43675 MS5.1.1 cover - Login Department of Educationlrr.cli.det.nsw.edu.au/legacy/Mathematics/43675_03_05.pdfUnit overview iii Unit overview In this unit you will explore perimeter

M9/10_5-1 43675

MathematicsStage 5

MS5.1.1 Perimeter and area

Number: 43675 Title: MS5.1.1. Perimeter and Area

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Extract from Mathematics Syllabus Years 7-10 ©Board of Studies, NSW 2002 Unit overview pp iii, iv

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Unit overview i

Unit contents

Unit overview ....................................................................................... iii

Outcomes ................................................................................. iii

Indicative time............................................................................v

Resources..................................................................................v

Icons .........................................................................................vi

Glossary................................................................................... vii

Part 1 Straight-sided figures ............................................... 1–100

Part 2 Figures with curved sides......................................... 1–60

Unit evaluation ...................................................................................61

ii MS5.1.1 Perimeter and area 03/05

Unit overview iii

Unit overview

In this unit you will explore perimeter and areas of a variety of shapes.

You will look for common links between areas of figures and develop

formulas to describe some areas.

You will be given the opportunity to apply your understanding of

perimeter and area to practical situations, exploring ways of dividing

shapes into familiar two-dimensional figures. You will integrate your

geometry skills with your knowledge of measurement and units.

OutcomesBy completing the activities and exercises in this unit, you are working

towards achieving the following outcomes.

You have the opportunity to learn to:

• developing and using formulae to find the area of quadrilaterals:

– for a kite or rhombus, Area = � 12xy where x and y are the lengths

of the diagonals;

– for a trapezium, Area =� 12h(a + b) where h is the perpendicular

height and a and b the lengths of the parallel sides

• calculating the area of simple composite figures consisting of two

shapes including quadrants and semicircles

• calculating the perimeter of simple composite figures consisting of

two shapes including quadrants and semicircles.

You have the opportunity to learn to:

• identify the perpendicular height of a trapezium in different

orientations (Communicating)

• select and use the appropriate formula to calculate the area of a

quadrilateral (Applying Strategies)

iv MS5.1.1 Perimeter and area 03/05

• dissect composite shapes into simpler shapes (Applying Strategies)

• solve practical problems involving area of quadrilaterals and simple

composite figures (Applying Strategies).Source: Extracts from outcomes of the Maths Years 7–10 syllabus

<www.boardofstudies.nsw.edu.au/writing_briefs/mathematics/mathematics_710_syllabus.pdf > (accessed 04 November 2003).© Board of Studies NSW, 2002.

Unit overview v

Indicative timeThis unit has been written to take approximately 8 hours.

Each part should take approximately 4 hours.

Your teacher may suggest a different way to organise your time as you

move through the unit.

ResourcesResources used in this unit are:

For part 1 you will need:

• match sticks or toothpicks

• Internet

• a scientific calculator

• a ruler

• string or wool

• scissors.

For part 2 you will need:

• Internet

• a scientific calculator

• a ruler

• a pair of compasses.

vi MS5.1.1 Perimeter and area 03/05

IconsHere is an explanation of the icons used in this unit.

Write a response or responses as part of an activity.

An answer is provided so that you can check your

progress.

Compare your response for an activity with the one in the

suggested answers section.

Complete an exercise in the exercises section that will be

returned to your teacher.

Think about a question or problem then work through the

answer or solution provided.

Access the Internet to complete a task or to look at

suggested websites. If you do not have access to the

Internet, contact your teacher for advice.

Unit overview vii

GlossaryThe following words, listed here with their meanings, are found in the

learning material in this unit. They appear in bold the first time they

occur in the learning material. For these words and their meanings

including pronunciation see the online glossary on the CLI website at

http://www.cli.edu.au/Kto12 and follow the links to Stage 5 mathematics.

adjacent Another word for ‘next to’.

These two sides are adjacent.

algebra A branch of mathematics that uses letters to representnumbers or concepts.

arc Part of a circle or curve.

area The amount of surface. It is usually measured in squareunits, such as square centimetres, square millimetres,square metres, or square kilometres.

bisect To cut exactly in half.

circumference The distance around the outside of a circle. The namegiven to the perimeter of a circle.

composite Another word for ‘more than one’. For example: acomposite shape is made up of several simpler shapes

diagonal A line drawn from one corner to another corner of aplane shape.

dimension Length, breadth, width, height, thickness, depth.These are all dimensions.

equilateral All sides of equal length. Often used to describetriangles that have three equal sides.

figure Another word for shape.

formula A rule or principle stated in the language ofmathematics. Often written using algebraic symbols.

isosceles At least two sides equal.

perimeter The distance around the boundary of a two-dimensionalshape.

perpendicular At right-angles to. Meeting at 90°.

plane A flat surface.

product The result of multiplying eg the product of 2 and 3 is 6.

quadrant A quarter of a circle.

viii MS5.1.1 Perimeter and area 03/05

quadrilateral A plane shape with four straight sides.

rhombus A quadrilateral with four equal sides.

semicirculararc

Half the circumference of a circle.

substitute To replace one thing with another. Insertion of aninteger or number for a pronumeral.

tessellate To cover a plane surface with repeated shapes leavingno gaps and with no overlaps.

Mathematics Stage 5


Part 1 Straight sided figures

Part 1 Straight sided figures 1

Contents – Part 1

Introduction – Part 1..........................................................3

Indicators ...................................................................................3

Preliminary quiz – Part 1 ...................................................5

Area and perimeter..........................................................11

Simple composite figures ................................................17

Triangles and parallelograms..........................................25

Parallelograms.........................................................................30

Area of a rhombus and a kite ..........................................33

Kites .........................................................................................37

Area of a trapezium.........................................................41

Area of composite figures................................................47

Suggested answers – Part 1 ...........................................51

Additional resources – Part 1 ..........................................69

Exercises – Part 1 ...........................................................77

2 MS5.1.1 Perimeter and area 03/05


Introduction – Part 1

In this part you will review the perimeter and area of some basic

quadrilaterals and the triangle. You will also develop the formulas for

the area of a rhombus and a trapezium. This will enable you to find the

perimeter and area of simple composite figures consisting of two shapes.

Indicators

By the end of Part 1, you will have been given the opportunity to work

towards aspects of knowledge and skills including:

• developing and using the formula for the area of a kite and rhombus

• developing and using the formula for the area of a trapezium

• calculating the perimeter and area of simple composite figures

consisting of two shapes that have no curved sides.


mathematically by:

• identifying the perpendicular height of a trapezium in different

orientations

• selecting and using the appropriate formula to calculate the area of a

quadrilateral

• dissecting composite shapes into simpler shapes

• solving practical problems involving area of quadrilaterals and

simple composite figures, with straight sides.



Preliminary quiz – Part 1

Before you start this part, use this preliminary quiz to revise some skills

you will need.

Activity – Preliminary quiz

Try these.

1 Write the best name for these quadrilaterals, by using the markings

on their sides and in their angles to help you.

a

_______________________

b

_______________________

c

_______________________

d

_______________________


2 Each of the formulas can be used to determine the area of a type of

figure. Match the area formula to the correct figure, by drawing a

line to link them.

A = length × breadth

A =1

2of the base × height

A = length of side2

3 A square is 5 metres long.

a Calculate its area.

___________________________________________________

___________________________________________________

b Calculate its perimeter.

___________________________________________________

___________________________________________________


4 The rectangle below is 6 metres long and 4 metres wide.

It is not drawn to scale.

6 m

4 m

a Calculate its area

___________________________________________________

___________________________________________________

b Draw a sketch of another rectangle on the grid below that has

the same area but different dimensions. Write the dimensions

on your diagram in the correct place.

What made you choose these dimensions?__________________

___________________________________________________

___________________________________________________


5 a Calculate the perimeter of the figure below.

8 m

2 m

___________________________________________________

___________________________________________________

b Draw a sketch of another rectangle in the space below that has

the same perimeter as the figure above. Write dimensions on

your figure.

6 Calculate the area of a triangle that has a base of 8 centimetres and a

perpendicular height of 3 centimetres.

_______________________________________________________

_______________________________________________________

_______________________________________________________

7 Complete the following statements that describe what is true about

each shape. (These statements are called properties.) Use the word

list: perpendicular, right angles, diagonals.

a The diagonals of a square meet at ________________________

b The ________________ of a rhombus meet at right angles.

c The diagonals of a kite are ______________________ to

each other.


8 Explain what is meant by the statement ‘The diagonals of a square

and rhombus bisect each other but the diagonals of a kite do not.’

(You may use diagrams to assist your explanation.)

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

Check your response by going to the suggested answers section.



Area and perimeter

Use a blue pencil to colour the area of the shape below.

Use a red pencil to show its perimeter.

Did you colour it correctly? You should have blue shaded inside the

shape and there should be a thin line of red around the outside of the

blue area.

Remember that area is the amount of space inside a plane (or flat) shape

and perimeter is the total length around the outside of that shape.

The figure in the activity below has no numbers. You will need to count.


Activity – Area and perimeter

Try these.

1 Use the diagram below to answer the following questions.

a Calculate the area of this rectangle.

___________________________________________________

___________________________________________________

b Calculate the perimeter.

___________________________________________________

___________________________________________________


You can calculate area by either counting the little squares inside a shape

or you can use the formula for the area of that shape.

You can find the perimeter of a shape by either adding up the total of all

the lengths around the outside of the shape or by recognising that because

there are some equal sides in your shape, you can multiply as well as add.


But I can’t remember which is area andwhich is perimeter?

The last part of the word perimeter lookslike the word metre. A metre is a length, justlike perimeter.

In the next activity you will investigate figures that have the same

perimeter and figures that have the same area.


Try these.

2 A square is 3 cm long.

a Draw an accurate diagram of this shape. Write the length of

each side on your diagram.

b Calculate the perimeter of the square.

Write the correct units next to your answer.

___________________________________________________

___________________________________________________

c Calculate the area of this square, writing the correct units next to

your answer.

___________________________________________________

___________________________________________________


d Draw a rectangle that has the same area as the square above.

e Is the perimeter of your rectangle the same as the perimeter of

your square? _________________________________________

Explain _____________________________________________

___________________________________________________

f Draw a rectangle that has the same perimeter as the

square above.

g Is the area of your second rectangle the same as the area of the

square above? ________________________________________

Explain _____________________________________________

___________________________________________________


Shapes that have the same area can have different perimeters.


Also, shapes with the same perimeter can have different areas.

A good way of understanding this last idea is to tie two ends of a piece of

string together. This gives you a fixed perimeter. Put the string around

four of your fingers. Move your fingers together first and then apart,

like this.

Thin rectangle

Wider rectangle

The thinner rectangle has a smaller area than the wider one.

Make your string into any shape you like. All these shapes have the

same perimeter.

Use the exercise below to make sure you understand the difference

between area and perimeter.

Go to the exercises section and complete Exercise 1.1 – Area and

perimeter.



Simple composite figures

Composite plane figures can have two or more shapes glued together to

make a new shape.

You already know that different areas can have the same perimeter.

You are going investigate this further using tessellated (repeated) shapes

that create composite figures.

The series of pictures below show the steps used to build up a shape.

figure 1 figure 2 figure 3

figure 4

figure 5

The final diagram contains five repeated rhombuses. Each rhombus has

the same area and the same perimeter.

The area of each rhombus is 0.9 cm2 .

1 cm

Area = 0.9 cm2.

Use the diagrams above when doing the following activity.


Activity –Simple composite figures

Try these.

1 a Use figure 5, above, to complete the last row of this table.

Number ofrhombuses

Area in cm2 Perimeter in cm

1 0.9 4

2 1.8 6

3 2.7 8

4 3.6 10

5

b Another rhombus is added to figure 5, above, like this:

Use this new shape to complete the first row of the table below.

Number ofrhombuses


6

7

8

c Add the seventh and eighth rhombus next to the shaded rhombus

in the picture above. Use these to complete the table.

d What do you notice about the perimeter of the shape, even

though you are increasing its area.

___________________________________________________



The perimeter of some shapes remains the same even when you cut bits

out of them. For example, the following two shapes both have the

same perimeter.

8 cm

5 cm

8 cm

5 cm

The opposite sides of the first shape are the same. In the second shape,

the two sides opposite the 5 cm side have a total of 5 cm. Perhaps these

two sides may be 1 cm and 4 cm, or 2 cm and 3 cm, or any other pair of

numbers that add to 5 cm. You can also see that the two sides opposite

the 8 cm side add to 8 cm.

There are many lengths that these four unknown sides may be.

Below are two possibilities. However, the total of each part of one side

is always equal to the length of its opposite side.

8 cm

5 cm

4 cm

1 cm

3 cm

5 cm

8 cm

5 cm

3 cm

2 cm

1 cm

7 cm

Go to the two web activities called Magical Matches by visiting the CLI

webpage below. Select Mathematics then Stage 5.1 and follow the links

to resources for this unit MS5.1.1 Perimeter and Area Part 1.

<http://www.cli.nsw.edu.au/Kto12>


Below is the calculation for the perimeter of a rhombus and a right-

angled triangle.

1 cm

Perimeter = 5 + 5 + 5 + 5 or

= 5 × 4 (because all the sides are the same)

= 20 cm

5 cm

4 cm

3 cm

Perimeter = 4 + 3 + 5

=12 cm

(Remember, calculate perimeter the same way every

time. Start at the top of the triangle and move

around clockwise.)

The class teacher drew this rhombus and right-angled triangle together.

The new shape looked like this:

start here5 cm

4 cm

3 cm

What is the perimeter ofthe new shape?

32 cm. Just add 20 and 12.

No. The 5 cm side of the triangle isnow not part of the new perimeter.

And neither is oneside of the rhombus!

Perimeter = 5 + 4 + 3 + 5 + 5

= 22 cm


(For a quicker calculation, you could triple the five and then add the four

and three.)

If you are not getting the correct answers, try starting at the top left hand

corner of the shape and then move clockwise around the shape adding the

lengths as you go. Do this for all composite figures, every time.

Activity – Simple composite figures

Try these.

Answer the following questions about the composite figures below.

These figures have not been drawn to scale.

2 This figure contains a rectangle and a right-angled triangle.

x cm

8 cm

10 cm

20 cm 6 cm

a How long is x? ______________________________________

b Calculate the perimeter. _______________________________

___________________________________________________


3 This figure has a parallelogram sitting on top of a rectangle.

5 mm

6 m

m

8 mm

a Write down the missing lengths on the diagram above.

b Calculate the perimeter. _______________________________

___________________________________________________

4 All the corners in this figure are right angles.

5 m

2 m

9 m

a The bottom of this figure has no length written on it.

Write its length on the diagram.

b What is the sum of the lengths of the two sides opposite the

9 m side? ___________________________________________

c Choose appropriate lengths for these two sides and write them

on the diagram above. (There are many possible answers.)

d What is the perimeter of this figure? _____________________

___________________________________________________

___________________________________________________



Before you find the perimeter of a plane (flat) figure, you need to make

sure that you know all the lengths around the outside.

Remember, always look for sides that are equal in length such as the

opposite sides of a rectangle or the adjacent sides (sides next to each

other) of a square.

You have been increasing area and finding that perimeter can remain the

same. You have also been practising finding the perimeter of

composite figures. Now check that you can solve these kinds of

problems by yourself.

Go to the exercises section and complete Exercise 1.2 – Simple composite

figures.



Triangles and parallelograms

Triangles have three sides. The base is not necessarily on the bottom of

the triangle. However, the height is always perpendicular (at 900 ) to its

base, as in the diagrams below.

height

base

height

bas

e

Remember:

Area of triangle =1

2× base × perpendicular height

or A =1

2bh

Explore why this formula works in the following activity.


Activity – Triangles and parallelograms

Try these.

1 Answer the following questions about the diagram below.

A

B

a The length of the rectangle is 5 units.

How wide is the rectangle?

___________________________________________________

b Calculate the area of the rectangle. ______________________

___________________________________________________

c i Draw a line from A to B so you divide your rectangle and

triangle into two parts. Compare the shaded section of each

part with the unshaded section. What do you notice?

________________________________________________

ii Complete this sentence.

The area of the triangle is _______________ the area of the

rectangle.

d Use the fact that the area of a triangle = 1

2× area of rectangle to

calculate the area of the triangle above.

___________________________________________________


e Complete these sentences.

i The width of the rectangle is the same as the ____________

of the triangle.

ii The length of the rectangle is the same as the ___________

of the triangle.

iii To find the area of a triangle you must ______________ the

area of the rectangle. To do this you must multiply the

length of the rectangle by the _____________ and

___________ by two. This is the same as multiplying the

length of the triangle’s ____________ by its height and

dividing by two.

f Complete for the figure above: If b = ____ and h = ______ ,

then

A = ___ × b × h

= ___× 4 × 5

= 10 square units


To calculate the area of any triangle, you can:

• calculate the area of the rectangle surrounding it, then halve this

answer to find the area of the triangle

or

• use the formula for the area of a triangle, A =1

2bh .

The activity below, shows you different ways of thinking about this

area formula.



Try these.

2 The base of a triangle is 8 metres long and its perpendicular height is

12 metres. By doing the calculation inside the brackets first, show

that the following three expressions give the same answer for the

area of the triangle.

a1

2× (b × h) __________________________________________

___________________________________________________

b (1

2× b) × h __________________________________________

___________________________________________________

c (1

2× h) × b __________________________________________

___________________________________________________

3 There are four lengths given to you in the triangle below.

9 m7.6 m

7 m

3 m

a How long is the:

i base ____________________________________________

ii perpendicular height _______________________________


b Use A =1

2bh to calculate the area of this triangle.

___________________________________________________

___________________________________________________

___________________________________________________

c What are the two lengths you did not need when you calculated

the area of this triangle? ________________________________


When calculating the area of any triangle, using A =1

2bh make sure you

use its base and perpendicular height and no other lengths.

Also, try to choose the easiest way to do your calculations.

• If the base is an even number, then halve the base.

• If the perpendicular height is an even number, then halve the height.

• If neither dimension is an even number, then divide your answer to

(base × height) by two.

• If both dimensions are even numbers, just remember to halve only

one number.


Parallelograms

The class teacher writes the following formula for the area of a

parallelogram on the board.

Area = length of parallelogram × perpendicular height

She then hands out the following diagram.

base length

perpendicular height

Use a pair of scissors to show why this isthe correct way to calculate the areaof a parallelogram.

Use scissors to cut the parallelogram provided for you in

additional resources.

John shows the class that if you cut the parallelogram along the dotted

perpendicular height, you get a right-angled triangle. This triangle can

be moved to the other side of the parallelogram to form a rectangle.

length of parallelogram

perpendicularheight

The area of this rectangle must be the same as the area of

the parallelogram.


Use the diagrams and information above to answer the questions in the

next activity.


Try these.

4 Complete these sentences:

a The width of the rectangle is the same as the _______________

of the parallelogram.

b The length of the rectangle is the same as the _______________

of the parallelogram and its opposite side.

5 One pair of parallel sides of a parallelogram measure 10 cm.

The height perpendicular to these sides is 7 cm long.

a Write these dimensions on the parallelogram below.

b Calculate the area of the parallelogram.

___________________________________________________

___________________________________________________

___________________________________________________


6 Explain why you cannot do this area calculation for the

parallelogram below.

7 m

m

3 mm

Area of parallelogram = length × height

= 3 × 7

= 21 mm2

_______________________________________________________

_______________________________________________________

_______________________________________________________


If you are given the sides of a parallelogram, you can find the perimeter

but you cannot find its area. You can only find its area if one dimension

is perpendicular to the other. The four sides of a parallelogram are not

perpendicular. (If they are, then you call it a rectangle.)

You have been revising finding the area of triangles and parallelograms.

The following exercise also reviews finding areas of squares

and rectangles.

Go to the exercises section and complete Exercise 1.3 – Triangles and

parallelograms.


Area of a rhombus and a kite

A rhombus has four equal sides. If you push a square it becomes a

rhombus. (This is because a square is just a special type of rhombus.)

Apart from having four equal sides, the diagonals inside a rhombus and

a square meet at 900 . These diagonals also bisect each other.

(This means that they cut each other in half.)

When you push the square, one diagonal gets longer and the other gets

shorter, but the two diagonals always remain at 900 , and always cut each

other in half.

You will investigate the area of a rhombus in the next activity.

Use the rhombus provided for you in the additional resources.

This rhombus has a rectangle around the outside of it.


Activity – Area of a rhombus and a kite

Try these.

1 Fold the rhombus along the dotted line and look at only half of it.

a Describe what you are looking at by naming shapes and stating

their position.

___________________________________________________

___________________________________________________

b What area is the shaded triangle compared to the rectangle?

___________________________________________________

c Open your rhombus. Both halves of your diagram are the same.

Complete this sentence: The area of the rhombus is

____________ the area of the rectangle that surrounds it.

2 a Your rhombus and rectangle are on grid paper.

If your rectangle is 14 metres long, what does each square on the

grid paper represent? _________________________________

b How wide is your rectangle? ___________________________

c Calculate the area of the rectangle.

___________________________________________________

___________________________________________________

d Show how you can now calculate the area of the rhombus.

(Show your working. Don’t just give the answer.)

___________________________________________________

___________________________________________________


3 a i How long are the two diagonals of the rhombus?

________________________________________________

ii Complete this sentence.

The two __________________ of the rhombus are the same

length as the ______________ and ________________ of

the rectangle that surrounds it.

b i Multiply the lengths of these two diagonals together.

________________________________________________

ii Halve this answer. ________________________________

What do you notice? ______________________________

________________________________________________

iii Complete this sentence.

The product of the two diagonals is the same as the area

________________________________________________

iv Complete these sentences.

Area of rhombus = 1

2 of the ______________ of its two

diagonals.

Area of rhombus = ___ × ___________× other diagonal.



You can write area formulas in a shorter form using algebra.

The last formula is a shorter way of writing the area of a rhombus.

You know the first four already.

l

l

Area of square = l × l

= l2

b

l

Area of rectangle = lb

h

b

Area of triangle = 12bh

h

l

Area of parallelogram = lh

x

y

Area of rhombus = 12xy


There is one thing that all the dimensions above have in common.

What is it?

Look at the diagrams above. Both dimensions of each shape are

perpendicular (at 900 ) to each other. You can only multiply

perpendicular lengths together to get area.

Kites

Imagine that two of the sides of your rhombus are made of elastic and

that you could stretch them like this.

What shape is it now?

It’s got two different isosceles trianglesin it.

It’s a kite.


Look carefully at the area of the new triangle.

126

12

The area of this rectangle above is 72 m2 (6×12 ) and the area of the

triangle is 12

×12 × 6 = 12

of 72. So the area of the triangle is half the

area of the rectangle, just like before.

How does the area of the kite compare to the whole new rectangle?

It’s just like the rhombus. The kite is halfthe new rectangle.

This means that the formula for a kite is the same as a rhombus.

x

y

Area of kite = 12xy


The class checks that this is right, by using the formulas above.

Rectangle around kite has l = 19, b = 6

Area of new rectangle = lb

= 19 × 6

= 114 m2

Diagonals of kite are x = 6 and y = 19 (or x = 19 and y = 6)

Area of kite =1

2xy

=1

2× 6 × 19

= 3 × 19 (by halving 6)

= 57 m2

57 is half of 114, so it’s true.

Use the formula for the area of a kite in the following activity.


Try these.

4 Complete the following using the kite below.

8 cm

25

cm

a Copy the lengths of the two diagonals below:

x = ________________________________________________

y = ________________________________________________


b Substitute these two numbers into the formula, A =1

2xy and

then calculate the area of the kite.

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________


Remember that this formula can also be used for rhombuses.

To calculate areas of both kites and rhombuses, you only need the

lengths of the two diagonals. Also, remember to halve only once when

doing your calculations.

Use the formula A =1

2xy in the following exercise.

Go to the exercises section and complete Exercise 1.4 – Area of a rhombus

and a kite.


Area of a trapezium

A trapezium has four sides and at least one pair of parallel sides.

It may look like this:

Of course most people call the last figure a rectangle, because rectangles

are special kinds of trapeziums.

You will investigate the area of a trapezium in the next activity.

Use the two trapeziums provided for you in the additional resources.

Activity – Area of a trapezium

Try these.

1 Cut out only one of these trapeziums and place it on top of the other.

a Are both these trapeziums the same size? __________________

b Do they have the same area? ____________________________

c Place the cut trapezium next to the other trapezium, so that you

form a large parallelogram. (You will need to rotate your cut

trapezium.)

d Carefully trace around three of the sides of your cut trapezium

so you can clearly see the parallelogram you have formed.

Remove your cut trapezium.


e Complete this sentence.

The area of the trapezium is _______________ the area of the

parallelogram.

f i Imagine that each square on your grid paper is one metre by

one metre, so that each square has an area of 1 m2 .

(If this is difficult to do, try imagining that you are on the

top of a building looking down at these square metres on the

grass below. From that height they would look much

smaller than metres.)

ii Complete these sentences.

Length of parallelogram, l = ________________________

Height of parallelogram, h = ________________________

Area of parallelogram = l __

= __ × __

= 70 ___

g Show how you can calculate the area of the trapezium.

___________________________________________________

___________________________________________________

2 Look at only one trapezium.

a How far apart are its parallel sides? _____________________ .

b Choose the best words from the list below and write them in the

sentences:

from between parallelogram trapezium

The height of the parallelogram is the same as the distance

_________________ the parallel sides of the trapezium.

This distance is also the height of the ____________________ .


c i How long is the shorter parallel side? _________________

ii How long is the longer parallel side? __________________

iii Add the lengths of the two parallel sides together.

________________________________________________

iv Complete this sentence.

The sum of the two _________________ sides in the

trapezium = the _______________ of the parallelogram.

v Complete this sentence.

Area of trapezium =1

2 × ______ of the ___________ sides × height .

vi Show how you can find the area of this trapezium by first

finding the sum of the parallel sides.

________________________________________________

________________________________________________

________________________________________________

________________________________________________


To find the area of a trapezium you can:

1 Add the lengths of the two

parallel sides together.

This is the same as the length of the

parallelogram.

2 Multiply this sum by the

perpendicular height of the

trapezium.

The perpendicular height of the

trapezium is the distance between

the two parallel sides and the same

as the height of the parallelogram.

3 Halve your answer The area of the trapezium is half the

area of the parallelogram.


So area of trapezium =1

2 the sum of the parallel sides × height .

The class try this with the trapezium below.

12 cm

7 cm

8 c

m

12 + 7 = 19

1

2of 152 is 76 cm2

press 151 ÷ 2 = .5 cm2Just use your calculator and

What if it’s an oddnumber, like 151?

19 × 8 = 152

75

Mathematicians use algebra to write down formulas. If you call the

parallel sides a and b, and the height of the trapezium h, then you get a

picture like this:

b

a

h

(Remember that h is always 900 to the parallel sides.)


The formula that goes with this diagram is:


2× (a + b) × h or

A =1

2h(a + b)

Use the methods shown above in the next activity.

Activity – Area of trapezium

Try these.

3 Calculate the area of the trapezium below.

7 m

m

10 m

m

5 mm

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________


4 If a and b are the lengths of the parallel sides of a trapezium and h is

its perpendicular height, calculate the area of this trapezium, using

the information below.

a = 9 cm

b = 5 cm

h = 10 cm

A =1

2h(a + b)

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________


Using the formula is often faster because you can choose to halve the

larger even number.

If you use your calculator you must remember to add the lengths of the

parallel sides together first (and press “equals” to get the answer.)

Alternatively, you could use brackets, remembering to use ( as well as

) . You can also use the fraction key ab⁄c instead of dividing by two.

Use your calculator in the question above. Can you get the same answer?

You have been practising finding the area of trapezium.

Now check that you can solve these kinds of problems by yourself.

Go to the exercises section and complete Exercise 1.5 – Area of a

trapezium.


Area of composite figures

Composite figures can have more than one shape inside them.

They also may have shapes cut out of them. Here are two examples.

The composite areas are shaded.

To calculate the area of the first figure, you must add the triangular area

and trapezium area together. To calculate the area of the second figure

you must subtract the smaller rectangular area from the larger

rectangular area.

Some figures can be divided different ways.

Can you think of the first figure above in a different way?


There are many different shapes in composite figures.

However, when calculating area it is always best to think carefully about

what shapes you are going to choose. Choosing the smallest number of

shapes means you have less calculations to do.

Activity – Area of composite figures

Try these.

1 Answer the following questions about the figure below.

2 m 2 m

5 m 5 m

4 m

The rectangle above has a triangle cut out of it.

Draw a dotted line along the base of the triangle so you can see it

more clearly.

a How long are the base and height of this triangle?

___________________________________________________

Indicate exactly where these dimensions are, on the

diagram above.


b Calculate the area of the triangle.

___________________________________________________

___________________________________________________

___________________________________________________

c What are the dimensions of the rectangle? __________________

d Calculate the area of the rectangle.

___________________________________________________

___________________________________________________

e Calculate the area of the shaded composite figure by subtracting

one area from the other.

___________________________________________________

2 A farm property was surveyed along its length from corner to corner.

The diagram below shows the results of the survey.

2750 m

2900 m

630

0 m

2500

m

(This is called a traverse survey.)

a There are many shapes inside this figure. However, there are

two that make up the whole of this property. Write the names of

the two shapes.

___________________________________________________


b Calculate the area of the property by finding the area of these

two shapes first, then adding these areas together.

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________


Areas of composite shapes are often needed in many aspects of real life.

For example, in the building industry, area needs to be calculated for

such things as painting, laying of bricks, or ordering decking materials.

Area is also needed in land management. For example, after a bushfire,

to stop soil from being blown or washed away, seed is sometimes

scattered by aeroplanes. You need to know the area of land the seed is

covering in order to plan the amount of seed to purchase.

Use your knowledge of the areas of composite shapes to complete the

exercise below.

Go to the exercises section and complete Exercise 1.6 – Area of composite

figures.


Suggested answers – Part 1

Check your responses to the preliminary quiz and activities against these

suggested answers. Your answers should be similar. If your answers are

very different or if you do not understand an answer, contact your teacher.


1 a square. All the sides are equal and each vertex (corner) is a

right angle.

b rhombus. All the sides are equal.

c trapezium. At least one pair of opposite sides are parallel.

(The two right angles are not an important feature of a

trapezium. These angles just mean that the two parallel sides

both meet another side at 900 .)

d kite.

A kite is the shape that flies in the sky.

It has two different isosceles triangles in it.


2

A = length x breadth

A = 12

of base x height

A = length of side 2

3 a 25 m2 .

Area = length of side2

= 52

= 5 × 5

= 25 m2

b 20 m.

5 m5 m

5 m

5 m

start here

Starting from the top left corner and going clockwise, add

5 +5 + 5+ 5 = 20. Alternatively, you will notice that all the

sides are equal, so you can simply calculate 4 × 5 = 20 .


4 a 24 m2

Area = length × breadth

= 6 × 4

= 24 m2

b There are many rectangles you can draw. Here are two.

8 m

3 m

12 m

2 m

(If you are not sure if your diagram is correct, contact

your teacher.)

These dimensions were chosen because they multiply together

to give the same area of 24 m2 .

5 a 20 m. Again, starting at the top left of your diagram and going

clockwise, Perimeter = 8 + 2 + 8 + 2 = 20 m. Alternatively, you

will notice that the opposite sides have the same length, so:

Perimeter = 2 × (8+2)

= 2 × 10

= 20 m

b There are many rectangles you can draw. Here are two.

9 m

1 m

6 m

4 m

One length and one breadth must add to a total of 10 m.

(If you are not sure whether your diagram is correct, contact

your teacher.)

6 12 cm2 .

Area =1

2 of base × height

=1

2 of 8 × 3

= 4 × 3 (by halving 8)

= 12 cm2


7 a right angles b diagonals c perpendicular

8 This means that in a square and a rhombus, each diagonal cuts the

other diagonal on half. However in a kite, only one diagonal cuts

the other diagonal in half, so you cannot use the words ‘each other’

when talking about the diagonals of a kite.

For example: if the diagonals of a square are 6 cm long then,

3 cm each

If the diagonals of a rhombus are 6 cm and 8 cm long then,

3 cm each

4 cm each

If the diagonals of this kite below are 4 cm and 10 cm, then

2 cm each

The other diagonal may be divided into lengths of 1 cm and 9 cm or

2 cm and 8 cm or 3 cm and 7 cm (or any two lengths that have a total

of 10 cm.)



1 a Area = 35 square units (or 35 u2 )

Area is the total space inside the rectangle. There are 35 little

squares inside this rectangle. However, the answer is not just

35. You must write down the name of the units you are using.

For example: if you were in an aeroplane and looking down at a

large piece of land perhaps these units are square kilometres.

If this was under a microscope, perhaps they are even smaller

than square millimetres.

As there are no lengths in the diagram you can only write a

general term. That is 35 square units (or 35 u2 .)

Alternatively, because it is a rectangle, you can count the spaces

between the marks on two of the sides of the rectangle and get:

length = 7 units

width = 5 units

(Remember to use the word ‘units’ because you

do not know what each mark is.)

Area of rectangle = length × width

= 7 × 5

= 35 square units

b Perimeter = 7 + 5 + 7 + 5

= 24 units

If you use the this method, try to always start at the same place,

like the top left corner of the shape, counting the spaces between

the marks. Continue going around clockwise until you get back

to your starting position.

start here1 2 3

2324

This way you will never miss out a side.


Alternatively, you could do this:

perimeter = 2 × (length+width)

= 2 × (7+5)

= 2 × 12

= 24 units

This second method is faster. Both pairs of opposite sides are

the same. Once you know the length of one side, you know the

opposite side is the same.

2 a 3 cm

3 cm

3 cm

3 c

m

b Perimeter = 3+3+3+3

= 12 cm

or because all four sides are equal you can do this:

perimeter = 4 × 3

= 12 cm

c Area of square = 3 × 3 (or 32 )

= 9 cm2

d There are other more difficult rectangles you can draw.

This is the simplest:

9 cm

1 cm

(Another rectangle, for example, may be 0.9 cm by 10 cm.)

e No.

The rectangle above has a perimeter of 9 + 1 + 9 + 1 = 20 cm,

whereas the square has a perimeter of 12 cm.


f There are many answers. Here are two:

4 cm

2 cm

2 cm

4 cm

5 cm

1 cm

5 cm

1 cm

Notice that one pair of adjacent sides add up to half the

perimeter. So to decide the length and width of your rectangle,

you must first divide the perimeter of your square by 2.

12 ÷ 2 = 6. This means that the length and width of your

rectangle must add up to 6, so the sides may be 5 and 1, or 4 and

2 or 3 and 3 (just like the square,) or 5.5 and 0.5. The list is

endless.

g No, the area of the rectangle is not 9 cm2 . It is different:

area of first rectangle, above = 2 × 4

= 8 cm2

area of second rectangle = 5 × 1

= 5 cm2

Activity – Simple composite figures

1 a5 4.5 12

Area = 0.9 × 5

= 4.5

(You may use a calculator if you wish or simply multiply 9 by 5

in your head and change it to 4.5. As there is one decimal place

in the question, there must be one decimal place in the answer.

This is because both question and answer are one tenth the size

of 9 × 5.)

Be careful when counting area. Always start at the top left of

the shape and count around clockwise.


b and c

Number ofrhombuses


6 5.4 12

7 6.3 12

8 7.2 12

d The perimeter of the shape remains the same, even though you

are increasing its area.

Isn’t that amazing!

2 a x = 14 cm.

The bottom of the rectangle = 20 − 6

= 14 cm

As the opposite sides of a rectangle are the same length, then the

top length is also 14 cm.

b Staring at the top left and moving around the shape in a

clockwise direction:

Perimeter = 14 + 10 + 20 + 8

= 52 cm

3 a

5 mm

6 m

m

8 mm

6 m

m

5 m

m

8 mm

The opposite sides of parallelograms and rectangles are all

equal.

b Starting at the top left again,

Perimeter = 8 + 6 + 5 + 8 + 5 + 6

= 2 × (8 + 6 + 5)

= 38 mm


4 a 7 m. (5 + 2 = 7 m – see diagram below.)

In figures like this, always add the lengths that are parallel to the

side you are trying to find.

b 9 m. The two unknown sides must have the same total length as

the other side opposite them.

c You could choose 1 and 8 or 2 and 7 or 3 and 6 metres, etc.

You can write whatever you like so long as their sum is 9.

One possible set of lengths are:

5 m

2 m

9 m

1 m

8 m

7 m

d Perimeter = 5 + 1+ 2 + 8 + 7 + 9

= 32 m

Alternatively, its perimeter is twice the length plus width.

Perimeter = 2 × (9 + 7)

= 2 × 16

= 32 m


1 a 4 units. (Remember to count the spaces, not the marks.)

b 20 square units.

You can draw lines in your rectangle, and then count the squares

or you can just multiply five by four because there are five rows

each with four squares in it.

Area = 5 × 4

= 20 square units


c i In each part, the shaded section is the same as the unshaded

section.

A

B

dia

gona

l

diagonal

Each part, on either side of AB, is divided into two sections

by a diagonal. In each part, both sides of each diagonal are

the same shape and have the same area.

ii half

d Area of triangle = 1

2× 20

= 10 square units

You can simply halve twenty or divide the two into the twenty.

e i base

ii height

iii halve, width, divide, base.

f b = 4 and h = 5, then

A = 1

2× b × h

=1

2× 4 × 5

= 10 square units

2 Make sure you substitute 8 into b and 12 into h, so that you get three

different ways of doing the same question.

a1

2× (bh) =

1

2× (8 ×12)

=1

2× 96

= 48 (by halving 96)


b (1

2b) × h = (

1

2× 8) × 12

= 4 × 12 (by halving 8)

= 48

c (1

2h) × b = (

1

2× 12) × 8

= 6 × 8 (by halving 12)

= 48

3 a i 3m.

ii 7 m.

The base is always at right angles to its height. It is not always

on the bottom of the triangle. (Practise visualising this by

turning the triangle around so you can look at it in different

directions.

b 10.5 m2

A =1

2bh

=1

2× 3 × 7

=1

2× 21

Note that it as 3 and 7 are odd numbers, it is easier to simply

multiply them. To calculate the answer, you could either

mentally halve 21, or use a calculator like this:

• 1 ab⁄c 2 21 = or

• 21 ÷ 2 =

c 7.6 m and 9 m. Neither of these lengths are either the base or

the perpendicular height of the triangle.

4 a height b length (or base), opposite

5 a

10 cm

7 cm


b Area of parallelogram = length × height

= 10 × 7

= 70 cm2

6 The height of a parallelogram must be perpendicular to its length.

This is not true in the example given, so you cannot multiply three

by seven to get area. The two sides are not at right angles to each

other. You need the height in the diagrams below to find its area.

7 m

m

3 mm

height

7 m

m

3 mm

height


1 a You are looking at an isosceles triangle (two sides equal) inside

a rectangle that surrounds it.

b Half. (Divide the shape into two parts, from the top of the

triangle (the apex) to the base of the triangle. Each of the two

shaded sections are the same size as the two unshaded sections.)

c half

2 a 1 m2 Each square is one metre by one metre and this equals one

square metre.

b 6 m. (Remember to count the spaces along the width.)

c Area of rectangle = length × width

= 14 × 6

= 84 m2

Alternatively, you could use

A = lb

= 14 × 6

= 84


d Area of rhombus = 1

2× area of rectangle

= 1

2× 84

= 42 m2

3 a i 14 m and 6 m

ii diagonals, length, width (or breadth)

b i 14 × 6 = 84

ii1

2 of 84 = 42 . The answer is the same as halving the area

of the rectangle.

iii of the rectangle

iv product, 1

2, diagonal.

4 a x = 8 cm, y = 25 cm (or x = 25 cm, y = 8 cm because x and y are

used for both diagonals. )

b 100 cm2

A =1

2xy

=1

2× 8 × 25

Now halve the even number, 8. You get 4 .

= 4 × 25

= 100

Alternatively, after substituting, you could simply use a

calculator, like this: 1 ab⁄c 2 8 25 =

or this: 8 25 ÷ 2 =


Activity – Area of a trapezium

1 a Yes b Yes

c and d

e half

f i l = 14 m , h = 5 m .

ii Area of parallelogram = lh

= 14 × 5

= 70 m2

g Area of trapezium =1

2of area of parallelogram

=1

2× 70

= 35 m2

2 a 5 metres

count thisdistance

(“How far apart?” always means the shortest distance.

This distance is usually at 900 to at least one of the lines.)

b between, trapezium.


c i 5 m ii 9 m

shorter parallel side

longer parallel side

(Remember, you must be careful to count the spaces.)

iii 5 + 9 = 14 metres.

iv parallel, length.

v sum, parallel.

vi Sum of parallel sides = 14 m and height of trapezium = 5 m.


2× 14 × 5

=1

2× 70

= 35 m2

(Notice that this is the same area as before when you halved

the area of the parallelogram.)

3 42.5 mm2 .

Sum of parallel sides = 7 + 10

= 17 mm

height = 5 mm

(Remember, the shortest distance

between the parallel sides is

the height.)

Area =1

2× 17 × 5

=1

2× 85

= 42.5 mm2

(Remember that halving a number

is the same as dividing it by two.

Using a calculator, you can press:

• 7 + 10 = × 5 ÷ 2 =. (You must press equals after adding.)

Alternatively, you can use brackets: ( 7 + 10 ) × 5 ÷ 2 =

• 1 ab⁄c 2 × ( 7 + 10 ) × 5 =


4 Always write the formula first, then substitute.

(Remove a letter and put a number in its place.)

A = 1

2h(a + b)

=1

2× 10 × (9 + 5)

=1

2× 10 × 14

At this point, you can choose to halve the 10 or halve the 14 or halve

the answer to 10 ×14 . If you are using your head, which is better?

Always try to reduce the size of your numbers, so halve 14, because

14 is the biggest number. (Halving 10 means you will get 5×14 and

this is more difficult to do in your head than when you halve 14.

Halving 14 means you need to only calculate 10 × 7 .

This is much easier.)

So A = 10 × 7 (by halving 14)

= 70

So area of trapezium is 70 cm2

Activity – Area of composite figures.

1 a base = 6 m, height = 4 m.

base = (5 + 5) − (2 + 2)

= 6 m

4 m

2m 2m

5 m 5 m

4 m

6 m

b 12 m2

Area of triangle = 1

2× base × height

= 1

2× 6 × 4

= 3 × 4 (by halving 6)

= 12 m2


c 10 m by 4 m.

d 40 m2

Area = length × breadth

= 10 × 4

= 40 m2

e 28 m2

Shaded area = rectangular area − triangular area

= 40 − 12

= 28 m2

2 a trapezium, triangle

triangletrapezium

b 21 235 000 m2

Trapezium: h = 2750 m, a = 2500 m, b = 6300 m


2h(a + b)

= 1

2× 2750 × (2500 + 6300)

=1

2× 2750 × (8800)

= 2750 × 4400 (by halving 8800)

= 12 100 000 m2

Triangle: b = 6300 m, h = 2900 m


Area of triangle =1

2bh

=1

2× 6300 × 2900

= 3150 × 2900 (by halving the larger number)

= 9 135 000 m2

Total area = area of trapezium + area of triangle

= 12 100 000 + 9 135 000

= 21 235 000 m2


Additional resources – Part 1

To be cut out and used in Activity – Triangles and parallelograms,

question 4.



To be cut out and used in Activity – Area of a rhombus and a kite.

fold

her

e



Cut out only one figure and use both pieces in Activity – Area of a

trapezium.



Use this map in Exercise 1.6 – Area of composite figures.

Na

ttai

R

i v e r

Win

ge c a r r i b e e R i v e r

Shoa l h a v e n

Rive

r

Mul

war

eeR

iver

Wo

l lo nd i l l

y R i ve r

Walla

Campbellt

BMarulan

Goulburn

Crookwell

LAKEGEORGE

CANBERRA

Moss Vale

Mittagong

Picton

AvoDa

LAKEBURRAGORANG

WarragambaDam

Katoomba

Blackheath

Wi

Lithgow

Penr

Ko

wm

un

gR

i v

e r

Co

xs

R i ve

rG ro s e

R i ve r

Nap

ea

nR

ive

r

OH

WingeReser

Fitzroy FallsReservoir

LAKE YARRUNGA

Tallowa Dam

Warragambacatchment

NapeanDam

135 km

56 km

12 km

45 km

40 km

59 km



Exercises – Part 1

Exercises 1.1 to 1.6 Name ___________________________

Teacher ___________________________

Exercise 1.1 – Area and perimeter

1 Match the word ‘perimeter’ or ‘area’ next to each of the

following situations.

a You are painting a wall. _______________________________

b You wrap some string around some nails __________________

c Planting a crop of wheat. ______________________________

d Drawing the outline of a shape. _________________________

e Fencing a property. ___________________________________


2 Below are two rectangles.

a Check that they both have the same area.

Show your thinking below.

___________________________________________________

___________________________________________________

b Discuss Ivan’s statement below.

Rectangles with the same area alwayshave the same perimeter.

___________________________________________________

___________________________________________________

___________________________________________________


3 a Find the area and perimeter of the following shaded shapes.

i

________________________________________________

________________________________________________

ii

________________________________________________

________________________________________________

iii

________________________________________________

________________________________________________

b What do all these shapes above have in common?

___________________________________________________

c Look at the three shapes above. What is the name of the shape

that has the biggest area?

___________________________________________________


d Look at the two shapes below.

The first shape does not have the same perimeter as the

rectangle above. However, the second shape does.

Explain why? _______________________________________

___________________________________________________

___________________________________________________

4 a Use the grid paper provided to draw a square that has the same

area as the shaded rectangle below.

rectangle

grid paper

b Name the shape above that has the smallest perimeter.

___________________________________________________


5 (Harder) Use the diagrams below to answer the following questions.

a

The perimeter of this square is 20 m. How long is each side?

___________________________________________________

b7 m

i The perimeter of this rectangle is 20 m. What is its width?

________________________________________________

________________________________________________

ii How many more rectangles have this same perimeter?

________________________________________________

Draw as many as you can on a blank piece of paper and

send it to your teacher.


Exercise 1.2 – Simple composite figures

1 Use match sticks to make this rectangle.

Each�match�is�one�unit�long

a Write the correct number in the spaces below.

i The perimeter of this rectangle is ____ units long.

ii The area of this rectangle is ____ square units.

b Reduce the area of this rectangle by two square units, without

changing its perimeter. Draw your answer in the space below.


2 a What is the perimeter of the trapezium, below?

___________________________________________________

5 m

6 m

4.1 m

10 m

This trapezium above has been copied, turned around and joined to

the first trapezium to make the parallelogram below.

b Write correct lengths against all four sides of the parallelogram

above. What is the perimeter of this parallelogram?

___________________________________________________

c Explain why the perimeter of the parallelogram is not double the

perimeter of the trapezium.

___________________________________________________

___________________________________________________


3 An area of land on an inside bend of the Murray River is

shown below.

120 km

130 km

This land contains a rectangular and triangular plot. The total river

frontage is 100 km and both distances along each side of the bend

are the same.

a On the diagram above, write lengths against the remaining

four sides.

b If no fencing is needed either along the river frontage or

between the two plots, how many kilometres of fencing is

required to enclose the remainder of the property?

___________________________________________________

___________________________________________________

4 A large canvas awning for a paved area is made with two pieces of

material.

7 m

20 m

One of the pieces is shaped like a parallelogram and the other is an

equilateral triangle (three sides equal). The awning needs to be

sewn along the outside edge. How many metres of sewing is

there to do?

_______________________________________________________


5 Use 12 matchsticks to make the following shapes. (You are only

allowed to put the matchsticks at 900 to each other. This means thatyou cannot put the matchsticks together like this ⟨ or this <).

a Make the biggest possible area you can, using all

12 matchsticks. Draw your answer below.

b Make the smallest possible area you can using all

12 matchsticks. Draw your answer below.

6 This is the front view of a house.

15 m

3 m

x

The owner has a length of Christmas lights that he uses every year to

highlight the outside of his house. These lights run along the

ground, up the 3 m walls and over the roof with the two ends

meeting. The string of lights are 36.6 m long. How long is x?

_______________________________________________________

_______________________________________________________


Exercise 1.3 – Triangles and parallelograms

1 There are many lengths given to you in the following diagrams.

7.2 cm

4 cm

5 cm

9 cm

7 mm

24 m

m25 m

m

A B

a Complete the table below by choosing the correct base and

perpendicular height of each of these triangles.

Triangle Base Perpendicular height

A

B

b i In triangle B, which length do you not need to calculate

its area? ________________________________________

ii Calculate the area of triangle B.

________________________________________________

________________________________________________

________________________________________________

c Calculate the area of triangle A.

___________________________________________________

___________________________________________________

___________________________________________________


d i In triangle A, which length do you not need to calculate its

perimeter? _______________________________________

ii Calculate the perimeter of triangle A.

________________________________________________

________________________________________________

________________________________________________

2 a What is the length of the base and the perpendicular height of

the parallelogram below? ______________________________

___________________________________________________

8 m

m

10 m

m

12 mm

b Calculate the area of the parallelogram above.

___________________________________________________

___________________________________________________

c Calculate its perimeter. ________________________________

___________________________________________________


3 An orange orchard has a river that is 200 m wide flowing through it.

The property has a 1000 m river frontage.

200 m

Ri v

er

600

m

1500 m

1000 m

a Assuming the river is perfectly straight, what is the name of the

shape that the river cuts through the orchard?

___________________________________________________

b Calculate the area of land that the river covers.

___________________________________________________

___________________________________________________

c Calculate the area of the rectangle.

___________________________________________________

___________________________________________________

d Calculate the area of dry land that can be used for

growing oranges.

___________________________________________________

___________________________________________________

e If one orange tree needs 20 m2 of land to grow properly, how

many orange trees need to be planted so that all of the dry land

is used?

___________________________________________________


Exercise 1.4 – Area of rhombus and a kite


3 mm

7 mm

4 mm4 mm

5 mm

8 mm

a What shape is it? _____________________________________

b What are the lengths of both diagonals? ___________________

c i Which lengths do you not need to calculate the area of

this figure? ______________________________________

ii Calculate the area of this figure, using the formula A = 12xy ?

________________________________________________

________________________________________________


d i The figure above contains two isosceles triangles.

Label the base and height of both these triangles on the

diagrams below.

ii Calculate the area of both these triangles.

________________________________________________

________________________________________________

________________________________________________

________________________________________________

________________________________________________

iii Show that the total of these two areas is equal to the area of

the figure at the beginning of this question.

________________________________________________

________________________________________________


2 a The diagonals of a rhombus measure 14 cm and 20 cm.

Which diagram is most correctly drawn?

A 8 cm

9 cm

11 cm

6 cm

B 7 cm

10 cm

10 cm

7 cm

C 10 cm

7 cm

7 cm

10 cm

b Choose the best word from this list to complete the sentence

below: cut, bisect, meet.

The diagonals of a rhombus _________________ each other.

3 a The diagonals of a rhombus measure 12 mm and 7 mm.

Write the correct dimensions on the diagram below.

b Calculate the area of this rhombus by using the formula A = 12xy .

___________________________________________________

___________________________________________________

___________________________________________________


4 The rhombus below can be cut up into triangles.

9 cm

12 c

m

12 c

m9 cm

a Draw a sketch of one of these triangles in the space below, then

calculate its area.

___________________________________________________

___________________________________________________

___________________________________________________

b Add the areas of the two triangles together to get the total area

of the rhombus.

___________________________________________________

___________________________________________________

c Show that the formula for the area of a rhombus, A = 12xy ,

gives the same area.

___________________________________________________

___________________________________________________

___________________________________________________


Exercise 1.5 – Area of a trapezium

1 a Write the correct lengths on this trapezium, using the

information provided below.

The parallel sides of this trapezium measure 7 cm and 12 cm.

The perpendicular height of the trapezium is 8 cm.

b Complete the following for the information above.

a = __

b = __

h = __

c Calculate the area of this trapezium above, by using the formula,

A =1

2h(a + b) .

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________


2 The regular hexagon below is divided into two trapeziums.

5 cm

13 cm

3 cm

a How long is each side of the hexagon? ____________________

b These two trapeziums are called isosceles trapeziums.

Why do you think they have this name? ___________________

___________________________________________________

c Calculate the area of one of these trapeziums.

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________

d Calculate the area of the regular hexagon.

___________________________________________________

___________________________________________________


3 The shaded side of this bus shelter below is to be used for

advertisements.

4 m

2 m

3 m

a Calculate the area of this ‘advertisement’ side of the bus shelter.

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________

b If it cost $100 per square metre to advertise, how much does it

cost to cover this whole side with an advertisement?

___________________________________________________

___________________________________________________


Exercise 1.6 – Area of composite figures.

1 Calculate the area of the shaded part of the shape below by

answering the following questions.

8 m

2 m

1 m

a Complete, to find the area of the triangle.

b =________, h =__________

Area =12bh

___________________________________________________

___________________________________________________

___________________________________________________

b Calculate the area of the square. (Remember, both sides are the

same.) ______________________________________________

___________________________________________________

c Calculate the area of the shaded part by subtracting one area

from the other.

___________________________________________________

___________________________________________________


2 The diagram below is a sewing pattern for an awning that is to go

over a boat.

2.5 m

2.5

m

1.5

m

a This sewing pattern can be divided into a bottom part and a top

part. Name the shape in each part.

___________________________________________________

b What dimensions do you need to calculate the area of the:

i bottom part ______________________________________

ii top part _________________________________________

c Calculate the area of:

i the bottom part ___________________________________

________________________________________________

ii the top part ______________________________________

________________________________________________


3 The rectangular wall below is to have a mural painted on it.

The shaded section will have a sky blue background.

1 m

1.1 m

0.5

m

2.5 m

2 m

a Divide the shaded area above into two shapes.

(The shapes must be from the list: square, rectangle, triangle,

rhombus, parallelogram, trapezium because these are the area

formulas you know.)

b Calculate the area of both these shapes.

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________

c Find the total area of the shaded section.

___________________________________________________

d The wall is 2 metres high by 2.6 metres wide.

Calculate the area of the whole rectangular wall.

___________________________________________________

___________________________________________________


e i The remaining part of the wall is to be painted apple green.

Describe how you would find this area.

________________________________________________

________________________________________________

ii Calculate this area._________________________________

________________________________________________

4 Use the map supplied in the additional resources to answer the

following question.

The area of Warragamba catchment that supplies Sydney’s water

can be approximated into the area of two trapeziums.

The distances supplied to you on the map are actual distances.

Use these two trapeziums to calculate the approximate area of the

Warragamba catchment.

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________


5 a Calculate the area of the side of this step and landing that needs

to be tiled.

90 cm

120 cm15 c

m

15 c

m

Area to be tiled

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________

b If tiles cost $42 per square metre, how much will it cost to buy

the tiles?

___________________________________________________

___________________________________________________

Mathematics Stage 5


Part 2 Figures with curved sides

Part 2 Figures with curved sides 1

Contents – Part 2

Introduction – Part 2..........................................................3

Indicators ...................................................................................3

Preliminary quiz – Part 2 ...................................................5

Circumference and perimeter..........................................13

Areas of circular shapes..................................................17

Quadrants .......................................................................21

Area of a quadrant ..................................................................25

Curved composite figures................................................29

Suggested answers – Part 2 ...........................................35

Exercises – Part 2 ...........................................................47



Introduction – Part 2

In this part you will review the formulas for finding the area and

circumference of a circle. You will also learn to choose appropriate

formulas to find the area and perimeter of composite shapes with curved

sides. This enables you to use the area and circumference formulas for a

circle, semicircle and quadrant.

Indicators


towards aspects of knowledge and skills including:

• calculating the area of simple composite figures consisting of two

shapes, where at least one of them is a circle, semicircle or quadrant

• calculating the perimeter of simple composite figures consisting of

two shapes, where at least one of them is a circle, semicircle

or quadrant.


mathematically by:

• dissecting composite shapes into simpler shapes

• solving practical problems involving area of simple composite forms

• asking questions which can be solved by using mathematics.



Preliminary quiz – Part 2

Before you start this part, use this preliminary quiz to revise some skills

you will need.


Try these.

1 To what decimal do you think the arrows are pointing?

Write the correct decimal above the arrow on the number

lines below.

a

0 1

b

0.1 0.2


2 a To what decimals are the following arrows pointing?

2.3 2.4

i ii iii

i ________________________________________________

ii ________________________________________________

iii ________________________________________________

b Which of the decimals, i, ii and iii, on the number line above,

are approximately equal to (closer to):

2.3? ________________________________________________

2.4? ________________________________________________

3 Round off 23.76 to one decimal place. _______________________

4 Approximate 44.44 correct to one decimal place. _______________

5 Approximate 75.398 to two decimal places. ___________________

6 How many right angles are there in one revolution? ____________

7 Rewrite the following fractions as decimals.

a1

2 _________________________________________________

b1

4 _________________________________________________

c3

4 _________________________________________________


8 Square the following numbers. Check that you are correct by using

the square key x2 on your calculator.

a 11 _________________________________________________

b 12 _________________________________________________

c 15 _________________________________________________

9 a Write the calculator keys that you must press in order to put six

into the memory of your calculator and then to recall it.

___________________________________________________

b Write the calculator keys that you must press in order to put six

into the memory of your calculator, add seven to this memory

and then recall the answer (13).

___________________________________________________

c Write the keys you must press in order to clear the memory of

your calculator.

___________________________________________________


10 Write the area formula for each of the plane figures below.

Draw a diagram of each shape showing where each pronumeral, used

in the formula, lies on your diagram. The first one has been done

for you.

Square: A = l2

l

l

rectangle: _______________________________________________

triangle: ________________________________________________

parallelogram: ___________________________________________


rhombus: _______________________________________________

kite: ___________________________________________________

trapezium _______________________________________________

11 Calculate the area of:

a a square that has a side length of 7 cm.

___________________________________________________

___________________________________________________

b a rectangle that has a length of 9 mm and a breadth of 4 mm.

___________________________________________________

___________________________________________________

c a triangle that has a perpendicular height of 15 mm and a base

length of 8 mm.

___________________________________________________

___________________________________________________

___________________________________________________


12 Calculate the area of the following figures.

a

9 m

6 m

___________________________________________________

___________________________________________________

b 7 m

m

12 mm

___________________________________________________

___________________________________________________

___________________________________________________

c 17 mm

6 m

m

11 mm

___________________________________________________

___________________________________________________

___________________________________________________


13 Calculate the perimeter of the following figure.

5 cm

3 cm

7 cm

_______________________________________________________

_______________________________________________________

14 Calculate the area and perimeter of the following figure.

6 m2

m

10 m

8 m

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________




Circumference and perimeter

The circumference of a circle is the length of the outside of a whole

circle. It is a length, just like perimeter. Can you remember how to find

the circumference of a circle?

Circumference = π × diameter

or = 2 × π × radius

circumference

diameter

centre

radius

(Remember, π is pronounced ‘pie’, as in ‘meat pie’, but it is actually

spelt, ‘pi’. π is the letter ‘p’ of the Greek alphabet.)

π is approximately equal to three (π ≈ 3 ), so if the diameter of a circle

is equal to 10 mm, then its circumference is approximately equal to

3×10 = 30 mm.

You can use this quick check method to either check if your calculator

has given a reasonable answer or as an approximation before you do a

more accurate calculation.


Follow through the steps in this example. Do your own working in the margin

if you wish.

A circle has a radius of 12 mm.

a What is the length of its diameter?

b Using π ≈ 3 , give a rough estimate of the length of the

circumference.

c Using the pi key, , on your calculator, find the

circumference of the circle correct to one decimal place.

d Explain why your more accurate answer above is always

going to be bigger than your rough estimate.

Solution

a The diameter of a circle is twice the radius. So the diameter is

2 × 12 = 24 mm long.

b Using, π ≈ 3 then,


or C = πd

≈ 3× 24

C ≈ 72 mm

c C = πd

= π × 24

≈ 75.398 mm

∴ Circumference ≈ 75.4 mm. (Remember, when

approximating this answer you must look at the nine next to the

three. This number is closer to 75.4, not 75.3

d The actual value of π is just a little bigger than three, so π × d

is always going to bigger than triple the diameter.

After calculating the circumference of a circle, you can always do a quick

check by multiplying its diameter by 3.

Use these skills in the following activity to help you calculate the

perimeter of a semicircle.


Activity – Circumference and perimeter

Try these.

1 A circle has a diameter of 7 mm.

a Use the formula, C = πd and the pi key, to calculate the

length of its circumference.

_______________________________________________________

_______________________________________________________

_______________________________________________________

b What is the approximate length of the circumference if you use

π ≈ 3 ?______________________________________________

Was your first answer close to this approximation?___________

2 This semicircle has the same diameter as the circle above.

7 mm

a i Divide your more accurate value above, by 2 and write your

answer here.______________________________________

ii Explain why this calculation gives the length of the

semicircular arc.

________________________________________________

________________________________________________

________________________________________________


b i To calculate the perimeter of this semicircle you must add 7

to your answer above. Explain why. __________________

________________________________________________

________________________________________________

ii Complete the equation below.

_____________ = semicircular arc+ diameter

= _____________________

= _____________________


Always remember to halve the circumference when calculating the length

of the semicircular arc. When calculating the perimeter of a semicircle

you must remember to add the length of the diameter.

You have been revising finding the circumference of circles and the

perimeter of semicircles. Now check that you can solve these kinds of

problems by yourself.

Go to the exercises section and complete Exercise 2.1 – Circumference

and perimeter.


Areas of circular shapes

Can you remember how to calculate the area of a circle?

Area of a circle = π × radius × radius

= πr2or A

How can you calculate the area of a semicircle?

Or you could halve it.

Isn’t that the same thing.

First you have to find thearea of the whole circleand then divide it by two.

Yeah. You could multiplythe whole circle area byzero point five. That’s thesame thing as well.

radius

Area of semicircle =1

2× area of whole circle

=1

2× π × radius× radius

=1

2πr2 or πr2 ÷ 2 or 0.5πr2

Use the following activity to review your understanding of calculating

areas of circles and semicircles.


Activity – Areas of circular shapes.

Try these.

1 a Calculate the area of a circle that has a radius of 20 mm.

___________________________________________________

___________________________________________________

___________________________________________________

b Use a pair of compasses to draw a semicircle that has a radius of

20 mm. Sit your semicircle anywhere on the line below.

c Calculate the area of this semicircle.

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________


You must be careful when using your calculator to find areas of circles

and semicircles. Try not to reuse approximated answers.

Put answers that you may need in memory or perhaps do all your

calculations at the end. This will help you get the most accurate answers.


You have been revising finding areas of circles and semicircles.


Go to the exercises section and complete Exercise 2.2 – Areas of circular

shapes.



Quadrants

Remember, a quadrant is just another name for a quarter of a circle.

The teacher draws the following diagram on the board.

centre of circle5 cm

arcquadrant

The shaded part of this diagram is called a quadrant.

The curved part is called an arc.

Explain how you would calculatethe perimeter of this quadrantcorrect to one decimal place.

The little square in the cornermeans it is 90°, so it is a quarterof a circle.

The class checks that 900 × 4 = 3600 .

Ivan then doubles 5 cm to get the diameter and then writes these two

calculations down.


= π ×10

Length of arc = π ×10 ÷ 4


He then uses his calculator like this: 10 ÷ 4 = and gets

7.8539816….

The perimeter is 7.9 cm.

But isn’t the perimeter thetotal of all the sides?

Oh, I forgot. 7.9 + 10 is 17.9.The perimeter is 17.9 cm.

Ivan added 10 to 7.9 because each radius on the quadrant above is 5 cm

long. Ivan’s perimeter calculation is shown below.

Perimeter = arc length + radius+ radius

or P = arc+ 2 × r

= 7.9 + 2 × 5

= 7.9 +10

=17.9 cm

The activity below helps you to remember to add all three sides of the

quadrant together to find its perimeter.


Activity – Quadrants

Try these.

1 Answer the following questions about the quadrant below.

4.2 mm

a How long is the diameter of the circle that this quadrant came

from?_______________________________________________

b Write the correct numbers in the calculation below.


= π × _______

Length of arc = π × ______÷ ________

c Use a calculator to find the length of the arc of this quadrant.

___________________________________________________

d You are asked to find the perimeter of the quadrant above.

Write the missing words and numbers below.

P__________ = _____ length + _________ + radius

= ______+ _______+ _______

e What is the perimeter of the quadrant? (You can use a

calculator or your head.) ________________________________


In the activity above you calculated the length of the arc of a quadrant by

dividing the circumference of a whole circle by four.


There are two other ways you could have done this:

• Length of arc = 1

4× π × 8.4

1 ab⁄c 4 8.4 =

or

• Length of arc = 0.25 × π × 8.4

0.25 8.4 =

(Remember, 0.25 is the same as 1

4)

Use one of these methods in the following activity.


Try these.

2 A colourful awning at the entranceway of a building is supported by

wire that is anchored to the building at both ends.

6 m

awning

wire support

anchor points

How long is the wire, correct to one decimal place?

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________



The question above did not ask for perimeter. Always read the question

carefully, so you do not do extra work.

Area of a quadrant

When calculating the area of a quadrant, you can use similar methods to

the above.

So, for a quadrant, you could firstfind the area of the whole circle andthen divide by four … or multiply byone quarter … or multiply by zeropoint two five.

radius

Area of quadrant = 1

4× area of whole circle

= 1

4× π × radius× radius

= 1

4πr2 or πr2 ÷ 4 or 0.25πr2

The activity below helps you to practise using all three methods so you

can decide on the one you like best.



Try these.

3 Answer the following questions about the quadrant below.

82 m

m

a Write the correct numbers and symbols to match the calculator

buttons used to find the area of the quadrant.

i Area of quadrant = ________________________________

1 ab⁄c 4 82 82 =

ii Area of quadrant = ________________________________

0.25 82 x2 =

iii Area of quadrant = ________________________________

82 x2 ÷ 4 =

b Write the answer next to each calculator method above. Do they

all give the same answer?_______________________________


It doesn’t really matter which method you use as they all give the same

answer. The important thing is to remember to divide the area of the

whole circle by four or to multiply it by 1

4 or 0.25. If you forget to do

this you will get the wrong answer.

Use one of the methods above in this activity.



Try this.

4 Calculate the area of the figure below.

3.2 cm

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________


Although the formula for the area of a circle is different to the formula

for its circumference, you must remember that for a quadrant, both

formulas must be altered by:

• ÷ 4 or

• × 1

4 or

• × 0.25

You have been calculating the perimeter and area of quadrants.


Go to the exercises section and complete Exercise 2.3 – Quadrants.



Curved composite figures

Combining different shapes together in the one diagram produces

composite figures. Look at the diagrams below. Two figures are used

to create each shape. What are they?

In the first diagram you have a semicircle on top of an isosceles

trapezium and in the second you have a square with a quadrant

(1

4 or 0.25 of a circle) removed from it.

You may be asked to find both area and perimeter of these types

of shapes.

The following example shows you how to calculate both the area and

perimeter of the second diagram above.


Follow through the steps in this example. Do your own working in the

margin if you wish.

Two equal perpendicular sides and an arc enclose the area

below. The radius of the arc is the same length as the two

perpendicular sides.

10 mm

a Calculate the area of this figure correct to one

decimal place.

b Calculate its perimeter correct to one decimal place.

Solution

a The area of this figure looks like it is going to be difficult

to calculate. However, it is only a square with a quadrant

cut out of it. You must subtract the area of the quadrant

from the area of the square. (Complete the 10 mm square

on the diagram above as it will help you visualise the two

figures. Your diagram should look like the previous

diagram.)

Area of square = side× side

=10 ×10

=100 m2


Area of quadrant = 1

4× area of circle

= 1

4× πr2

=1

4× π ×102 (Note: 102 =10 ×10)

= π ×100 ÷ 4

≈ 78.5 m2

(Remember, you can multiply ten by ten or use the square

key x2 like this: 10 x2 ÷ 4 = )

Area of figure = square area − quadrant area

≈ 100 − 78.5

≈ 21.5 m2

b The perimeter of the figure is easier to visualise.

Perimeter = side + side + length of arc.

First find the diameter because you need it to find the

circumference of the circle. (Diameter = 20 m)

Next calculate the length of the arc.

Length of arc = circumference of circle ÷ 4

= π × diameter ÷ 4

= π × 20 ÷ 4

= π × 5 (because 20 ÷ 4 = 5)

≈ 15.7 m

Finally add the three sides together.

Perimeter = side+ side+ length of arc

=10 +10 +15.7

= 35.7 m

It is important to identify your calculations by using English. Label them

as ‘length of arc =’, or ‘area =’ or ‘perimeter =’. If you don’t, you can

easily make a mistake by adding or subtracting the wrong numbers.

In the next activity, make your mathematics clear by using English to

label your calculations.


Activity – Curved composite figures

Try these.


8 cm

2 cm

4 cm

5 cm

A semicircle and an isosceles trapezium form the diagram above.

a Tick (�) the correct answer. The top section of the diagram is:

a circle with radius 8 cm

a semicircle with radius 8 cm

a circle with radius 4 cm

a semicircle with radius 4 cm.

b Calculate the area of the figure by completing these steps.

i Calculate the area of the top section of the diagram above.

________________________________________________

________________________________________________

________________________________________________

________________________________________________

ii The bottom section of the diagram is a trapezium.

Write down the values of a, b and h, where a and b are the

lengths of the parallel sides and h is the perpendicular

height between the two parallel sides.

a = b = h =


iii Calculate the area of the trapezium above using the formula

for the area of a trapezium, A =1

2h(a + b) .

________________________________________________

________________________________________________

________________________________________________

iv What is the total area of this composite figure?

________________________________________________

c Calculate the perimeter of the shape by completing these steps.

i Calculate the length of the arc.

________________________________________________

________________________________________________

________________________________________________

ii Calculate the perimeter of the figure.

________________________________________________

________________________________________________

________________________________________________

iii There are many lengths given in the diagram above.

Which lengths did you not need to use in order to calculate

the perimeter of the figure? __________________________



When calculating area and perimeter of composite shapes you must:

• recognise the type of shapes you have in the diagram

• recall the formulas needed for calculations

• link the correct dimension with each part of your formula

• decide what to add and what to subtract

• decide when to add and when to subtract.

Practise finding area and perimeter of composite shapes in the

following exercise.

Go to the exercises section and complete Exercise 2.4 – Curved composite

figures


Suggested answers – Part 2

Check your responses to the preliminary quiz and activities against these

suggested answers. Your answers should be similar. If your answers are

very different or if you do not understand an answer, contact your teacher.


1 a about 0.7. Halfway between 0 and 1 is 0.5. The arrow is to the

right of 0.5.

b about 0.17. Halfway between 0.1 and 0.2 is 0.15. (You can use

a calculator to check if you wish using (0.1 + 0.2) ÷ 2 = 0.15.)

To the right of this is 0.16, 0.17, 0.18 and 0.19. The next

decimal after this is 0.20 (or simply 0.2).

2 a i 2.33 ii 2.35 iii 2.38

b 2.33 is approximately equal to 2.3 because it is closer to that end

of the number line shown.

It is written like this: 2.33 ≈ 2.3 or 2.33 2.3

2.35 and 2.38 are approximately equal to 2.4 because they are

closer to that end. Although 2.35 is exactly in the middle of 2.3

and 2.4, all mathematicians have agreed that this will always be

approximately equal to the decimal above (in this case 2.4)

rather than the decimal below.

3 23.8

23.7 23.8

23.76

4 44.4

The last four is less than five, so remove the last four and do nothing.

Just leave it alone.

Note: ‘round off’ and ‘approximate’ mean exactly the same thing.


5 75.40 (Do not remove the zero. This answer is correct to two

decimal places.)

To approximate 75.398 to two decimal places, you must look at the

third decimal place. As eight is bigger than or equal to five, then the

0.39 is changed to 0.40.

Another way of looking at this more difficult number is to imagine

the decimal part of the answer as 398. The closest ten to this number

is 400, not 390.

6 Four

7 a 0.5 Think of 1

2 as 1÷ 2 = 0.5 (Use a calculator to check this.)

Alternatively you can look at the number line at the beginning of the

preliminary quiz. 0.5 is halfway between 0 and 1, just like a half is

also halfway between 0 and 1.

b 0.25 Similarly 1

4 is the same as 1÷ 4 = 0.5

c 0.75 (Same as 3÷ 4 )

8 a 121. When you use a calculator: press 11 x2 =

b 144

c 225

15 ×15 =15 ×10 +15 × 5

=150 + 75

= 225

15 × 5 may also be broken down the same way, like this:


15 × 5 =10 × 5 + 5 × 5

= 50 + 25

= 75

9 a Some calculators require different key sequences.

Check your calculator manual. One method is shown below.

6 STO M+ C RCL M+

You do not need to press C . This is only included so you can

see the six come and go from your display.

b 6 STO M+ 7 M+ RCL M+

c 0 STO M+

This is how your calculator works:

M+STO C RCL M+6 7 M+ RCL M+ 0 STO M+

replaces memorywith 6

clears the display only

recalls the numberin memory and putsit into the display

adds 7to memory

recalls 6 + 7

replaces memorywith 0

Your calculator may use different keys. If you cannot do this,

check how to use your calculator with your teacher.

When you reuse an answer it is best to put this into your

calculator memory so you don’t lose it.

If you choose to put answers into your calculator memory, you

need to practise these skills.


10 Rectangle: area = length × breadth

A = lb

l

b

Triangle: area =

1

2× base× height

A =1

2bh

b

h

Parallelogram: area = length × height

A = lhh

l

Rhombus: area =1

2× diagonal× other diagonal

A =1

2xy

x

y

Kite: area =1

2× diagonal× diagonal

A =1

2xy

x

y

Trapezium: area =1

2× height × sum of parallel sides

A =1

2h(a + b)

a

b

h


11 a 49 cm2

l = 7 Area = l2

= 72

= 7 × 7

= 49 cm2

b 36 mm2

l = 9, b = 4 Area = lb

= 9 × 4

= 36 mm2

c 60 mm2

b = 8, h = 15 Area =1

2bh

=1

2× 8 ×15

= 4 ×15 (by halving 8)

= 60 mm2

12 a 54 m2

l = 6, h = 9 Area = lh

= 6 × 9

= 54 m2

b 42 mm2

x = 12, y = 7 Area =1

2xy

=1

2×12 × 7

= 6 × 7 (by halving 12)

= 42 mm2


c 84 mm2

h = 6, a = 11, b = 17 Area =1

2h(a + b)

=1

2× 6 × (11+17)

=1

2× 6 × 28

=14 × 6 (by halving 28, the bigger number)

= 84 mm2

13 18 cm. Two sides of this figure are equal. Starting at the top left

corner and going clockwise around the shape,

Perimeter = 3+ 3+ 7 + 5

= 18 cm

14 Area = 56 m2

Perimeter = 36 m

6 m

2 m

10 m

8 m 8 –

2 =

6 m

10 – 6 = 4 m

A

B

You can calculate the area of this shape many ways. Two methods

are explained below.

The first way is to divide the figure into two different areas, A and

B. Calculate the area of each, making sure you work out the

dimensions of each rectangle first.

Area A has dimensions 6 m by 6 m, and area B has dimensions 10 m

by 2 m. (It is not necessary to know the length 10 – 6 = 4 m.)

Total area = area A+ area B

= (6 × 6)+ (10 × 2)

= 36 + 20

= 56 m


Alternatively you can calculate the area of the big rectangle (area A)

below and take the area of the little rectangle from it.

area A = 80 m2

10 m

8 m

area B = 24 m2

4 m6

m

Area = 80 − 24

= 56 m2

Perimeter = 6 + 6 + 4 + 2 +10 + 8

= 36 m

Make sure you check that you have added six numbers together.

Alternatively you can just double 8 + 10. (This is explored further in

part 1.)

Activity – Circumference and perimeter

1 a about 22 mm

C = πd

= π × 7

You might press these keys: 7 =

Your answer is approximately 21.99 mm.

b 21mm. ( 3× 7 )

If your original answer is not close to 21, you need to recalculate

it. Perhaps you accidentally pressed the wrong keys!

2 a i 10.995. ( 21.99 ÷ 2 ) Your answer could also be 11 mm.

( 22 ÷ 2 ) but because you are reusing answers the more

accurate answer is preferable at this stage.

Note: if you prefer you could use the fraction key on your

calculator like this:

1 ab⁄c 2 21.99 =


ii A semicircle is half a circle, so the length of the

semicircular arc = 1

2 the whole circumference.

Finding half of something is the same as dividing by two.

b i Perimeter is the total length around the outside of a plane

shape, so you must add the length of the diameter to the

length of the semicircular arc.

ii The first answer below is better because it is always best to

approximate at the end of a calculation than in the middle.

In this question it did not matter but sometimes it does and

then you will get the wrong answer.

Perimeter = semicircular arc+ diameter

=10.995 + 7

=17.995

≈ 18 mm

or

perimeter = semicircular arc+ diameter

=11+ 7

=18 mm

Activity – Areas of circular shapes

1 a 1256.6 mm2 (or perhaps 1257 mm2 )

r = 20 mm, A = ?

Area = πr2

= π × 202

= π × 20 × 20

≈ 1256.6 mm2

You may either simply multiply your pi key, , by 20 then by

20 again or use the square key x2 by pressing 20 x2 =

(Note: it would be a good idea to put this answer into your

calculator memory – see preliminary test answers.)


b You must open the compass to a radius of 20 mm by pushing the

pencil and the point of the compass against a ruler to help you

measure this radius.

0 10 20 30 40

29 28 27 2630

Put the compass point anywhere in the middle of the line and

draw a big arc that meets the line twice. (See diagram below.)

20 mm

c Area of semicircle = 628.3 mm2 . If you had put the area for the

whole circle, above, into your calculator memory, all you would

need to do is to divide it by two. Otherwise, you may do the

calculation below:


2πr2

=1

2× π × 202

There are many ways you can use your calculator to get the

above answer. Here is one. 1 ab⁄c 2 20 x2 =



1 a 8.4 mm (Double 4.2)

b Circumference = π × diameter

= π × 8.4

Length of arc = π × 8.4 ÷ 4

c about 6.6 mm. The number in your display should be

6.597344573…. To approximate to one decimal place, you

need to look at the nine next to the five. This nine makes the

number closer to 6.6, not 6.5.

d Perimeter = arc length + radius + radius

= 6.6 + 4.2 + 4.2

e 15 mm

2 9.4 m

First double the radius, so

diameter = 2 × 6

=12 mm

Length of arc = π × diameter ÷ 4

= π ×12 ÷ 4

= 9.42477796....

≈ 9.4 m

3 a i1

4× π × 82 × 82 =

ii 0.25 × π × 822 =

iii π × 822 ÷ 4 =

b All answers are the same.Area of quadrant ≈ 5281 square millimetres (mm2 )

4 8.04 cm2 (or 8.0 cm2 )

Area of quadrant = π × radius2 ÷ 4

= π × 3.22 ÷ 4

3.2 x2 ÷ 4 =


You could also use:

area of quadrant =1

4× π × 3.22

1 ab⁄c 4 3.2 x2 =

or

area of quadrant = 0.25 × π × 3.22

0.25 3.2 x2 =

Activity – Curved composite figures

1 a � a semicircle with radius 4 cm.

b i 25 cm2 (You can choose to write this answer to one

decimal place if you wish.)


2πr2

=1

2× π × 42

=1

2× π × 4 × 4

= (1

2× 4 × 4)× π

= 8π

≈ 25 cm2

ii a = 2, b = 8, h = 4

iii 20 cm2


2h(a + b)

=1

2× 4 × (2 + 8)

=1

2× 4 ×10

= 2 ×10 (by halving 4)

= 20 cm2

Alternatively, you can do this in your head.

First add the two parallel sides together (8 + 2 = 10)


Then halve either 10 or the height, 4. (Halving 10 is better

because it is bigger – It is always easier to work with

smaller numbers in your head.) So 5 × 4 is 20.

iv about 45 cm2

Total area = semicircle+ trapezium

≈ 25 + 20

≈ 45 cm2

c i 12.6 cm

Semicircular arc length =1

2πd (Remember, d=diameter)

=1

2× π × 8

= 4π (by halving 8)

≈ 12.6 cm

(Remember, you could also divide the whole circle

circumference by two.)

ii 24.6 cm

Perimeter = semicircular arc+ three sides of trapezium

≈ 12.6 + 5 + 2 + 5

≈ 24.6 cm

iii 4 cm (height of trapezium) and/or

8 cm (diameter of semicircle).

You need the diameter in order to calculate the length of the

arc, so if you didn’t include 8 cm in your answer, it is still

correct.


Exercises – Part 2

Exercises 2.1 to 2.5 Name ___________________________

Teacher ___________________________

Exercise 2.1 – Circumference and perimeter

1 A circle has a radius of 22 mm.

a How long is its diameter? _______________________________

b Triple this diameter to get a rough estimate of the length of the

circumference.________________________________________

c Calculate a more accurate value for the circumference of this

circle by using the pi key, , on your calculator.

Write your answer correct to one decimal place.

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________


2 Use this semicircle to complete the task below.

9 cm

a Calculate the length of the semicircular arc.

___________________________________________________

___________________________________________________

___________________________________________________

b Calculate the perimeter of the semicircle.

___________________________________________________

___________________________________________________

3 A circular running track is 119 m wide at its inner most point.

The inner circumference is the smallest distance that you can run if

you complete one lap of this track.

119 m

a The circumference of the inner circle is about 374 m.

Write down the correct working for this answer.

___________________________________________________

___________________________________________________


b A 1500 m walk is to be organised. Use your answer above to

calculate how many laps of this track competitors must walk.

___________________________________________________

___________________________________________________

c i If the track is 7 m wide, what is the diameter of the outside

edge of the track? (Remember, the track is on both sides if

the inner diameter.) ________________________________

ii Calculate the circumference of the outside edge of the track?

________________________________________________

________________________________________________

________________________________________________

d Why do you think competitors in a long race like to crowd

along the inner edge of running track?

___________________________________________________

___________________________________________________

4 The circumference of a circle is 47 metres long.

Approximately how long is its diameter?

_______________________________________________________

_______________________________________________________


Exercise 2.2 – Areas of circular shapes

1 a Calculate the area of a circle that has a radius of 12 m.

___________________________________________________

___________________________________________________

___________________________________________________

b Another circle has a diameter of 12 m.

i Tick (�) the answer you think is correct.

The area of this circle is:

the same as the area of the circle above.

half as big as the area of the circle above.

a quarter of the area of the circle above.

ii Check your answer above by calculating the area of this

circle.

________________________________________________

________________________________________________

________________________________________________

________________________________________________

Did you tick the correct statement?____________________

You may change your answer above, if you wish.


2 The fan below is semicircular. It has a diameter of 0.3 m and it is

made out of a beautiful ivory-coloured lace.

a What is the radius of the fan? ____________________________

b Ignoring the extra material needed to anchor the fan to its arms,

how much material in square metres is needed to make:

i one fan. (Write your answer correct to three

decimal places.)

________________________________________________

________________________________________________

________________________________________________

ii 500 fans. (Write your answer to the nearest square metre.)

________________________________________________

________________________________________________

________________________________________________

c If this special lace costs $150 per square metre, how much

would it cost to buy the material for all 500 fans?

___________________________________________________

___________________________________________________

(Note: this calculation represents the minimum about of material

needed. You would always need to buy more to allow for wastage.)


3 Young children who do not like getting water in their eyes can use a

special plastic bath hat that protects their face. This hat has a hole in

the middle so that the child’s hair can be washed.

15 cm 15 cm

The diameter of the hole is 15 cm and it protects the eyes with a

15 cm brim.

a i What is the diameter of the hat? (Remember, the hat has

the same brim all the way around it.

________________________________________________

ii What is the radius of the hat? ________________________

iii This hat is first made as a complete circle (without the

hole.) What is the total area of plastic needed to make this

complete circle?

________________________________________________

________________________________________________

________________________________________________

b i What is the radius of the hole where the child’s hair gets

washed?_________________________________________

ii What is the area of this hole?

________________________________________________

________________________________________________

________________________________________________


c Describe how you would calculate the area of plastic that is in

the finished bath hat?

___________________________________________________

___________________________________________________


Exercise 2.3 – Quadrants

1 a Use a pair of compasses to draw a quadrant that has a radius of

4 cm. Draw this figure as accurately as you can below.

b Estimate the length of the arc of this quadrant. ______________

c How long is the diameter of the circle that this quadrant

came from?__________________________________________

d i Write the correct numbers and symbols below to show the

method you are going to use.

Circumference of circle = π × diameter

Length of arc = _____________________

ii Follow your method above by pressing the appropriate keys

on your calculator. Write your answer correct to one

decimal place here. ________________________________

e Calculate the perimeter of this quadrant by completing the

following:

perimeter = length of arc + radius + radius

___________________________________________________

___________________________________________________


2 The dimensions of a colourful awning for an entrance to a building

are shown below.

6 m

awning

Complete the calculation for the amount of material needed to make

this awning.

Area of whole circle = π × radius × radius

Area of quadrant = ________________________________________

_______________________________________________________

_______________________________________________________


8 mmx°

a Draw a dotted line on the diagram below to complete the

missing part of the whole circle.

b What is the diameter of this circle? _______________________

c i Calculate the circumference of the whole circle, using the

formula: circumference of whole circle = π × diameter

________________________________________________

________________________________________________

________________________________________________


d i What fraction of the whole circle is missing in the diagram?

________________________________________________

ii Express this fraction as a decimal.

________________________________________________

iii Calculate the length of the ‘dotted’ arc in the diagram

above, using either the decimal or the fraction above.

________________________________________________

________________________________________________

e Calculate the length of the big arc by completing the method

below. (Choose the correct numbers from above.)

Length of big arc = circumference of circle − length of dotted arc

___________________________________________________

___________________________________________________

f Calculate the perimeter of the figure.

___________________________________________________

___________________________________________________


Exercise 2.4 – Curved composite figures

1 a Divide this figure into two shapes by drawing one line.

The two shapes must be from this list: triangle, rectangle,

square, rhombus, kite, trapezium, circle, semicircle, quadrant.

AB

C D

b Name the two shapes __________________________________

c Name the dimensions you would need to be given to be able to

find the area of these two shapes. _________________________

___________________________________________________

Draw them on the diagram above.


2 A circular stained glass window for a church is shown in the

diagram below.

To highlight the cross, the kite surrounding it is made of yellow

stained glass and the rest of the window is blue stained glass.

a If the cross is 1.6 m high and 1 m wide, what is the area of

yellow glass in the kite? (Use the formula, A=1

2xy , where x

and y are the diagonals of the kite.)

___________________________________________________

___________________________________________________

___________________________________________________

b The vertical part of the cross, which is 1.6 m high, divides the

window into two equal sections. What length is the:

i diameter of the circle?______________________________

ii radius of the circle?________________________________

c What is the area of the whole circular window?

Show your working below.

___________________________________________________

___________________________________________________

___________________________________________________

d Calculate the area of blue glass in the window by subtracting the

kite area from the circular area.

___________________________________________________

___________________________________________________

___________________________________________________


3 A tablecloth is to be made to fit a dining table for a special event.

It will be an enlarged shape of the table because the owner wants it

to hang equally over the edge of the table.

0.6 m1.2 m

0.9 m

a The tablecloth has three shapes inside it. What are the names of

the three shapes? (Two of the shapes have the same name.)

___________________________________________________

b i Two of these shapes can be combined to form one shape.

What is the name of this shape? ______________________

ii Draw a diagram of this shape showing its dimensions.

iii Calculate the area of this shape.

________________________________________________

________________________________________________

________________________________________________

c Calculate the total area of material needed to make this

tablecloth.

___________________________________________________

___________________________________________________

___________________________________________________


d A lace border is to be sewn around the edge of the table cloth.

Describe in words, the lengths you need to add together, in order

to calculate the amount of lace that will be used.

___________________________________________________

___________________________________________________

e Calculate the amount of lace needed for the tablecloth edging.

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________

43675 MS5.1.1 Perimeter and area 61

We need your input! Can you please complete this short evaluation to

provide us with information about this module. This information will

help us to improve the design of these materials for future publications.

1 Did you find the information in the module clear and easy to

understand?

_______________________________________________________

_______________________________________________________

_______________________________________________________

2 What sort of learning activity did you enjoy the most? Why?

_______________________________________________________

_______________________________________________________

3 Name any sections you feel need better explanation (if any).

_______________________________________________________

_______________________________________________________

4 Were you able to complete each part in around 4 hours? If not

which parts took you a longer or shorter time?

_______________________________________________________

_______________________________________________________

5 Do you have access to the appropriate resources? This could include

a computer, graphics calculator, the Internet, equipment and people

to provide information and assist with the learning.

_______________________________________________________

Centre for Learning InnovationNSW Department of Education and Training

Documents

43675 MS5.1.1 cover - Login Department of Educationlrr.cli.det.nsw.edu.au/legacy/Mathematics/43675_03_05.pdfUnit overview iii Unit overview In this unit you will explore perimeter