surface.)

Identify the different-sized common figures and set up a sum of the surface areas. The length of the rectangle is the circumference of the circle.

TSA = A1 + A2

A1 = 2πr × H= 2 × π × 4 × 12= 301.59 cm2

A2 = π × r2

= π × 42

= 50.27 cm2

Sum the areas. TSA = A1 + A2

= 301.59 + 50.27= 351.86 cm2

Write your answer. The total surface area of the open cylindrical can is 351.9 cm2.

8 cm

12 c

m

1 2 rπ

A1

12 c

m

A24 cm

2

3

4

11WORKEDExample



Find the surface area of the square pyramid at right.

THINK WRITE

Calculate the area of the square base. A = S2

= 62

= 36 cm2

Calculate the area of a triangular side. Note each side is identical and the height of each triangular side is 5 cm.

A = × b × h

= × 6 × 5

= 15 cm2

Calculate the total surface area.(Note: There are 4 identical triangular sides.)

SA = 36 + 4 × 15= 96 cm2

6 cm4 cm

5 cm

1

212---

12---

3

12WORKEDExample

Find the total surface area of a size 7 basketball with a diameter of 25 cm. Give your answer to the nearest 10 cm2.THINK WRITE

Use the formula for the total surface area of a sphere. Use the diameter to find the radius of the basketball and substitute into the formula.

Diameter = 25 cmRadius = 12.5 cmTSA of sphere = 4πr2

= 4 × π × 12.52

= 1963.495Write your answer. Total surface area of the ball is approximately 1960 cm2.

1

2

13WORKEDExample

A die used in a board game has a total surface area of 1350 mm2. Find the linear dimensions of the die (to the nearest millimetre).THINK WRITE

A die is a cube. We can substitute into the total surface area of a cube to determine the dimension of the cube. Divide both sides by 6.

TSA = 6 × S2

TSA = 1350 mm2

1350 = 6 × S2

S2 =

= 225Take the square root of both sides to find S.

S = = 15 mm

Write your answer. The dimensions of the die are:15 mm × 15 mm × 15 mm

1

13506

------------

2 225

3

14WORKEDExample



The diagram shows the proposed shape for a new container for takeaway Chinese food. The shape will be used if the TSA of the container is less than 750 cm2. If the TSA is greater than or equal to 750 cm2 then production and manufacturing costs are too expensive and the takeaway shop will have to stay with the old cylindrical container. Next time I order my Chinese takeaway, could it come in the new design?THINK WRITE

Identify the distinct shapes that make up the total object: these are a square-based pyramid and a cube. The base of the pyramid and one face of the cube are not on the surface and therefore their area should not be included.

TSA= square pyramid (no base)+ 5 faces of a cube

Calculate the TSA of a square-based pyramid with no base.(a) Alter the general square-based

pyramid formula so as not to include the square base.

(b) Allocate a value to the pronumerals.(c) Substitute the values into the formula

and evaluate.

TSA of square-based pyramid:

A = 4 × × b × h

b = 10, h = 10

A = 4 × × 10 × 10

= 200

Calculate the TSA of a cube (5 faces only).(a) Alter the general cube formula to

include 5 faces instead of 6.(b) Allocate a value to the length.(c) Substitute the value of the side length

into the formula and evaluate.

TSA of a cube (5 faces only):

A = 5S2

S = 10A = 5 × 102

= 500Add the individual TSA together to find the TSA of the whole object.

TSA= 200 + 500= 700 cm2

Write an answer sentence. Next time I order Chinese takeaway there could be a newly designed container with a surface area of only 700 cm2.

10 cm

10 cm

1

2

12---

12---

3

4

5

15WORKEDExample

remember1. The TSA is the sum of the areas of the outside surfaces of a 3-dimensional object.2. Formulas for all types of objects are not possible. For those objects without a

formula you will need to follow these steps.(i) Draw the net of the object.(ii) Work out the different shapes that make up the net.(iii)Calculate their individual areas.(iv) Add all the individual parts together.

3. Do not include in the TSA the surfaces of contact of the distinct shapes that make up a composite figure.

remember



129

Total surface area

1

Name each solid in the top row then match it with a net in the bottom row.

2

Draw the net of each of the following solids.

3

Identify the solids from the nets below. Draw the solid for each.

4

Find the total surface area of each of the figures below.

5

Oliver is making a box in the shape of a rectangular prism. The box is to be 2.5 m long, 1.2 m wide and 0.8 m high. Calculate the surface area of the box.

a b c

i ii iii

a b c

a b c

a b c

d e f

4C

5 cm9 cm

32 cmWORKEDExample

10

3.9 cm4.1 cm

4 cm

20 cm

13 cm14 cm

42 mm

7 mm7 mm


130


6

Calculate the surface area of an open box in the shape of a cube, with a side length of 75 cm. (

Hint

: Since the box is open there are only five faces.)

7

A room is in the shape of a rectangular prism. The floor is 5 m long and 3.5 m wide. The room has a ceiling 2.5 m high. The floor is to be covered with slate tiles, the walls are to be painted blue and the ceiling is to be painted white.

a

Calculate the area to be tiled.

b

Each tile is 0.25 m

2

. Calculate the number of tiles needed.

c

Calculate the area to be painted blue.

d

Calculate the area to be painted white.

e

One litre of paint covers an area of 12 m

2

. How many litres of paint are needed to paint the room?

8

Find the total surface area of the following cylinders.

a

b

c

9

Calculate the surface area of the square pyramid at right.

10

A triangle-based pyramid has four equal faces as shown at right.Calculate the surface area to the nearest cm

2

. (Recall Heron’s formula.)

WORKEDExample

11

Radius = 410 mmLength = 1.5 m

90 cm

28 cm

250 mm

250 mm

(Answer to nearest cm2)

WORKEDExample

12

10 cm

13 cm

4 cm



131

11

Two cubes are drawn such that the side length on the second cube is double the sidelength on the first cube. The surface area of the larger cube will be:

A

twice the surface area of the smaller cube

B

four times the surface area of the small cube

C

six times the surface area of the small cube

D

eight times the surface area of the small cube.

12

Calculate the surface area of the triangular prism at right.

13

Calculate the surface area of the following cones.

14

Find the total surface area of the following spheres and hemispheres.

15

A cube has a total surface area of 24 cm

2

. What is the length of each side?

16

Another cube has a total surface area twice that of the one in question

15

. Is the sidelength of this cube twice that of the one in the previous question? Explain.

17

Calculate the surface area of this prism.

18

A concrete swimming pool is a rectangular prism with the following dimensions: length of 6 metres, width of 4 metres and depth of 1.3 metres. What surface area of tiles is needed to line the inside of the pool? (Give answer in m

2

.) Remember there is no top on the pool.

a b c

a b c


4 cm

3 cm

5 cm

2 cm

2.4 cm

2.9 cm

40 cm

32 cm

10 cm

18 cm

WORKEDExample

13

43 mm(Answer in cm2.)

5 m

Open

(Answer tonearest mm2.)

Closed

6.3 mm

WORKEDExample

14

WORKEDExample

15 3.2 m

1 m

2 m

4 m6 m

MQ Maths A Yr 11 - 04 Page 131 Friday, July 6, 2001 8:40 AM

132


19

What is the total area of canvas needed for the tent (including the base) shown in the diagram at right?Give the answer to the nearest m

2

.

20

A ball used in a game of pool has a diameter of 48 mm. The total surface area of the ball is closest to:

A

1810 mm

2

B

2300 mm

2

C

7240 mm

2

D

28 950 mm

2

E

115 800 mm

2

21

The total surface of a golf ball of radius 21 mm is closest to:

22

The formula for the total surface area for the object shown is:

A

abh

B

2

×

bh

+

ab + 2 × ah

C 3( bh + ab) D bh + 3ab

E bh + 3ab

23

The total surface area of a poster tube that is 115 cm long and 8 cm in diameter isclosest to:

24 A baker is investigating the best shape for a loaf of bread. The shape with the smallestsurface area stays freshest. The baker has come up with two shapes: a rectangular prismwith a 12-cm-square base and a cylinder with flat, round ends that have a 14-cmdiameter.a Which shape stays fresher if they have the same overall length of 32 cm?b What is the difference between the total surface areas of the two loaves of bread?

A 550 mm2 B 55 cm2 C 55 000 mm2 D 0.055 m2 E 5.5 cm2

A 3000 cm2 B 2900 cm2 C 1500 mm2 D 6200 m2 E 23 000 cm2

4.5 m6.5 m

2.5 m

1.5

m

1.0

m




h

b

a

12--- 1

2---

12--- 1

2---




1 Calculate the area of a rectangle with a length of 0.4 m and a width of 1.1 m.

2 Calculate the area of a triangle with a base of 12.3 m and a height of 4.8 m.

3 Calculate the area of the trapezium at right.

Name the solids below.

4 5 6

7 Find the surface area of the cube shown at left.

8 Find the surface area of a rectangular prismwith a length of 8 cm, a width of 5 cm

and a height of 6 cm.

9 Find the surface area of thetriangular prism below.

10 Find the surface area of thesquare pyramid below.

2

32 m

96 m

56 m

10 cm 8 cm

20 cm

6 cm

6 cm

8 cm

9 cm



Volume and capacityVolumeThe volume of a 3-D object represents the amount of space contained in, or occupiedby, the object. The units used to measure volume are those of cubic measure: mm3,cm3, m3.

Constructing our conversion ladder, as before, will enable us to convert from oneunit to another.

To obtain the conversion factors for cubic measure, the linear measure conversionfactors are cubed. The same procedure applies as before: multiply when converting to asmaller unit and divide when converting to a larger unit.

The capacity of a 3-D object refers to the quantity of solid, liquid or gas it couldhold. The units used to measure capacity are millilitres (mL), litres (L) and kilolitres(kL). The conversion ladder for capacity units is as follows:

In capacity units, 1 mL represents the amount of liquid which a 1-cm cube could hold. Two useful conversion relationships are:

1 cm3 ≡ 1 mLand, for larger quantities 1 m3 ≡ 1 kL

mm3

10 × 10 × 10 = 1000

cm3

100 × 100 × 100 = 1 000 000

m3

× ÷

mL

1000

L

1000

kL

× ÷

EXCEL

Spreadsheet

Capacity

Mathca

d

Unit con-versions

Convert 1.12 cm3 to mm3.

THINK WRITE

The conversion from centimetres to millimetres is 1 cm = 10 mm.The conversion factor for cm3 to mm3 is to multiply by 103 or 1000; that is, 1 cm3 = 1000 mm3.

1.12 cm3 = 1.12 × 1000 mm3

= 1120 mm3

Write the answer in correct units.

1

2

3

16WORKEDExample

1 cm

1 cm

1 cm

Holds 1 mL



In our calculations of volume, we shall consider only three classes of 3-D figures:prisms, pyramids and spheres.

Volume of prismsThe volume of a prism is given by the following formula:

Volume of a prism = cross-sectional area × height of the prism

The height is the dimension perpendicular to the cross-sectional area.

Shape Cross-sectional shape Volume

Cylinder

Area = πr2

V = area of a circle× height

= πr2 × H

Triangularprism

Area = bh

V = area of a triangle × height

= bh × HNote: Lower-case h

represents the height of the triangle.

Rectangularprism

Area = L × W

V = area of a rectangle × height

= L × W × H

Cube

Area = S2

V = area of a square × height

= S2 × H= S2 × S= S3

(since in a cube, H = S)

Mathcad

Capacity

Convert:a 400 cm3 into mL b 1200 cm3 into mL and into L c 2 kL into m3.

THINK WRITE

a Since 1 cm3 is equivalent to 1 mL, then 400 cm3 is equivalent to 400 mL.

a 400 cm3 = 400 mL

b Each 1 cm3 will hold 1 mL of liquid. Therefore, 1200 cm3 will hold 1200 mL of liquid.

b 1200 cm3 = 1200 mL

To change mL to L, divide by 1000 (since there are 1000 mL in 1 L).

= 1.2 L

c One kL is equivalent to 1 m3. Therefore, 2 kL is equivalent to 2 m3.

c 2 kL = 2 m3

1

2

17WORKEDExample

H

r

r

bHh b

h

12---

12---

EXCEL Spreadsheet

VolumeWL

HW

L

Mathcad

VolumeformulasS

H

S



Volume of pyramidsAs we have seen previously, a pyramid has a flat base at one end, and tapers to a pointat the other. Some examples of pyramids are shown below. A cone is really a circle-based pyramid.

Cone Square pyramid Rectangular pyramid Triangular pyramid

Comparing volumes of pyramids and prisms

Resources: Set of 3-D volumetric shapes of pyramids and prisms with same base area and height; water (or rice).

For the following investigation, the volumes of pairs of open 3-D containers are compared by considering the amount of water (or rice) each can hold. Each 3-D pair should have the same base area and the same perpendicular height.

Consider the following pairs of containers:square-based pyramid and cuberectangle-based pyramid and rectangular prismtriangle-based pyramid and triangular prismcone and cylinder

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

Find the volume of the shape shown correct to 1 decimal place.THINK WRITE

Identify the shape. Triangular prismWrite down the appropriate formula for the volume.

V = bh × H

Allocate values to the pronumerals keeping in mind that b and h are the base and height of a triangular cross-section or base of the prism, while H is the height of the prism.

b = 2.6, h = 2.3, H = 3.2

Substitute and evaluate, rounding the answer to 1 decimal place.

V = × 2.6 × 2.3 × 3.2

= 9.568≈ 9.6 m3

2.6 m

3.2 m2.3

m

12 1

2---

3

4 12---

18WORKEDExamplein

vestigation

investigation



A pyramid does not have a uniform cross-section. The cross-sectional area becomessmaller as it nears the apex (point). The internal capacity or volume of a tapered objectis a fraction of the volume of a prism. Mathematicians found this fraction to be a third( ). They defined the base of a pyramid to be the flat end opposite the apex. To calcu-late the volume of a pyramid we find the area of the flat end, multiply this by the heightof the pyramid (which must be perpendicular to the base) and then multiply by (ordivide by 3).

Volume of a pyramid = × area of base × height of object

The following table shows the formulas for the volume of some common pyramids.

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

3 Having previously considered the general formula for the volume of a prism, suggest a general formula for the volume of a pyramid.

4 Draw labelled diagrams and deduce specific formulas for the volume of each of the following:a square-based pyramid b rectangle-based pyramidc triangle-based pyramid d cone (circle-based pyramid).

ShapeFlat end (base)

shape Volume

Cone V = × area of a circle

× height

V = πr2 × H

Squarepyramid

V = × area of a square

× height

V = S2 × H

Rectangularpyramid

V = × area of a

rectangle × height

= L × W × H

Triangularpyramid

V = × area of a triangle

× height

V = ( bh) × H

Note: Lower-case h represents the height of the triangle.

13---

13---

13---

Mathcad

Mensura-tion

r

H r

13---

13---

H

S

S

13---

13---

H

LW

W

L

13---

13---

Mathcad

VolumeformulasH

hb

h

b

13---

13--- 1

2---



Volume of spheresThe volume of a sphere of radius r is given by the following formula:

Volume of a sphere = πr3

A hemisphere is half of a sphere. Its volume, therefore, is half of the volume of asphere.

Volume of hemisphere = (volume of sphere)

= × πr3

= πr3

Find the volume of the pyramid at right (to the nearest m3).

THINK WRITE

Write the equation. V = × area of base × height

The pyramid has a square base. It is a square pyramid. The area of the base is given by S2.

Area of base = S2

Calculate the volume. Volume = × S2 × H

Volume = × 302 × 40

Volume = 12 000 m3

Write your answer. The volume of the square pyramid is 12 000 m3.

30 m 30 m

Height of pyramid = 40 m

113---

2

313---

13---

4

19WORKEDExample

Find the volume of the cone at right, correct to 2 decimal places.

THINK WRITE

Write the formula. V = πr2H

Substitute the radius and height. = × π × 3.22 × 8.5

Calculate the volume. = 91.15 cm3

3.2 cm

8.5 cm

113---

213---

3

20WORKEDExample

Mathca

d

Volume formulas

r 43---

12---

r12--- 4

3---

23---



Note that the volume and capacity of a 3-D object do not depend on whether the objectis open or closed. An open rainwater tank could hold the same quantity of water as aclosed one. The surface area, however, varies depending on whether the object is openor closed.

Cross-sections of solid 3-D shapesResources: Plasticine (or play dough), thread.

In this activity we investigate the surfaces exposed when solid 3-D shapes are cut in different directions.

1 Using plasticine, mould 3-D solids in the shape of a:a cube b rectangular prism c triangular prismd cylinder e sphere.

2 Investigate slicing these solids in various directions with a piece of thread to see whether it is possible to obtain sections with faces in the shape of a:a square b rectangle c triangled ellipse e circle.

3 Copy and complete the following table, showing the direction of the sectional cut required for each particular face to be exposed. Not all face shapes are possible.

Find the volume of a sphere with a radius of 9.5 cm, correct to the nearest cm3.

THINK WRITE

Write the formula. V = πr3

Substitute the radius. = × π × 9.53

Calculate the volume. = 3591 cm3

143---

243---

3

21WORKEDExamplein

vestigation

investigation

Section face

SolidCube Rectangular

prismTriangular

prismCylinder Sphere

Square

Rectangle

Triangle

Ellipse

Circle



Volume of composite objectsOften, an object can be identified as comprising two or more different common prisms,pyramids or spheres. Such figures are called composite objects. The volume of a com-posite object is found by adding the volumes of the individual common figures ordeducting volumes. A grain silo can be modelled as the sum of a cylinder and a largecone, less the tip of the large cone.

The volume of a composite object is equal to the sum (or difference) of theindividual common prisms, pyramids or spheres.

Vcomposite = V1 + V2 + V3 + . . . (or Vcomposite = V1 − V2)

Find the capacity of the object shown at right (to the nearest litre).

THINK WRITE

The object is a composite of a cylinder and a square prism.

The volume of the composite object is the sum of volumes of the cylinder plus the prism.

Vcomposite = volume of cylinder + volume ofsquare prism

= (πr2 × Hc) + (S2 × Hs)

= (π × 62 × 20) + (182 × 25)= 2261.946 711 + 8100= 10 361.946 711 cm3

Convert to litres using the conversion of 1 cm3 = 1 mL1000 mL = 1 L

10 362 cm3 = 10.362 litres

Write your answer. The capacity of the object is 10 litres.

25 c

m

12 cm

18 cm20

cm

1 r = 6 cm

18 cm 25 c

m

18 cm

H =

20

cm

2

3

22WORKEDExample



Volume and capacity

1 Convert the volumes to the units specified.

2 Convert the following units as indicated.

3 Calculate the volume and capacity of each of the prisms below.

a 0.35 cm3 to mm3 b 4800 cm3 to m3 c 56 000 cm3 to litres

d 1.5 litres to cm3 e 1.6 m3 to litres f 0.0023 cm3 to mm3

g 0.000 57 m3 to cm3 h 140 000 mm3 to litres i 250 000 mm3 to cm3

a 750 cm3 = mL b 800 cm3 = Lc 2500 cm3 = mL d 40 000 cm3 = Le 6 m3 = cm3 = mL = L f 12 m3 = Lg 4.2 m3 = kL h 7.5 m3 = kL = Li 5.2 mL = cm3 j 6 L = cm3

k 20 L = mL = cm3 l 5.3 kL = m3

a b c

remember1. Volume is measured in cubic units: mm3, cm3, m3.2. Capacity is measured in mL, L, kL.3. To convert volume units into capacity units:

1 cm3 = 1 mL1 m3 = 1 kL

4. The volume of a prism is found using the formulaV = area of base × height of prism

5. The volume of a pyramid is found using the formula

V = area of base × height of pyramid

6. The volume of a sphere is calculated using

V = πr3

and for a hemisphere,

V = πr3

7. The volume of a container is not dependent on whether it is open or closed.8. The volume of a composite object can be found by adding or subtracting

volumes of individual prisms, pyramids or spheres.

13---

43---

23---

remember

4D

WorkS

HEET 4.2WORKEDExample

16

WORKEDExample

17

5 cm

2.4 m13 m



4 For each of the triangular prisms below find:i the area of the base of the prism ii the volume of the prism.

a b

c d

5

The shape at right could be described as a:

6

The area of the base of a prism is 34.67 cm2, and the height is 3.6 cm. The volume ofthe prism is:

7

The dimensions of a rectangular prism are all doubled. The volume of the prism willincrease by a factor of:

8 A refrigerator is in the shape of a rectangular prism. The internal dimensions of theprism are 60 cm by 60 cm by 140 cm.a Find the volume of the refrigerator in cm3.b The capacity of a refrigerator is measured in litres. If 1 cm3 = 1 mL, find the

capacity of the refrigerator in litres.

d e f

g h i

A cube B square prismC rectangular prism D both B and C

A 38.27 cm2 B 38.27 cm3 C 124.12 cm2 D 124.812 cm3

A 2 B 4 C 6 D 8

4.2 m3.2 m

50 mm

9 mm9 mm

12.5 m

20.5 m16.5 m

6 cm

12 cm12 m

3 m

13 cm

27 cm

WORKEDExample

18

8 cm

3 cm 5 cm

6 cm

8 cm12 cm

3.4 m

2.7 m 1.5 m3.2 m

12.5 m

7.8 m





C h a p t e r 4 L e n g t h , a r e a a n d v o l u m e 1439 A semitrailer is 15 m long, 2.5 m wide and 2.7 m high in the shape of a rectangular

prism. Find the capacity of the semitrailer. (Ignore the thickness of the walls.)

10 A petrol tanker is shown at right.The tank is cylindrical in shape. The radius of the tank is 2 m and the length is 12 m. Ignoring the thickness of the material, calculate:a the volume of the tank, correct to 3 decimal

placesb the capacity of the tank, to the nearest

100 litres. (1 m3 = 1000 L).

11 At right is a diagram of a concrete slab for a house.a Calculate the area of the slab.b The slab is to be 10 cm thick. Calculate the volume of

concrete needed for the slab. (Hint: Write 10 cm as 0.1 m.)c Concrete costs $145.50/m3 to lay. Calculate the cost of this slab.

12 A flat rectangular roof is 14 m long and 8 m wide. When it rains, the water is col-lected in a cylindrical tank.a Calculate the volume of water collected

on the roof when 25 mm of rain falls.b How many litres of water does the roof

collect?c The cylindrical tank has a radius of 1.8 m

and is 2.4 m high. What is the capacity of the tank, in litres?

d By how much does the depth of water in the tank rise when the rain falls? Answer in centimetres, correct to 1 decimal place.

13 For each of the following pyramids, calculate the volume by first calculating the areaof the base shape.

a b

c d

12 m2 m

10 m

10 m

2.5 m15 m

WORKEDExample

19

8 cm6 cm

15 cm 8 cm

14 cm

10 m

12 m

6 m

12 cm

5 cm

8 cm

6 cm



14 Find the volume of each of the following cones, correct to the nearest whole number.a b

c d

15 A cone has a base with a diameter of 9 cm and a height of 12 cm. Calculate thevolume of that cone, correct to 1 decimal place.

16 Calculate the volume of each of the following spheres, correct to 1 decimal place.a b

c d

17 Calculate the volume of a sphere with a diameter of 2.3 cm. Answer correct to 2decimal places.

18Which of the following solids could not be described as a pyramid?A B

C D

19A triangular pyramid and a square pyramid both have a base area of 20 cm2 and aheight of 15 cm. Which has the greater volume?A the triangular pyramid B the square pyramidC both have equal volume D this can’t be calculated

WORKEDExample

20

5 cm

10 cm

12 cm

12 cm

8 mm

33 mm

42 cm

42 cm

WORKEDExample

216 cm 8 cm

12.5 m3.2 m





A spherical balloon has a volume of 500 cm3. It is then inflated so that the diameter ofthe balloon is doubled. The volume of the balloon will now be:

21 In each of the following, the prism’s front face is made up of a composite figure. Foreach:

i calculate the area of the front face ii find the volume of the prism.a b

c d

22 Find the volume of the solid at right. Answer correct to 1 decimal place.

23 A hollow rubber ball is to be made with a radius of 8 cm, and the rubber to be used is 1 cm thick.a What would be the radius of the hollow inside?b Calculate the volume of the ball.c Calculate the volume of space inside the ball.d Calculate the amount of rubber (in cm3) needed to make the ball.

24 The figure at right is a truncated cone, that is, a cone with the top cut off.a Calculate the volume of the cone before it was truncated.b The portion cut off was itself a cone. Calculate its volume.c Calculate the volume of the truncated cone.

25 Use the same method as in question 24 to find the volume of the truncated pyramid shown at right.

26 The figure at right is of an ice-cream cone, containing a spherical scoop of ice-cream (a whole sphere).a Calculate the volume of the cone.b Calculate the volume of the scoop of ice-cream.c Calculate the total volume of the shape. (Hint: Only half the

sphere sits above the cone.)

A 1000 cm3 B 2000 cm3 C 3000 cm3 D 4000 cm3


10 cm

10 cm20 cm

4 cm

16 cm

8 m9 m

4 m

6 cm20 cm

15 cm8 cm

12 cm

3 m

18 m12 m

12 m

6 m

WORKEDExample

22 4 cm

12 cm

6 cm

3 cm15 cm

6 cm

1 cm

5 cm

3 cm

3 cm

8 cm

2.5 cm



The optimum swimming poolResources: Pen, paper, calculator.

You are considering having a swimming pool constructed in your back yard. In order to get the best value for money, you need to research the advantages of differently shaped pools. Approach this investigation in a formal manner and summarise your research under the headings:

AimProcedureResultsConclusion(s).

For the purpose of this investigation, you are to consider four factors:

1. A major cost in the construction of the pool lies in tiling the interior surface. For this reason, you wish your pool to have a minimum surface area.

2. Your back yard allows you only a 10-metre by 5-metre area of land.3. The depth of the pool must lie within the range 1.5 metres and 2 metres.4. The pool must (subject to the above restrictions) have a maximum water

capacity.

It is obviously not possible to satisfy all these requirements with one particular shape and size of pool. There must be compromises. Your task is to investigate differently shaped pools and decide on a shape and size which best satisfies the above requirements.

1 AimBegin by summarising the above information to define the aim of your investigation.

2 ProcedureExplain how you intend to collect data that would enable you to make a decision in light of the above restrictions.

3 ResultsIn order to approach this in a methodical manner, draw up a table with the following headings.

4 ConclusionsStudy the two right-hand columns of your table. Decide on a shape that offers the best compromise between surface area and volume. Write your recommendations. (You may consider that there are two shapes which would be just as suitable.)

inve


n

Shape Surface area Volume

Within the above restrictions, draw at least 5 shapes here. Label figures with dimensions.

Calculate the surface area of the base and walls of each shape (the top is open).

Determine the volume of water each pool would contain.



Minimising surface areaGo to the Maths Quest CD and download the spreadsheet ‘Volume’.

Task 1A cylindrical drink container is to have a capacity of 1 litre

(volume = 1000 cm3). We are going to calculate the most cost-efficient dimensions to make the container. To do this, we want to make the container with as little material as possible; in other words we want to minimise the surface area of the cylinder. The spreadsheet should look as shown below.

1. In cell B3 enter the volume of the cylinder, 1000.

2. In cell A6 enter a radius of 1. In cell A7 enter a radius of 2 and so on up to a radius of 20.

3. The formula that has been entered in cell B6 will give the height of the cylinder corresponding to the radius for the given volume.

4. The surface area of each possible cylinder is in column D. Use the charting function on the spreadsheet to graph the surface area against the radius.

5. What is the most cost efficient dimensions of the drink container?

Task 2Use one of the other worksheets to find the most efficient dimensions to make a rectangular prism of volume 1000 cm3 and a cone of volume 200 cm3.

EXCEL Spreadsheet

Volumeinve


n



Upkeep of above-ground circular poolsResources: Computer spreadsheet.

Above-ground pools are a popular alternative to in-ground pools. They can be dismantled when the children outgrow them and the lawn can be re-established. The cost of maintaining a pool depends on the volume of water it contains (and also on its use). The chemicals added to the water help to kill germs and to maintain a healthy environment.

In this investigation we shall ignore the contribution of frequency of use and consider only the effects of the diameter and depth of a circular pool on the upkeep cost. As the diameter (and radius) of a circular pool increases, the volume of water it contains also increases. If the radius doubles, does the volume also double (for a given depth)? This is the basis of our investigation. Let us consider two situations:1. circular swimming pools of varying radii and the same depth2. circular swimming pools of varying depths and the same radius.

Part 11 Set up a spreadsheet with the following headings:

Radius Depth Surface area Volume

2 Enter values for the radius from 5 m to 15 m in steps of 1 m.

3 Under the heading ‘Depth’, enter a figure of 2 m down the entire column.

4 Enter the formula for surface area (area of a circle) in column 3 and copy it down the column.

5 In column 4, enter the formula for volume (of a cylinder) then copy it down the column.

6 Enter the graphing section of the spreadsheet and plot Radius on the x-axis and Volume on the y-axis. Add suitable headings and print out a copy.

7 What are your conclusions about the variation of volume with radius for a given depth?

Part 21 Amend your spreadsheet, keeping the following headings:

Radius Depth Surface area Volume

2 Enter a value of 10 m for the radius down the entire length of column 1.

3 Under the heading ‘Depth’, enter a figure of 1 m to 2 m in steps of 0.1 m.Your spreadsheet should automatically recalculate the surface area and volume for these new figures.

4 Enter the graphing section of the spreadsheet and plot Depth on the x-axis and Volume on the y-axis. Add suitable headings and print a copy.

5 What are your conclusions about the variation of volume with depth for a given radius?

Part 3The cost of upkeep for a pool depends largely on the volume of water it contains. Write a report to outline your findings on pool maintenance costs for circular pools.

EXCEL

Spreadsheet

Upkeep circular pools

inve


n



The optimum size for a rainwater tank

Resources: Spreadsheet.

It is becoming more common these days to have a rainwater tank in the back yard of a suburban residential property. Because space is limited in these situations, certain restrictions must be placed on size.

Kirsten and Daniel have recently purchased their first home. They wish to install a cylindrical rainwater tank. Their constraints are as follows:1. the base diameter plus the height of the tank must not exceed 5 metres2. within the above restrictions, the volume of the tank must be as large as

possible.

1 Set up a spreadsheet with the following headings:

2 In a systematic manner, investigate the volume obtained from a variety of combinations of diameter and height. Remember that the sum of these two measurements must not exceed 5 metres.

3 What sized tank would you recommend for Kirsten and Daniel?4 How much rainwater would it hold?5 Prepare a report for Kirsten and Daniel, supporting your recommendations with

mathematical evidence.

Top-dressing lawnsResources: Pen, paper, calculator.

Len owns a landscaping business. At the moment he has four small jobs on his books. Each one requires top-dressing a lawn. They are all jobs where the owners have removed sporting facilities and now want to establish a lawn.Job 1 A tennis court has been dismantled and the area requiring top-dressing is

14 m by 6 m.Job 2 A 10-m diameter circular pool has been removed.Job 3 A children’s sand pit 10 m square is no longer required.Job 4 A triangular play area 12 m by 16 m by 20 m is to be top-dressed and

turned into lawn.Without thinking too much about the jobs, Len quoted to supply 4 m3 of top-soil for each job.

EXCEL Spreadsheet

Rain-watertank

inve


n

Diameter Radius Height Volume

EXCEL Spreadsheet

Top-dressing

lawns

inve


n



Draw up the table below and complete the blank spaces.

Which job receives the greatest depth of soil? Justify your answer with mathematical evidence.

Developing islands and canalsResources: Pen, paper, calculator.

A tourist enterprise is considering developing land by creating artificial islands surrounded by canals. It is envisaged that pleasure cruises and water sports would take place on the canals, while tourist accommodation would be established on the islands. The land in question has an area of 2 km square.

The constraints for the project are:1. The canal must go completely around the perimeter of the land.2. The islands created must be circular and each must be no smaller than 5000 m2.3. There must be at least 4 islands in the development.4. In order that the pleasure cruisers can navigate the islands, each canal must

have a minimum width of 50 metres and a depth of 10 metres across its entire width.

1 Draw a sketch of the area of land and investigate options within the above constraints. Organise your investigations in the form of a table.

The developer has to bear in mind that the islands provide accommodation for guests, while the canals provide entertainment.

2 Consider the results from your table above. Recommend a development which you consider would provide an estate with the optimum balance between land and water. Provide a plan (with measurements indicated) and justify your decision with sound mathematical reasoning.

Job Diagram Area Volume of topsoil

Depth of topsoil

1 4 m3

2 4 m3

3 4 m3

4 4 m3

inve


n

Sketch Total area of islandsVolume of water

in canals

Draw at least 4 sketches showing proposals for the positions and shapes of the islands.

Calculate the total area of the exposed land on the islands.

Find the volume of soil that would be removed to form the canals. This approximates the volume of water that would fill the canals.



Units of measurement• Measurements of length

10 mm = 1 cm100 cm = 1 m1000 m = 1 km

• Measurements of area100 mm2 = 1 cm2

10 000 cm2 = 1 m2

1 000 000 m2 = 1 km2

10 000 m2 = 1 ha

• Measurements of volume1000 mm3 = 1 cm3

1 000 000 cm3 = 1 m3

• Measurements of capacity1000 mL = 1 L

1000 L = 1 kL

• Conversion of volume to capacity1 cm3 = 1 mL1 m3 = 1 kL

Perimeter• Perimeter is the distance around an enclosed figure.

1. Perimeter formulas for common shapes encountered areSquare P = 4SRectangle P = 2(L + W)Circle C = 2πr or πD

Sector C = × 2πr + 2r

Other figures P = sum of lengths of all sides.2. Perimeter is measured in linear measure.

Area• Area is the amount of space within the boundary of a closed figure.• Area formulas for common shapes encountered are:

Square A = S2

Rectangle A = L × WParallelogram A = base × perpendicular height

Trapezium A = (a + b) × h

Triangle A = bh

(when 3 sides are known) A = (Heron’s formula)

where s = (a + b + c)

summary

θ°360°-----------

12---

12---

s s a–( ) s b–( ) s c–( )12---



Circle A = πr2

Sector A = × πr2

Composite figures A = sum or difference of areas of individual shapes• Area is measured in square measure.

Total surface area (TSA)• The surface area of a 3-D object represents the total area of all its exposed surfaces.• Total surface area formulas for common shapes encountered are:

Cube TSA = 6S2

Rectangular prism TSA = 2(WH + LW + LH)Cylinder TSA = 2πr2 + 2πrHSquare pyramid TSA = S2 + 4 × ( bh)Cone TSA = πr2 + πrSSphere TSA = 4πr2

Open hemisphere TSA = 2πr2

Closed hemisphere TSA = 3πr2

Composite figures TSA = sum of areas of all exposed faces

Volume• Volume represents the amount of space contained in, or occupied by, an object.• Volume formulas for common shapes encountered are:

Prisms V = area of base × heightPyramids V = area of base × height

Spheres V = πr3

Hemispheres V = πr3

Composite figures V = sum or difference of volumes of individual shapes• While volume is represented in cubic measure, capacity is represented in mL, L or

kL.• The volume of an object does not depend on whether it is open or closed.

θ°360°-----------

12---

13---

43---

23---



1 Copy and complete each of the following.a 190 mm = _____ cm b 190 mm2 = _____ cm2 c 190 mm3 = _____ cm3

d 500 mL = _____ L e 500 mL = _____ kL f 50 m3 = _____ Lg 0.2 m3 = _____ cm3 h 0.2 m2 = _____ cm2 i 0.2 m = _____ cmj 120 cm3 = _____ mL k 120 cm3 = _____ L l 0.3 kL = _____ cm3

2 Find: i the area and ii the perimeter of the following shapes.

3 Calculate i the area and ii the perimeter of the following shapes. Give your answers correct to 1 decimal place.

4

Examine the diagram at right.a The circles cover an area of approximately:

b The shaded area is approximately:

5 Draw the net of each of the following solids.

6 Name the solid shape for which the net is given at right.

a b c

d e

a b c



a b c

4A

CHAPTERreview

4B4D4A4B

GC program

Mensuration

20 mm

14 mm

7 cm20° 12 cm

13 cm

5 cm 9 cm>>

>>

> >

16 cm24 cm

21.3

cm

4 cm18 cm 2.5 m

1.5 m

3.2 m

5.5 m

4A4B

30 cm

15 cm

10 cm

13 cm25 cm 10 m

6 cm

4Bmmultiple choiceultiple choice

32 cm

4C

4C



7 Find the surface area of each of the following solids.a b c d

8 Calculate the surface area of each of the figures below, by calculating the area of each face separately and adding them.

9 Calculate the surface area of each of the following cylinders.

10 Find the total surface area of the following pyramids.

11 Find the total exterior surface area of the following objects (to the nearest whole number).

12 At right is a diagram of an Olympic swimming pool.a Calculate the area of one side wall.b Use the formula V = A × h, to calculate the volume

of the pool.c How many litres of water will it take to fill the pool?

(1 m3 = 1000 L)d The walls and floor of the pool need to be painted.

Calculate the area to be painted.

a b c

a b c

a b c

a b c

4C

4.2 cm

3.9 m

2.1 m

0.8 m

1.8 m

0.9 m

4.6 m

4C

4 m

3 m 3.5 m

5 m

4 cm

12 cm

6 cm

10 cm

5 cm

3 m

2 m15 m

5 m

12 m

4C

6 cm

13 cm

3.8 m

1.6 m20 cm

32.5 cm

4C

8.4 m

10.5 m14 mm

42 mm

20 cm

18 cm

4C

12 cm

Closed

9 mm

10 m

Open

2 m

22 m

50 m

50.01 m

1 m



155

13

At right are the plans for a garage that Rob is building. All thewalls are bricked. (

Note

: The garage has an iron roof and is closed at one end.) Calculate the area that will need to be bricked.

14

Use the formulas to calculate the volume and capacity of each of the following cubes, rectangular prisms and cylinders.

15

A prism has a base area of 45 cm

2

and a height of 13 cm. Calculate the volume.

16

Use the formula

V

=

×

A

×

h

to calculate the volume and capacity of each of the pyramids below.

17

Calculate the volume of each of the pyramids, cones and spheres below.

18

Find the volume and capacity of each of the following shapes correct to 1 decimal place.

a b c

d e f

a b c

a b c

d e f

a b c

4CD

2.5 m 6 m

3 m 4D

6.5 cm29 mm 11.6 m

4.6 m3.8 m

3 cm8 cm

41 cm3 cm

13 cm32 mm

18 mm

4D4D

13---

A = 16 cm2

9 cm

A = 126 mm2

19 mm

A = 6.9 m2

2.3 m

4D

25 m36 m 7.9 m

3.2 m

2.6 m

19 mm

52 mm

23.5 mm

19.5 mm

23 mm 70 cm

4D

1.4

m

2 m 1.8 m

60 cm

113

cm

64 cm 22 cm

25 cm

10 cm

15 cm

MQ Maths A Yr 11 - 04 Page 155 Monday, September 24, 2001 7:14 AM


19 The diagram at right shows 3 tennis balls packed in a cylindrical container. Find:a the volume of each ballb the volume of the cylinderc the volume of space that remains free.

20 Calculate the area of a circle with a diameter of 8.6 cm, correct to 1 decimal place.

21 Calculate the area of the annulus (ring) shown at right, correct to 2 decimal places.

22 Calculate the area of the sector at right, correct to 1 decimal place.

23 Calculate the area of the figure at right.

24 Calculate the shaded area in the figure drawn at right,correct to 2 decimal places.

25 Calculate the area in the figure at right, correct to 2 decimal places.

26 Calculate the surface area of a closed cylinder with a radius of 10 cm and a height of 23 cm. Give your answer correct to the nearest whole number.

27 Calculate the surface area of a sphere with a radius of 1.3 m. Give your correct to 3 decimal places.

28 Calculate the volume of the prism drawn at right.

29 Calculate the volume of the solid at right, correct to the nearest whole number.

HC

7 cm

9 cm

3 cm

13.2 cm

85°

10 cm

28 cm

9 cm

29 c

m

9.7 cm

4.6 cm

5 cm

13.7 cm9.1 cm

13.4 cm

20.3

cm

testtest

CHAPTERyyourselfourself

testyyourselfourself

4

4 cm

8 cm


maths a - chapter 4

Technology