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Area and Perimeter

Area and Perimeter

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 different from what you expected.

Work through the book for a great way to do this

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

Football fields use rectangles, circles, quadrants and minor segments with specific areas and perimeters to mark out the playing field.

Write down the name of another sport that uses a playing field 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:

Area and Perimeter


2 9HSERIES TOPIC

How does it work?

Area using unit squares

Calculate the area of these shapes

Area and Perimeter

Area is just the amount of flat 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 counting 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

221

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

Little dashes on each side mean they are all the same length.

1 unit

1 cm

1 unit

1 unit

Area ( ) whole squares 2 half squares quarter squares

square cm square cm square cm

square centimetres

cm

A

.

.

2 2

2 221 2

41

2 1 0 5

3 5 2

# #

= + +

= + +

= + +

=

^ h

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 + 4

1# units2

= units2

Area = whole + half +

quarter squares

= units2 + 2

1# units2 +

41

# units2

= units2

Area = whole + quarter squares

= cm2 + 4

1# cm2

= cm2

Area = whole + half squares

= units2 + 2

1# units2

= units2

Area = whole squares

= mm2

1 unit

1 unit

1 m

1 unit

1 unit 1 mm

1 cm

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

b1 cm

1 m

1 unit

1 km

1 cm

1 mm

1 mm

1 unit

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

AREA USING UNIT SQUARES * ARE

A USING UNIT SQU

ARES *

Area and Perimeter


59HSERIES TOPIC


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

4

Number of different designs you found =

a

c

b

d


units8 26 @, using whole squares only.

mm3 26 @, include quarter squares in your shape.

units5 26 @, include half squares in your shape.

cm4.5 26 @, include halves and quarters.

An artist has eight, 1 m2, square-shaped panels which he can use to make a pattern.The rules for the design are:- the shape formed cannot have any gaps/holes.

i.e. or

- it must fit entirely inside the display panel shown,

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

How many different designs can you come up with?Sketch the main shapes to help you remember your count.

1 unit

1 mm 1 cm

1 m

1 unit

Area and Perimeter


6 9HSERIES TOPIC

How does it work? Area and Perimeter

Perimeter using unit squares

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

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

Start/end of path around the outside

1 unit

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, little 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 matter where you start/finish, 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

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 of 6 units2.Use your results in question 2 to help you explain briefly whether or not all shapes with thesame area have the same perimeter.

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

PERIMETER USING UNIT SQUARES * PER

IMET

ER USING UNIT

SQUAR

ES *

Area and Perimeter


8 9HSERIES TOPIC



4 Draw six patterns on the grid below which:

Draw another five patterns on the grid below which:

a

b

• all have an area of 5 units2 and,• have a different perimeter from each other.

• all have an area of 5 units2 and,• have a different perimeter than the shapes formed in part a .

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

All squares used for each pattern must share at least half of a common side

or a corner

point .

1 u

nit

1 unit

1 unit

1 u

nit

Area and PerimeterMathletics Passport © 3P Learning

99HSERIES TOPIC

Area and PerimeterWhere does it work?

Area: Squares and rectangles

Calculate the area of these shaded shapes

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

For squares and rectangles, just multiply 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

9 2

#

#

=

=

=

Area length width

m cm

cm cm

cm

2 60

200 60

12 000 2

#

#

#

=

=

=

=

(i)

(ii)

(iii)

All measurements (or dimensions) must be written in the same units before calculating 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

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 of 121 m2?

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 calculating the square root.

1.4 km

2 units

2 units

3 u

nits

3.2 mm

7 cm

43 mm

5 m

m

0.6 m

* AREA: SQUARES AND RECTANGLES * AREA: SQUARES

AND RECTANGLES

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

4 unitsArea = 28 units

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 h21 2# #

This rule works to find 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

21 3 4

6 2

# #

# #

=

=

=

Area base height

mm mm

mm

. 4

.

21

21 5 4

10 8 2

# #

# #

=

=

=

Area base height

units units

units

.

.

21

21 1 5 2

1 5 2

# #

# #

=

=

=

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 still multiply 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.

6 m

1.5 units

2 units

5.4 mm

5 m4 m

4 mm

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 = 21 # #

units2

= units2

Area = # # mm2

= mm2

Area = # # units2

= units2

Area = # # m2

= m2

Area = # # cm2

= cm2

Area = # # m2

= m2

Area = 21 # #

units2

= units2

base baseheight height

a

a

c

e

b

b

d

2 units

4 units8 units

12 mm

600 cm

4 m5 m

14 cm

8 mm

7.5 units 4.5 m

10 units

12 cm

Remember: same units needed.

* AREA: TRIANGLES * AREA: TRIAN

GLES

* AREA:

TRIANG

LES

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

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 distance between a pair of parallel sides is called the perpendicular height

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

height

Perpendicularheight (h)

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

= length # perpendicular height units2

= l # h units2

Parallelogram Rectanglemove triangle cut off

Calculate the area of these parallelograms

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

Area length height

mm mm

mm

30 15

450 2

#

#

=

=

=

Area length perpendicular height

m m

m

5 12

60 2

#

#

=

=

=

Area area of the triangle

m m

m

2 5 12

2

21

60 2

#

# # #

=

=

=

(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 flip both

vertically and horizontallyBring them together Parallelogram

20 mm

30 mm

15 mm

Area and Perimeter


14 9HSERIES TOPIC


1

2

3


Complete the area calculations for these parallelograms:

Calculate the area of the parallelograms formed using these triangles.

Fill the grid below with as many different parallelograms as you can which have an area of 4 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 cm2.2 cm

26 m 1.6 mm10 m 2 mm

24 m1.2 mm

* AREA: PARALLELOGRAM * ARE

A:

PARALLELOG

RAM

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

1 unit

1 u

nit

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 find 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 = 21 # 2 cm # 8 cm = 8 cm2

Area 1 = 4.5 m # 3.5 m = 15.75 m2

Area 1 = 8 m # 7 m = 56 m2

Area 2 = 8 cm # 8 cm = 64 cm2

Area 2 = 3.5 m # 7 cm = 24.5 m2

Area 2 = 3.5 m # 4.5 m = 15.75 m2

` Total area = Area 1 + Area 2

= 8 cm2 + 64 cm2

= 72 cm2

` Total area = 15.75 m2 + 24.5 m2

= 40.25 m2

` Total area = 56 m2 - 15.75 m2

= 40.25 m2

This next one shows how you can use addition or subtraction 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

12

8 m

7 m

3.5 m

4.5 m

4.5 m

3.5 m

8 m

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 putting together separate parts

Area and Perimeter


16 9HSERIES TOPIC



1 Complete the area calculations for these shaded shapes:

a

b

c

d

12

6 mm

4 m

m

2 m

m5

m5

m

4 m

m

6 m

6 m

3 m

3 m

11 m

11 m

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

1

1

1

2

2

2

2

1

2

2 cm

2.5 cm

6.5 cm

2 cm

2.5 cm

6.5 cm

3 m

3 m

5 m

4 cm

5 m

2 m

2 m

1

2

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

200 cm

4.5 m

300 cm

2 mm

5 units10

uni

ts

6 units

psst: this one needs three area calculations

AREA OF COMPOSITE SHAPES * AREA OF COMPOSITE

SHAPES *

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

psst: change all the units to metres first.


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 (z units)

side 1 (x units)

RectangleSquare Triangle

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

# #

= + + +

= + + +

= +

= +

^^ ^

hh 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 calculating 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 10

40

# #= +

= +

=

15 mm15 mm

5 mm 5 mm0.5 cm

15 mm

Sum of all the side lengths

Four lots of the same side length

All side lengths in mm

Opposite sides in pairs

Start/finish

Start/finish

Start/finish

Start/finish

Start/finish

Start/finish

Area and Perimeter


199HSERIES TOPIC



15 units8 units

17 units

1

2

a

a

b

b

c

c

d

d

Complete the perimeter calculations 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

9 mm

6 mm

11 cm

5 m

2.4 mm

5.8 cm

15 m

1.6 mm 5 m3 m

5 m

1.6 m 2.4 m

3.4 m

PERIMETER

OF SIMPLE SHAPES *

...../..

.../20.

...

PERIMETER

OF SIMPLE SHAPES *

Area and Perimeter


20 9HSERIES TOPIC



a b

3

4

Find the perimeter of each shape written using the smaller units of measurement in each diagram.

in cm. in mm.

3 m600 cm

550 cm

225 mm

16.5 cm

Perimeter = cm Perimeter = mm

Each shape below has its perimeter written inside and is missing one of the side length values.Rule a straight line between each shape and the correct missing side length on the right to answer:

How many straight sides does an icosagon have?

4.4 m

6.5 m

5.2 m

P = 18 m

1.6 m P = 12 m

P = 14 m

P = 24 m

9 m

P = 32 m

P = 12 m

8 m

2.4 m

3.5 m

380 cm

440 cm

2 m

7 m

650 cm

1.1 m

6 m

5 m

c

c

c

d

d

b

v

v

a

a

m

b

E

F V

N

LT

SW

H

N

E T

R

GY

a b c d m v


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

3.5 m 3.5 m

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-positioned to make the calculation easier

Start/finish

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

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

13 m

8 cm

2 cm

2.2

mm

1 m

m

b

b

ba

a

a

1.6 mm

3.4 mm

18 m

15 m

ba

5 cm

14 cm

15 cm

8 cm

4.8 cm

9.8 mm

4.1 cm

38 mm

2 m

1.2 cm

16 mm

Perimeter = # cm

= cm

Perimeter = 2 # a m + 2 # m

= m +

m

= m

Perimeter = 3 # mm +

mm +

mm

= mm

Perimeter = # 4.1 cm +

#

cm

= cm

4 m

20 mm

a m

Be careful with the units for these next two

PERIMETER OF COMPOSITE SHAPES * PERIMET

ER OF COMPOSITE SHAPES

*

312...../...../20....

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

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

3.6

cm 1200 m

1.5 km

a

c

b

d

mm6 @

cm6 @

m6 @

km6 @

2.2 m

Perimeter = mm

Perimeter = cm

Perimeter = m

Perimeter = km

Total perimeter of completed path = m

22 m

m

Earn an awesome passport stamp for this one!The incomplete geometric path shown below is being constructed using a combination of the following shaped pavers:

The gap in between each part of the spiral path is always 1 m wide.Calculate what the total perimeter of this path will be when finished.

1 m

1.41 m1 m

2 m

2 m

2 m

1.41 m

1 m

1 m 1 mCompleted path

AWESOME AWESOME AWESOME

AWESOME

AWESOME

AWESOME

psst: 1 km = 1000 m

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

Area and Perimeter


24 9HSERIES TOPIC



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

a

b

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

1 u

nit

1 unit

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

Area and Perimeter


259HSERIES TOPIC


Simple word problems involving area and perimeter

Sometimes we can only communicate ideas or problems through words.

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

For example,Miguel wants to paint a square. He has just enough paint to create a line 240 cm 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 240 cm.

So each side length cm

cm

240 4

60

'=

=

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

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

Here are some more examples

(i)

(ii)

A rectangular park is four times longer than it is wide. If the park is 90 m long, how much area does this park cover?

At a fun run, competitors run straight for 0.9 km before turning left 90 degrees to run straight for a further 1.2 km. The course has one final corner which leads back to the start along a straight 1.5 km long street. How many laps of this course do competitors complete if they run a total of 18 km?

90 m

m m22.590 4' =^ h

Area length width

m m

m

90 22.5

2025 2

#

#=

=

=

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

1.5 km 0.9 km

1.2 km

Start/finish

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 3.6 km

` Competitors must complete 5 laps of the course to finish

Area and Perimeter


26 9HSERIES TOPIC



1

2

3

Three equilateral triangles, each with sides of length 3 cm 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 fifth of its height. If the base of this triangle is 4.2 m, 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 relationship between area and perimeter.

You have been employed by a fabric design company called Double Geometrics. Your first task as a pattern maker is to design the following using all seven identical squares: “Closed shapes for a new pattern in which the value of their perimeter is twice the value of their area.” Draw five possible different patterns that match this design request.

SIMPLE WORD PROBLEMS INVOLVING AREA AND PERIMETER

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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 floor must be a rectangle shape with an area of 64 m2.

Another design rule is to try ensure a large perimeter so there is more space to hang paintings from. Use calculations to show which floor plan will have the largest perimeter.

Would the design with the largest possible perimeter be a good choice? Explain briefly 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 12.5 cm wide and 24.5 cm long if they all fit perfectly without any edges overlapping?

3.5 cm

2.5 cm

12.5 cm

24.5 cm

If only whole metre measurements can be used, sketch all the different possible floor dimensions.

Area and Perimeter


28 9HSERIES TOPIC



5

6

A fence used to close off a parallelogram-shaped area is being rearranged to create a square area with the same perimeter. The short side of the area is 34 m 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 after using the whole length of this fence?

A 500 mL tin of white paint has been purchased to paint the wall. The instructions on the paint tin say this is enough to cover an area of 11 500 cm2.

If the distance between the longer sides of the original area was 30 m and the length did not change, use calculations to show which fencing arrangement surrounded the largest area.

Use calculations to show that there is enough paint in the tin to cover side of the wall.

28 cm

16 cmEach brick =

34 m

If a beetle walked all around the outside of the wall (including along the ground), how many metres did it walk?

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

4 935 2

#

'

'

=

=

=

^

^

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 find the area:

Here are some examples:

(i)

(ii)

For this rhombus, WY = 12 cm and XZ = 16 cm.

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

length

heig

ht

B

X

C

Y

A

W

D

Z

B

B

C

C

A

A

D

D

10 cm

3.7 m

1.5 m

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 = 18 cm and QS = 52 cm BD = 1.8 mm and AC = 2.4 mm

Rhombus and Kite shapes

Q

S

P R

Area = # '

cm2

= cm2

Area = # #

21

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:

3.4 cm

5.1 cm

3.6 mm

41 cm

15 cm

6.5 cm

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

14 m

9 m

J

K

LM

HI

221#' =

RHOMBUS AND KITE SHAPES RHOMBUS

A

ND KITE SHAPES

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.

Area and Perimeter


319HSERIES TOPIC

Area and PerimeterWhat else can you do?

10.1 m

22 mm10 mm

20 mmArea sumof the parallel sides height

mm mm mm

mm mm

mm

16 2

32 16 2

2

22 10

256 2

#

#

#

'

'

'

=

= +

=

=

^

^

h

h

Area sumof the parallel sides height

m m m

m m

m

. .

.

.

2

6 7 14 5 2 2

21 2 2 2

21 2 2

#

#

#

'

'

'

=

= +

=

=

^

^

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)

16 mm

6.7 m

14.5 m

2.9 m 2 m

Trapezoids

A trapezoid is a quadrilateral which has only one pair of parallel sides.

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

Two common trapezoid 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 sumof the parallel sides height 2

2a b h#

: '

'

#=

= +

^

^

h

h

Perimeter AB BD CD AC: = + + +

B Baa

bbD D

A A

C C

height (h) height (h)

Area and Perimeter


32 9HSERIES TOPIC


a

a b

b

1

2

3

41 km8.2 m

15 km1.8 m

1 km 12.5 m

9 km

53 km 4.5 m

Area = + #

' 2 km2

= km2

Area = cm2

Area = m2

Area = mm2

Perimeter = m

Area = + #

' 2 m2

= m2

Perimeter = km Perimeter = m

5 cm

15 cm

16.7 m

240 cm

33.4 m

8 cm

1 mm 2.4 mm 14.3 mm

170 cm1 2

Trapezoids

Calculate the area and perimeter of these trapezoids:

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

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

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TRAPEZOIDS * TRAPEZOIDS * T

RA

PEZOIDS * TRAP

EZOIDS *

Area and Perimeter


339HSERIES TOPIC


Area challenge

1 u

nit

1 unit

Fill the grid below with as many different squares, triangles, rectangles, parallelograms, rhombi, kites and trapezoids as you can which all have the same area of 8 units2.

Area and Perimeter


34 9HSERIES TOPIC


Reflection Time

Reflecting 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 calculations to a real life situation.

• If you discovered or learnt about any shortcuts to help with calculating area and perimeter or some other cool facts/conversions, jot them down here:


359HSERIES TOPIC

Cheat Sheet Area and Perimeter

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




Area: Triangles


Area is just the amount of flat 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 fractions of squares to calculate the area.

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

Just multiply 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

unitsxy 2

#=

=

height height

base base

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

= unitsb h21 2# #

Perpendicularheight (h)

` Area of a parallelogram = length # perpendicular height units2

= l # h units2

heig

ht

base

side (x)

side (x)

Area length width

unitsx2 2

#=

=

Length (l)


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)

RectangleSquare Triangle

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 3 m

7 m 7 m

? m (3 + 3.5) m = 6.5 m

(7 - 2) m = 5 m? m

3.5 m 3.5 m

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 Area

Perimeter

( ) 2AC BD

AB AD2 2

#

# #

'=

= +

Area

Perimeter

( ) 2a b h

AB BD CD AC

# '= +

= + + +

Start/finish

Start/finish

Start/finish

Start/finish

perpendicularheight (h)

perpendicularheight (h)

The lengths of all unlabelled sides must be found in composite shapes before calculating their perimeter. It is easier to add them together if the lengths are all in the same units.

Rhombus, Kites and Trapezoids

Trapezoid

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