2535 18923329.nzl h area and perimeter nzl

8/22/2019 2535 18923329.NZL H Area and Perimeter NZL

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PASSPO

RT

PASSPO

RT

AreaandPerimeter

AREA ANDAREA AND

PERIMETERPERIMETER

www.mathlecs.co.nz


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Area and Perimeter

Mathletics Passport 3P Learning

19HSERIES TOPIC

Use all four squares below to make two shapes in which the number of sides is also equal to four.

Compare the distance around the outside of your two shapes.

Write down what you discovered and whether or not it was dierent from what you expected.

This booklet shows how to calculate the area and perimeter of common plane shapes.

Football elds use rectangles, circles, quadrants and minor segments with specic areas and perimeters

to mark out the playing eld.

Write down the name of another sport that uses a playing eld or court and list all the plane shapes

used to create them below (include a small sketch to help you out):

Q

Sport:

Shapes list:


4/40

Area and Perimeter


2 9HSERIES TOPIC

Howdoes it work?

Area using unit squares

Calculate the area of these shapes

Area and Perimeter

Area is just the amount of at space a shape has inside its edges or boundaries.

A unit square is a square with each side exactly one unit of measurement long.

Area (A) =1 square unit

=1 unit2 (in shorter, units form)

Area (A) =10 square units

=10 unit2

So the area of the shaded shape below is found by simply counng the number of unit squares that make it.

Here are some examples including halves and quarters of unit squares:

(i)

(ii)

Area ( ) whole square units half square units

square units 2 square units

square units

units

A 2 2

22

1

2 1

3 2

#

= +

= +

= +

=

^ h

When single units of measurement are given, they are used instead of the word units.

1 2 3

5

7 9

4

6

8 10

Lile dashes on each

side mean they are all

the same length.

1unit

1 cm

1 unit

1unit

Area ( ) whole squares 2 half squares quarter squares

square cm square cm square cm

square centimetres

cm

A

.

.

2 2

2 22

12

4

1

2 1 0 5

3 52

# #

= + +

= + +

= + +

=

^ h


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Area and Perimeter


39HSERIES TOPIC

How does it work? Area and PerimeterYour Turn


Calculate the area of all these shaded shapes:1

a

c

e

g

f

b

d

Area = whole squares

= units2

Area = whole + half squares

= m2+2

1# m2

= m2

Area = whole + quarter squares

= units2

+ 41

# units2

= units2

Area = whole + half+ quarter squares

= units2+2

1# units2 +

4

1# units2

= units2

Area = whole + quarter squares

= cm2

+ 41

# cm2

= cm2

Area = whole + half squares

= units2+2

1# units2

= units2

Area = whole squares

= mm2

1unit

1unit

1m

1unit

1 unit 1mm

1cm


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Area and Perimeter


4 9HSERIES TOPIC


Calculate the area of these shaded shapes, using the correct short version for the units:


2

Area = Area =

Area = Area =

Area = Area =

Area = Area =

a

c

e

g

d

f

h

b1cm

1 m

1 unit

1km

1cm

1mm

1mm

1unit

...../...../20....

AREAUSINGUNITS

QUARES*AR

EAU

SING

UNIT

SQUA

RES*


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Area and Perimeter


59HSERIES TOPIC


Shade shapes on these square grids to match the area wrien in square brackets.3

4

Number of dierent designs you found =

a

c

b

d


units8 26 @, using whole squares only.

mm326 @, include quarter squares in

your shape.

units5 26 @, include half squares in your shape.

cm4.526 @, include halves and quarters.

An arst has eight, 1m2, square-shaped panels which he can use to make a paern.

The rules for the design are:

- the shape formed cannot have any gaps/holes.

i.e. or

- it must t enrely inside the display panel shown,

- all the eight panels must be used in each design.

How many dierent designs can you come up with?

Sketch the main shapes to help you remember your count.

1unit

1mm 1cm

1 m

1 unit


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Area and Perimeter


6 9HSERIES TOPIC

How does it work? Area and Perimeter

Perimeter using unit squares

The word perimeter is a combinaon of two Greek words peri (around) and meter (measure).

So nding the perimeter (P) means measuring the distance around the outside!

Start/end of path around the outside

1unit

Perimeter unit unit unit unit

unit

units

P( )

4

1 1 1 1

4 1#

= + + +

=

=

Perimeter units

units

P( ) 1 2 1 2

6

= + + +

=

Perimeter units

units

P( ) 1 1 1 1 3 1 1 1

10

= + + + + + + +

=

Remember, lile dashes on

each side mean they are all

the same length.

Calculate the perimeter of these shapes formed using unit shapes

(i)

(ii)

2 units

2 units

3 units

1 unit

1 unit1 unit

1 unit1 unit

1 unit1 unit

1 unit

1 unit

1 unit

It does not maer where you start/nish, but it is usually easiest to start from one corner.



These examples shows that we only count all the outside edges.

Sides of unit squares inside the shape not included


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Area and Perimeter


79HSERIES TOPIC



1 unit

1 unit

1 unit

Calculate the perimeter of these shaded shapes:1

2

3

a

b

c

a b c d

Perimeter = + + + units

= units

Perimeter = + + + units

= units

Perimeter = + + + + + units

= units

P= units P= units P= units P= units

Write the length of the perimeter (P) for each of these shaded shapes:

The shaded shapes in 2 all have the same area of6 units2.

Use your results in queson 2 to help you explain briey whether or not all shapes with the

same area have the same perimeter.

...../...../20....

PERIMETERU

SING

UNIT

S

QUARES*PERIMETERUSING

UNIT

SQUAR

ES*


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Area and Perimeter


8 9HSERIES TOPIC



4 Draw six paerns on the grid below which:

Draw another ve paerns on the grid below which:

a

b

all have an area of5units2 and,

have a dierent perimeter from each other.

all have an area of5units2 and,

have a dierent perimeter than the shapes formed in part a .

All squares used for each paern must share at least one common side or corner point .

All squares used for each paern must share at least halfof a common side

ora corner

point .

1unit

1unit

1unit

1unit


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Area and Perimeter


99HSERIES TOPIC

Area and PerimeterWhere does it work?

Area: Squares and rectangles

Calculate the area of these shaded shapes

A simple mulplicaon will let you calculate the area of squares and rectangles.

For squares and rectangles, just mulply the length of the perpendicular sides (Length and width).

Square Rectangle

Length Length

Width Width Side (y)

Side (x)

Area length width

Side units Side units

units

units

x y

x y

xy

2

2

#

#

#

=

=

=

=

^ ^h h

Area length width

units units

units

units

4 4

4

16

2 2

2

#

#

=

=

=

=

Area length width

mm mm

mm

6 1.5

92

#

#

=

=

=

Area length width

m cm

cm cm

cm

2 60

200 60

12 0002

#

#

#

=

=

=

=

(i)

(ii)

(iii)

All measurements (or dimensions) must be wrien in the same units before calculang the area.

4 units

1.5 mm

6 mm

60 cm

2 m

So why units squared for area?

units units units units

units

units

4 4 4 4

4

16

2 2

2

# # #

#

=

=

=

Units of area match units of side length

Write both lengths using the same unit

Units of area match units of side length

Here are some examples involving numerical lengths:

Side (x)

Side (x)

Area length width

Side units Side units

units

units

x x

x x

x

2

2 2

#

#

#

=

=

=

=

^ ^h h


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Area and Perimeter


10 9HSERIES TOPIC

Area and PerimeterWhere does it work? Your Turn


Calculate the area of these squares and rectangles, answering using the appropriate units.

Calculate the area of these squares and rectangles. Round your answers to nearest whole square unit.

What is the length of this rectangle?

What are the dimensions of a square with an area of121m2?

1

2

3

4

a

a

c

b

b

d

Area = # units2

= units2

Area = # km2

= km2

. km2 (to nearest whole km2)

Area = # cm2

= cm2

. cm2 (to nearest whole cm2)

Area = # units2

= units2

Area = # mm2

= mm2

Area = # m2

= m2

length

length

length

length

width

width

width

width

length lengthwidth width

Psst: remember the opposite of squaring numbers is calculang the square root.

1.4 km

2 units

2 units

3

units

3.2 mm

7 cm

43 mm

5

mm

0.6 m

* AREA: SQUARESAND

RE

CTAN

GLE

S

*AREA:SQUARESAND

RE

CTAN

GLE

S

...../...../20....

4 unitsArea =28units


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Area and Perimeter


119HSERIES TOPIC


Area: Triangles

Base (width)

Height

(Length)

Look at this triangle drawn inside a rectangle.

The triangle is exactly half the size of the rectangle

` Area of the triangle = half the area of the rectangle units2

= 21

of width (base (b) for a triangle) # Length (height (h) for a triangle) units2

= unitsb h2

1 2# #

This rule works to nd the area for all triangles!

Here are some examples involving numerical dimensions:

Calculate the area of the shaded triangles below

(i)

(ii)

(iii)

Area base height

m m

m

21

2

13 4

62

# #

# #

=

=

=

Area base height

mm mm

mm

. 4

.

2

1

21 5 4

10 82

# #

# #

=

=

=

Area base height

units units

units

.

.

2

1

2

11 5 2

1 52

# #

# #

=

=

=

Height = use the perpendicular height

The rule also works for this next triangle which is just the halves of two rectangles combined.

For unusual triangles like this shaded one, we sll mulply the base and the perpendicular height

and halve it.

Here, we say the height

is the perpendicular

distance of the third

vertex from the base.

6m

1.5units

2units

5.4mm

5m4 m

4 mm


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Area and Perimeter


12 9HSERIES TOPIC


Calculate the area of the triangle that cuts these two shapes in half.

Calculate the area of these shaded triangles:

Area: Triangles

1

2

Area =2

1 # # units2

= units2

Area = # # mm2

= mm2

Area = # # units2

= units2

Area = # # m2

= m2

Area = # # cm2

= cm2

Area = # # m2

= m2

Area =2

1 # # units2

= units2

base baseheight height

a

a

c

e

b

b

d

2 units

4 units8 units

12mm

600cm

4 m5m

14cm

8mm

7.5units 4.5m

10units

12cm

Remember:

same units

needed.

*AREA:TR

IANGLES

*A

REA:TRIAN

GLES

*AR

EA:TRIANG

LES

...../...../20....


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Area and Perimeter


139HSERIES TOPIC


Area: Parallelograms

12 m13 m

5 m

12 m13 m

5 m

12 m 13 m

5 m

Parallelograms have opposite sides equal in length and parallel (always the same distance apart).

The shortest distancebetween a pair of

parallel sides is called

the perpendicular height

We can make them look like a rectangle by cung the triangle o one end and moving it to the other.

height

Perpendicular

height (h)

` Area of a parallelogram = Area of the rectangle formed aer moving triangle

= length # perpendicular height units2

=l#h units2

Parallelogram Rectanglemove triangle cut o

Calculate the area of these parallelograms

A parallelogram can also be formed joining together two idencal triangles.

Area length height

mm mm

mm

30 15

4502

#

#

=

=

=

Area length perpendicular height

m m

m

5 12

602

#

#

=

=

=

Area area of the triangle

m m

m

2 5 12

2

2

1

602

#

# # #

=

=

=

(i)

(ii) Find the area of the parallelogram formed using two of these right angled triangles:

OR

12 m13 m13 m

5 m

5 m

12 m

5 mCopy and ip both

vercally and horizontallyBring them together Parallelogram

20 mm

30 mm

15 mm


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Area and Perimeter


14 9HSERIES TOPIC


Area: Parallelogram

1

2

3

Complete the area calculaons for these parallelograms:

Calculate the area of the parallelograms formed using these triangles.

Fill the grid below with as many diferent parallelograms as you can which have an area of4 units2.

a

a

b

b

Area = # units2

= units2

Area = m2 Area = mm2

Area = # cm2

= cm2

length lengthheight height

4.5 mm

10 units

4.6 cm

3.9 cm2.2cm

26 m1.6 mm10 m 2 mm

24 m1.2 mm

*AREA:PARAL

LELO

GR

AM*AREA:P

ARALLELOGRA

M

...../...../20....

1unit

1unit


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Area and Perimeter


159HSERIES TOPIC


Area of composite shapes

When common shapes are put together, the new shape made is called a composite shape.

Just calculate the area of each shape separately then add (or subtract) to nd the total composite area.

Calculate the area of these composite shapes

(i)

(ii)

8 cm

8 cm

8 cm2 cm

10 cm

Split into a triangle 1 and a square 2 .

Area 1 =2

1 #2 cm #8cm =8cm2

Area 1 =4.5m #3.5m =15.75m2

Area 1 =8m #7m =56m2

Area 2 =8cm #8cm =64cm2

Area 2 =3.5m #7cm =24.5m2

Area 2 =3.5m #4.5m =15.75m2

` Total area = Area 1 + Area 2

=8cm2 +64cm2

=72cm2

` Total area =15.75m2 +24.5m2

=40.25m2

` Total area =56m2 -15.75m2

=40.25m2

This next one shows how you can use addion or subtracon to calculate the area of composite shapes.

method 1: Split into two rectangles 1 and 2

method 2: Large rectangle 1 minus the small 'cut out' rectangle 2

Add area 1 and 2 for the composite area

Add area 1 and area 2 together

Subtract area 2 from area 1

1

2

8 m

7 m

3.5 m

4.5m

4.5m

3.5m

8m

1

1

2

2

7 m

7 m

3.5 m

3.5 m

Common shape

(Rectangle)

Common shape

(Isosceles triangle)

Composite shape

(Rectangle + Isosceles triangle)

+ =Composite just means

it is made by pung

together separate parts


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Area and Perimeter


16 9HSERIES TOPIC



1 Complete the area calculaons for these shaded shapes:

a

b

c

d

1

2

6mm

4m

m

2m

m

5m

5m

4m

m

6m

6m

3m

3m

11m

11m

Area 1 = mm # mm

= mm2

Area 1 = # # m2

= m2

Area 1 = # cm2

= cm2

Area 1 = # # m2

= m2

` Composite area = + mm2

= mm2

` Composite area = + m2

= m2

` Composite area = - cm2

= cm2

` Composite area = - m2

= m2

Area 2 = mm # mm

= mm2

Area 2 = # m2

= m2

Area 2 = # # cm2

= cm2

Area 2 = # m2

= m2

1

2

2c

m

2.5cm

6.5cm

2c

m

2.5cm

6.5cm

3 m

3 m

5m

4cm

5m

2m

2m

1

2


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Area and Perimeter


179HSERIES TOPIC



2 Calculate the area of these composite shapes, showing all working:

a

b

c

d

12 cm

13 cm

5 cm

Area = cm2

Area = m2

Area = mm2

Area = units2

200cm

4.5m

300cm

2 mm

5 units10u

nits

6units

psst: this one needs three area calculaons

AREA OF COMPOSITESHAPE

S

*

AREAOFCOMPOSITESHAPE

S

*

...../...../20....

psst: change all the units to metres rst.


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Area and Perimeter


18 9HSERIES TOPIC


Perimeter of simple shapes

By adding together the lengths of each side, the perimeter of all common shapes can be found.

side 2

(y units)width

(y units)

side (x units) length (x units)

side 3

(zunits)

side 1 (x units)

RectangleSquareTriangle

side length

units

units

P 4

x

x

4

4

#

#

=

=

=

width length width length

units

units

units

P

y x y x

x y

x y

2 2

2 2

# #

= + + +

= + + +

= +

= +

^

^ ^

h

h h

side side side

units

P

x y z

1 2 3= + +

= + +

Here are some examples involving numerical dimensions:

Calculate the perimeter of these common shapes

(i)

(ii)

(iii)

11 units 8 units8 units 11 units

10 units10 units

2.3 cm

You can start/end at any

vertex of the shape

Perimeter units units units

units

11 8 10

29

= + +

=

Perimeter cm

cm

.

.

4 2 3

9 2

#=

=

2.3 cm

2.3 cm

2.3 cm2.3 cm

All measurements must be in the same units before calculang perimeter.

The perimeter for parallelograms is done the same as for rectangles. Calculate this perimeter in mm.

Perimeter mm mm

mm mm

mm

2 15 2 5

30 1040

# #= +

= +

=

15mm15mm

5mm 5mm0.5 cm

15mm

Sum of all the side lengths

Four lots of the same side length

All side lengths in mm

Opposite sides in pairs

Start/nish

Start/nish

Start/nish

Start/nish

Start/nish

Start/nish


21/40

Area and Perimeter


199HSERIES TOPIC



15 units8 units

17 units

1

2

a

a

b

b

c

c

d

d

Complete the perimeter calculaons for these shapes:

Calculate the perimeter of the shapes below, using the space to show all working:

Perimeter = units + units + units

= units

Perimeter = 2# mm +2# mm

= mm

Perimeter = 2# cm + cm

= cm

Perimeter = cm

Perimeter = mm

Perimeter = m

Perimeter = m

Perimeter = # m

= m

9mm

6mm

11cm

5 m

2.4mm

5.8cm

15m

1.6mm 5m3m

5 m

1.6m 2.4m

3.4 m

PERIMETEROF SIMPLE SHAPES *

.....

/.....

/20....

PERIMETEROFSIMPLESHAPES*


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Area and Perimeter


219HSERIES TOPIC


6 m

5 cm

3 m

3 m 3 m

9 cm +5 cm =14 cm

Perimeter of composite shapes

2 m 2 m

3 m 3 m

130 mm

13 cm

9 cm

9 cm12 cm

12 cm

7 m 7 m

? m (3 + 3.5) m =6.5 m

(7 - 2) m =5 m? m

3.5m 3.5m

Perimeter m m m m m m

m

7 6.5 2 3.5 5 3

27

` = + + + + +

=

Here are some more examples.

Calculate the perimeter of these composite shapes

(i)

(ii)

Calculate each side length of the shape in the same units

Perimeter cm cm cm cm

cm

9 13 14 12

48

` = + + +

=

Perimeter m m m

m

6 1.5 3 6

18

` #= + +

=

Perimeter m m

m

2 6 2 3

18

# #` = +

=

m m m2 1.56 3 '- =^ h

6 m 6 m

1.5 m +1.5 m =3 m

You can also imagine the

sides re-posioned to makethe calculaon easier

Start/nish

The lengths of the unlabelled sides must be found in composite shapes before calculang their perimeter.


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Area and Perimeter


22 9HSERIES TOPIC



1

2

a

c

a

b

c

d

b

d

Calculate the perimeter of these composite shapes:

a= cm

a= m

a= mm

a= cm

b= cm

b= m

b= mm

b= cm

13m

8cm

2cm

2.2m

m

1mm

b

b

ba

a

a

1.6mm

3.4 mm

18m

15m

b

a

5cm

14cm

15cm

8cm

4.8cm

9.8mm

4.1cm

38mm

2m

1.2cm

16mm

Perimeter = # cm

= cm

Perimeter =2#am +2# m

= m + m

= m

Perimeter =3# mm + mm + mm

= mm

Perimeter = #4.1cm + # cm

= cm

4 m

20mm

am

Be careful with the units for these next two

PERIMETEROFCOMP

OSITE

SHAP

ES

*PERIME

TERO

FC

OMPOSITE

SHAPES*

312

...../...../20....

Calculate the value of the sides labelled a andb in each of these composite shapes:


25/40

Area and Perimeter


239HSERIES TOPIC



3

4

Calculate the perimeter of these composite shapes in the units given in square brackets.

Show all working.

4 mm

48 mm

3.6c

m1200m

1.5km

a

c

b

d

mm6 @

cm6 @

m6 @

km6 @

2.2m

Perimeter = mm

Perimeter = cm

Perimeter = m

Perimeter = km

Total perimeter of completed path = m

22m

m

Earn an awesome passport stamp for this one!

The incomplete geometric path shown below is being constructed using a combinaon of

the following shaped pavers:

The gap in between each part of the spiral path is always 1m wide.

Calculate what the total perimeter of this path will be when nished.

1m

1.41m1m

2m

2m

2m

1.41m

1m

1m 1mCompleted path

AWESOMEAWESOMEAWESOME

AWES

OMEA

WESO

MEAWE

SOME

psst: 1km =1000 m

...../...../20....


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Area and Perimeter


24 9HSERIES TOPIC



5 The four composite shapes below have been formed using ve, unit squares.

a

b

Using your knowledge of perimeter and the grid below, combine all four pieces to create two

dierent shapes so that:

All shapes must be connected by at least one whole side of a unit square.

One shape has the smallest possible perimeter.

The other has the largest possible perimeter.

A shape with the smallest possible perimeter.

A shape with the largest possible perimeter.

1unit

1unit

Briey describe the strategy you used to achieve each outcome below:


27/40

Area and Perimeter


259HSERIES TOPIC


Simple word problems involving area and perimeter

Somemes we can only communicate ideas or problems through words.

So it is important to be able to take wrien/spoken informaon and turn it into something useful.

For example,

Miguel wants to paint a square. He has just enough paint to create a line 240cm long.

What is the longest length each side of the square can be if he wishes to use all of the paint?

To use up all the paint, the total perimeter of the square must equal 240cm.

So each side length cm

cm

240 4

60

'=

=

` The longest length each side of the square painted by Miguel can be is 60cm.

This is useful for Miguel to know because if he painted the rst side too long, he would run out of paint!

Here are some more examples

(i)

(ii)

A rectangular park is four mes longer than it is wide. If the park is 90m long, how much area

does this park cover?

At a fun run, competors run straight for 0.9km before turning le 90 degrees to run straight for a

further 1.2 km. The course has one nal corner which leads back to the start along a straight 1.5km

long street. How many laps of this course do competors complete if they run a total of18km?

90m

m m22.590 4' =^ h

Area length width

m m

m

90 22.5

20252

#

#=

=

=

Draw diagram to illustrate problem

Draw diagram to illustrate problem

Perimeter will be the length of each lap

Race distance divided by the length of each lap

1.5km 0.9km

1.2km

Start/nish

Perimeter of course km km km

km

0.9 1.2 1.5

3.6

= + +

=

Number of laps km km.18 3 6

5

` '=

=

` Length of each lap of the course is 3.6km

` Competors must complete 5 laps of the course to nish


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Area and Perimeter


26 9HSERIES TOPIC



1

2

3

Three equilateral triangles, each with sides of length 3cm have been placed together to make one

closed four-sided shape. Each triangle shares at least one whole side with another.

Calculate the perimeter of the shape formed.

The base length of a right-angled triangle is one h of its height. If the base of this triangle is

4.2m, calculate the area of the triangle.

a

b

Use all four squares below to make two shapes in which the number of

sides is also equal to four. Compare the distance around the outside of your

two shapes and explain what this shows us about the relaonship between

area and perimeter.

You have been employed by a fabric design company called Double Geometrics. Your rst task

as a paern maker is to design the following using all seven idencal squares:

Closed shapes for a new paern in which the value of their perimeter is twice the value of theirarea. Draw ve possible dierent paerns that match this design request.

SIMPLE WORD

PROBLEMS

INVOLVING

AREA AND

PERIMETER

...../...../20....


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Area and Perimeter


279HSERIES TOPIC


a

b

c

d


4 An architect is asked to design an art gallery building. One of the design rules is that the oor must

be a rectangle shape with an area of64m2.

Another design rule is to try ensure a large perimeter so there is more space to hang painngs

from. Use calculaons to show which oor plan will have the largest perimeter.

Would the design with the largest possible perimeter be a good choice?

Explain briey why/why not.

A small art piece at the gallery has one side of an envelope completely covered in stamps

like the one pictured below. How many of these stamps were needed to cover one side of an

envelope 12.5cm wide and 24.5cm long if they all t perfectly without any edges overlapping?

3.5cm

2.5cm

12.5cm

24.5cm

If only whole metre measurements can be used, sketch all the dierent possible oor dimensions.


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Area and Perimeter


28 9HSERIES TOPIC



5

6

A fence used to close o a parallelogram-shaped area is being rearranged to create a square area

with the same perimeter. The short side of the area is 34m long (half the length of the long side).

A wall is created by stacking equal-sized rectangular bricks on top of each other as shown.

The end of each rectangle sits exactly half-way along the long side of the rectangle underneath it.

a

a

b

b

How long will each side of the new square area be aer using the whole length of this fence?

A 500mL n of white paint has been purchased to paint the wall. The instrucons on the

paint n say this is enough to cover an area of11 500cm2.

If the distance between the longer sides of the original area was 30m and the length did not

change, use calculaons to show which fencing arrangement surrounded the largest area.

Use calculaons to show that there is enough paint in the n to cover side of the wall.

28cm

16cmEach brick =

34 m

If a beetle walked all around the outside of the wall (including along the ground), how many

metres did it walk?


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Area and Perimeter


299HSERIES TOPIC

Area and PerimeterWhat else can you do?

Rhombus and Kite shapes

The area for both of these shapes can be calculated the same way using the length of their diagonals.

Calculate the area and perimeter of these shapes

A rhombus is like a square parallelogram.

A kite has two pairs of equal sides which are adjacent (next to) each other.

Area diagonal lengths multiplied together 2

AC BD 2#

'

'

=

=

^

^

h

h

Area diagonal lengths multiplied together

cm cm

cm

12 2

96

2

16

2

#

'

'

=

=

=

^

^

h

h

Area diagonal lengths multiplied together

m m

m

. .

.

2

2 1 4 7 2

4 9352

#

'

'

=

=

=

^

^

h

h

Area diagonal lengths multiplied together 2

AC BD 2#

'

'

=

=

^

^

h

h

Perimeter length of one side4

4 AB#

#=

=

Perimeter length of sides

cm

cm

4

4 10

40

#

#

=

=

=

Perimeter short side long side2 2

AB AD2 2

# #

# #

= +

= +

Perimeter short side long side

m m

m

2 2

2 . 2 3.7

10.4

1 5

# #

# #

= +

= +

=

A Rhombus is a

parallelogram, so

we can also use

the same rule to

nd the area:

Here are some examples:

(i)

(ii)

For this rhombus, WY=12cm andXZ=16cm.

For the kite ABCD shown below,AC=4.7m andBD=2.1m.

length

height

B

X

C

Y

A

W

D

Z

B

B

C

C

A

A

D

D

10cm

3.7m

1.5m


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Area and Perimeter


30 9HSERIES TOPIC

Area and PerimeterYour TurnWhat else can you do?

1

2

3

a

a b

b

Calculate the area and perimeter of these shapes:

PR=18cm and QS=52cm BD=1.8mm andAC=2.4mm

Rhombus and Kite shapes

Q

S

P R

Area = # ' cm2

= cm2

Area = # #

2

1

mm2

= mm2

Perimeter = 2# +2# cm

= cm

Perimeter = m

Area = m2

Perimeter = cm

Perimeter = # mm

= mm

B

C

A

D

Calculate the perimeter of these composite shapes:

Calculate the area of this composite shape, showing all working when:

3.4cm

5.1cm

3.6mm

41 cm

15cm

6.5cm

HL=30m,IK=IM=16m and JL=21m

14m

9m

J

K

LM

HI

22

1#' =

RHOMBUSANDKITE

SHAP

ES

RHOMBUSAN

D

KITE

SHAPES

...../...../2

0....


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Area and Perimeter


319HSERIES TOPIC

Area and PerimeterWhat else can you do?

10.1m

22mm10mm

20mmArea sum of theparallel sides height

mm mm mm

mm mm

mm

16 2

32 16 2

2

22 10

2562

#

#

#

'

'

'

=

= +

=

=

^

^

h

h

Area sum of theparallel sides height

m m m

m m

m

. .

.

.

2

6 7 14 5 2 2

21 2 2 2

21 22

#

#

#

'

'

'

=

= +

=

=

^

^

h

h

Perimeter mm mm mm mm

mm

20 22 16 10

68

= + + +

=

Perimeter m m m m

m

. . . .

.

6 7 10 1 14 5 2 9

34 2

= + + +

=

Calculate the area and perimeter of these shapes

Here are some examples:

(i)

(ii)

16mm

6.7m

14.5m

2.9m2m

Trapeziums

B Baa

bbD D

A A

C C

A trapezium is a quadrilateral which has at least one pair of parallel sides.

Squares, rectangles, parallelograms and rhombi are all just special types of trapeziums.

So the area formula for a trapezium would also work on all of those shapes.

Two common trapezium shapes

In both shapes, the sides AB (a) and CD (b) are parallel ||( )AB CD .

The height is the perpendicular distance between the parallel sides.

Area sum of the parallelsides height 2

2a b h#

: '

'

#=

= +

^

^

h

h

Perimeter AB BD CD AC : = + + +

height (h) height (h)


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Area and Perimeter


32 9HSERIES TOPIC


a

a b

b

1

2

3

41km8.2m

15km1.8m

1km

12.5m

9km

53km 4.5m

Area = + # '2km2

= km2

Area = cm2

Area = m2

Area = mm2

Perimeter = m

Area = + # '2m2

= m2

Perimeter = km Perimeter = m

5cm

15cm

16.7 m

240cm

33.4 m

8cm

1mm 2.4mm 14.3 mm

170 cm

1 2

Trapeziums

Calculate the area and perimeter of these trapeziums:

Use the trapezium method to calculate the area of these composite plane shapes.

Use the trapezium method to calculate the area of this composite plane shapes.

TRAPEZIUMS*

TRAPE

ZIUMS*TRA

PEZIUMS

*TRAPEZ

IUMS*

...../...../20....


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Area and Perimeter


339HSERIES TOPIC


Area challenge

1unit

1unit

Fill the grid below with as many dierent squares, triangles, rectangles, parallelograms, rhombi, kites

and trapeziums as you can which all have the same area of8 units2.


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Area and Perimeter


34 9HSERIES TOPIC


Reflection Time

Reecng on the work covered within this booklet:

What useful skills have you gained by learning how to calculate the area and perimeter of plane shapes?

Write about one or two ways you think you could apply area and perimeter calculaons to a real

life situaon.

If you discovered or learnt about any shortcuts to help with calculang area and perimeter or some

other cool facts/conversions, jot them down here:


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Area and Perimeter


359HSERIES TOPIC

Cheat Sheet Area and Perimeter

Here is what you need to remember from this topic on Area and perimeter




Area: Triangles

Area: Parallelograms

Area is just the amount of at space a shape has inside its edges or boundaries.

A unit square is a square with each side exactly one unit of measurement long.

Count the total number of whole squares, or fracons of squares to calculate the area.

The perimeter with unit squares means count the number of edges around the outside of the shape.

Just mulply the length of the perpendicular sides (length and width).

Area (A) = 2 units2

Perimeter (P) = 6 units

Square Rectangle

length length

width width side (y)

side (x)

Area length width

unitsxy2

#=

=

height height

base base` Area of the triangle = (half the base mulplied by the perpendicular height) units2

= unitsb h2

1 2# #

Perpendicular

height (h)

` Area of a parallelogram =length # perpendicular height units2=l#h units2

height

base

side (x)

side (x)

Area length width

unitsx2 2

#=

=

Length (l)


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Area and Perimeter


36 9HSERIES TOPIC

Cheat Sheet Area and Perimeter




Add together the lengths of every side which make the shape.

Area 1

(Rectangle)

Area 2

(Isosceles triangle)

Composite Area = Area 1 + Area 2

(Rectangle + Isosceles triangle)

+ =

1 2 +1 2

side 2

(y units)width

(y units)

side (x units) length (x units)

side 3

(z units)

side 1 (x units)

RectangleSquareTriangle

side length

units

P

x

4

4

#=

=

width length width length

units

P

x y2 2

= + + +

= +

side side side

units

P

x y z

1 2 3= + +

= + +

2 m 2 m

3 m 3m

7 m 7 m

? m (3 + 3.5) m =6.5 m

(7 - 2) m =5 m? m

3.5m 3.5m

Rhombus Kite

Perimeter m m m m m m

m

7 6.5 2 3.5 5 3

27

` = + + + + +

=

B

D

A CB

C

A

D

Ba

bD

A

C

Ba

bD

A

C

Area

Perimeter

2

4

AC BD

AB

#

#

'=

=

^ h AreaPerimeter

( ) 2AC BD

AB AD2 2

#

# #

'=

= +

Area

Perimeter

( ) 2a b h

AB BD CD AC

# '= +

= + + +

Start/nish

Start/nish

Start/nish

Start/nish

perpendicular

height (h)

perpendicular

height (h)

The lengths of all unlabelled sides must be found in composite shapes before calculang their perimeter.

It is easier to add them together if the lengths are all in the same units.

Rhombus, Kites and Trapeziums

Trapezium


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PERIM

ETERUSINGU

NITSQUARES

*PERIMETER

USIN

GUN

ITSQ

UARES*

*ARE

A:PARALLELOGRA

LLEL

OGRAM

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AREAOFCOMPOSITE

SHAPES

*AR

EAO

FCO

MPOS

ITESHA

PES

*

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SIMPLE WORD

PROBLEMS

INVOLVING

AREA AND

PERIMETER

...../...../20....

PERIMETEROF SIMPLE SHAPES *

.....

/.....

/20....

PERIMETEROFSIMPLESHAPES*

* AREA: SQUARESAND

RE

CTAN

GLE

S

*AREA:SQUARESAND

RE

CTAN

GLE

S

...../...../20....

2535 18923329.nzl h area and perimeter nzl

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