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Area and Perimeter

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Area and Perimeter

ACMMG: 109, 159, 196, 216

Solutions

9HSERIES TOPIC

1ƌĞĂĂŶĚWĞƌŝŵĞƚĞƌ^ŽůƵƟŽŶƐ

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How does it work? Solutions Area and Perimeter

1 a

Page 3 questions

Page 4 questions

Area using unit squares

Area using unit squares

a

c

e

g

a

d

g

b

e

h

c

f

f

b

d

Area = ǁŚŽůĞƐƋƵĂƌĞƐ

= ƵŶŝƚƐ2

Area = ǁŚŽůĞ+ ŚĂůĨƐƋƵĂƌĞƐ

= m2 + 2

1# m2

= m2

Area = ǁŚŽůĞ+ ƋƵĂƌƚĞƌƐƋƵĂƌĞƐ

= ƵŶŝƚƐ2 + 4

1# ƵŶŝƚƐ2

= ƵŶŝƚƐ2

Area = ǁŚŽůĞ+ half +

ƋƵĂƌƚĞƌƐƋƵĂƌĞƐ

= ƵŶŝƚƐ2 + 2

1# ƵŶŝƚƐ2 + 4

1# ƵŶŝƚƐ2

= ƵŶŝƚƐ2

Area = ǁŚŽůĞ+ ƋƵĂƌƚĞƌƐƋƵĂƌĞƐ

= Đŵ2 + 4

1# Đŵ2

= Đŵ2

Area = ǁŚŽůĞ+ ŚĂůĨƐƋƵĂƌĞƐ

= ƵŶŝƚƐ2 + 2

1# ƵŶŝƚƐ2

= ƵŶŝƚƐ2

Area = ǁŚŽůĞƐƋƵĂƌĞƐ

= mm2

4 6

64

2

2

3 4 4

4 24

4

4

6

4

4

2

2

2

3 4 4

4 24

2

3

3

6

4.5

2 Area cm9 2= Area units12 2= Area m6 2=

Area mm4.5 2= Area units5 2=

Area km.12 5 2= Area cm14 2=

Area mm8 2=

2 ƌĞĂĂŶĚWĞƌŝŵĞƚĞƌ^ŽůƵƟŽŶƐ

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9HSERIES TOPIC

How does it work? Solutions Area and Perimeter

3

4

1

a

c

b

d

Page 4 questions

Page 7 questions

Area using unit squares

Perimeter using unit squares

Here are 42ƐŽůƵƟŽŶƐdŚĞƌĞĂƌĞŵŽƌĞ

a

b WĞƌŝŵĞƚĞƌ= + + +

ƵŶŝƚƐ

= ƵŶŝƚƐ

WĞƌŝŵĞƚĞƌ= + + +

ƵŶŝƚƐ

= ƵŶŝƚƐ

2

1 3 1 3

8

8

2 2 2

9HSERIES TOPIC

3ƌĞĂĂŶĚWĞƌŝŵĞƚĞƌ^ŽůƵƟŽŶƐ

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How does it work? Solutions Area and Perimeter

1

Page 7 questions

Page 8 questions

Perimeter using unit squares

Perimeter using unit squares

2

3

c

a

a

b

b c d

WĞƌŝŵĞƚĞƌ= + + +

+ +

ƵŶŝƚƐ

= ƵŶŝƚƐ

2

8

1 1 1 1 2

P = 12ƵŶŝƚƐP = 10ƵŶŝƚƐ P = 14ƵŶŝƚƐ P = 13ƵŶŝƚƐ

ǀĞŶƚŚŽƵŐŚƚŚĞƐŚĂƉĞƐĂůůŚĂǀĞƚŚĞƐĂŵĞĂƌĞĂƚŚĞLJĚŽŶŽƚĂůůŚĂǀĞƚŚĞƐĂŵĞƉĞƌŝŵĞƚĞƌůĞŶŐƚŚƐdŚŝƐƐŚŽǁƐƚŚĂƚƚŚĞƐŚĂƉĞƐǁŝƚŚƚŚĞƐĂŵĞĂƌĞĂǁŝůůŶŽƚŶĞĐĞƐƐĂƌŝůLJŚĂǀĞƚŚĞƐĂŵĞƉĞƌŝŵĞƚĞƌ ^ŽƚŚĞƉĞƌŝŵĞƚĞƌŝƐŶŽƚƌĞůĂƚĞĚƚŽƚŚĞĂƌĞĂŽĨƚŚĞƐŚĂƉĞ

4

P = 10ƵŶŝƚƐ

P = 11ƵŶŝƚƐ

P = 16ƵŶŝƚƐ

P = 17ƵŶŝƚƐ

P = 12ƵŶŝƚƐ

P = 13ƵŶŝƚƐ

P = 18ƵŶŝƚƐ

P = 19ƵŶŝƚƐ

P = 14ƵŶŝƚƐ

P = 15ƵŶŝƚƐ

P = 20ƵŶŝƚƐ

4 ƌĞĂĂŶĚWĞƌŝŵĞƚĞƌ^ŽůƵƟŽŶƐ

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9HSERIES TOPIC

Where does it work? Solutions Area and Perimeter

1

1

2

3

4

a

a

a

c

b

b

b

d

Area = # ƵŶŝƚƐ2

= ƵŶŝƚƐ2

Area = # ƵŶŝƚƐ2

= ƵŶŝƚƐ2

Area = # mm2

= mm2

Area = # m2

= m2

ůĞŶŐƚŚ

ůĞŶŐƚŚ

ůĞŶŐƚŚ

ůĞŶŐƚŚ

ǁŝĚƚŚ

ǁŝĚƚŚ

ǁŝĚƚŚ

ǁŝĚƚŚ

Area = # km2

= km2

. km2;ƚŽŶĞĂƌĞƐƚǁŚŽůĞŬŵ2)

ůĞŶŐƚŚ ǁŝĚƚŚ

Area = # Đŵ2

= Đŵ2

. Đŵ2;ƚŽŶĞĂƌĞƐƚǁŚŽůĞĐŵ2)

ůĞŶŐƚŚ ǁŝĚƚŚ

Page 10 questions

Page 12 questions

Area: Squares and rectangles

Area: Triangles

2 0.6

3

1.4 7

2 30

1.4 4.3

5

4 0.36

6

1.96 30.1

16

2 0.6

2 3.2

Length Area width

units units

units

The length of the rectangle is units7 .

28 4

7

2

'

'

`

=

=

=

Length of each side

units

121

11

=

=

Area = 21 # #

ƵŶŝƚƐ2

= ƵŶŝƚƐ2

Area = 21 # #

ƵŶŝƚƐ2

= ƵŶŝƚƐ2

ďĂƐĞ ďĂƐĞŚĞŝŐŚƚ ŚĞŝŐŚƚ

2 8

324

4 8

9HSERIES TOPIC

5ƌĞĂĂŶĚWĞƌŝŵĞƚĞƌ^ŽůƵƟŽŶƐ

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Where does it work? Solutions Area and Perimeter

Area = # Đŵ2

= Đŵ2

Area = # ƵŶŝƚƐ2

= ƵŶŝƚƐ2

10 2.2

Area = # # m2

= m2

Area = # # ƵŶŝƚƐ2

= ƵŶŝƚƐ2

Area = # # m2

= m237.5

10

13.5

Area = # # mm2

= mm2

Area = # # Đŵ2

= Đŵ2

2 a

a

a

c

b

b

b

d

Page 12 questions

Page 14 questions

Area: Triangles

Area: Parallelograms

48

12

10

5

4.5 4.6

450 10.12

6

14

84

8

7.5

4

4.5

12

e

21

21

21

21

21

1

2

3

Area m m

m

221 24 10

240 2

# # #=

=

Area mm mm

mm

. .

.

221 1 2 1 6

1 92 2

# # #=

=

6 ƌĞĂĂŶĚWĞƌŝŵĞƚĞƌ^ŽůƵƟŽŶƐ

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9HSERIES TOPIC

Where does it work? Solutions Area and Perimeter

Page 16 questions

Area of composite shapes

1 a

b

c

d

Area 1 = mm # mm

= mm2

Area 1 = # # m2

= m2

Area 1 = # Đŵ2

= Đŵ2

Area 1 = # # m2

= m2

`ŽŵƉŽƐŝƚĞĂƌĞĂ= + mm2

= mm2

`ŽŵƉŽƐŝƚĞĂƌĞĂ= + m2

= m2

`ŽŵƉŽƐŝƚĞĂƌĞĂ= - Đŵ2

= Đŵ2

`ŽŵƉŽƐŝƚĞĂƌĞĂ= - m2

= m2

Area 2 = mm # mm

= mm2

Area 2 = # m2

= m2

Area 2 = # # Đŵ2

= Đŵ2

Area 2 = # m2

= m2

4

0.5

6.5

0.5

16

24

13

20

4

55

4

4

2

11

0.5

16

24

13

20

20

79

9

16

4

55

4

4

4

6

2

8

8

5

2

5

4

2

2

2

9HSERIES TOPIC

7ƌĞĂĂŶĚWĞƌŝŵĞƚĞƌ^ŽůƵƟŽŶƐ

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Where does it work? Solutions Area and Perimeter

Page 17 questions

Page 19 questions

Area of composite shapes

Perimeter of simple shapes

2 a

b

c

d

Area Area of square area of triangle

cm cm cm cm

cm cm

cm

13 13 12 5 2

169 30

199

2 2

2

# # '

= +

= +

= +

=

^ h

Area Area of parallelogram area of triangle

m m m m

m m

m

. .

.

.

4 5 3 4 5 2

13 5 9

22 5

2 2

2

# #

= +

= +

= +

=

cm m and cm m300 3 200 2= =

Area Area of large square area of small square

mm mm mm mm

mm mm

mm

6 6 2 2

36 4

32

2 2

2

# #

= -

= -

= -

=

Area Area of rectangle area of square area of triangle

units units units units units units

units units units

units

5 10 5 5 5 6 2

50 25 15

90

2 2 2

2

# # # '

= + +

= + +

= + +

=

^ h

1 a

b

WĞƌŝŵĞƚĞƌ= ƵŶŝƚƐ+ ƵŶŝƚƐ+ ƵŶŝƚƐ

= ƵŶŝƚƐ

WĞƌŝŵĞƚĞƌ= 2 # mm + 2 # mm

= mm

15

9

40

30

17 8

6

8 ƌĞĂĂŶĚWĞƌŝŵĞƚĞƌ^ŽůƵƟŽŶƐ

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9HSERIES TOPIC

Where does it work? Solutions Area and Perimeter

Page 19 questions

Page 20 questions

Perimeter of simple shapes

Perimeter of simple shapes

2

3

c

a

a

b

b

c d

d WĞƌŝŵĞƚĞƌ= 2 # Đŵ+ Đŵ

= Đŵ

WĞƌŝŵĞƚĞƌ= # m

= m

1 4

11

20

27

5

5

Perimeter cm

cm

4 5.8

23.2

#=

=

Perimeter cm

cm

300 550 600

1450

` = + +

=

^ h Perimeter mm

mm780

2 225 2 165# #` = +

=

^ hm cm3 300= cm mm16.5 165=

Perimeter mm

mm

2 2.4 2 .1 6

8

# #= +

=

^ h

Perimeter m

m

3 15

45

#=

=

Perimeter m

m

. .

.

3 4 5 2 4

10 8

= + +

=

^ h

9HSERIES TOPIC

9ƌĞĂĂŶĚWĞƌŝŵĞƚĞƌ^ŽůƵƟŽŶƐ

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Where does it work? Solutions Area and Perimeter

Page 20 questions

Page 22 questions

Perimeter of simple shapes

Perimeter of composite shapes

4

a b c d m v

V

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

380Đŵ

440Đŵ

2 m

7 m

650Đŵ

1.1 m

6 m

5 m

c

c

c

d

d

b

v

v

a

a

m

b

E

F

N

L

T

S

W

H

N

E T

R

G

Y

T W E N T Y

1

2

a

a

cb d

WĞƌŝŵĞƚĞƌ= # Đŵ

= Đŵ

a = 4Đŵ a = 1.8 mm a = 9 m a = 5.2Đŵ

b = 5Đŵ b = 2.2 mm b = 6 m b = 11.2Đŵ

5 9.8

49

10 ƌĞĂĂŶĚWĞƌŝŵĞƚĞƌ^ŽůƵƟŽŶƐ

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9HSERIES TOPIC

Where does it work? Solutions Area and Perimeter

Page 22 questions

Page 23 questions

Perimeter of composite shapes

Perimeter of composite shapes

2 b

c

d

Perimeter = 2 # a m + 2 # m

= m +

m

= m

Perimeter = 3 # mm +

mm +

mm

= mm

Perimeter = # 4.1 cm +

#

cm

= cm

2

16

12

12

4

24

2 3.8

72

20 16

4

3

4

a

c

b

d

Perimeter mm

mm

12 4

48

#=

=

Perimeter cm

cm

2 4.8 2 3.6 2 .

21.2

2 2# # #` = + +

=

^ hPerimeter km

km

3 1.2 2 .

.

1 5

6 6

# #` = +

=

^ h

Perimeter m

m

. . .

.

5 2 2 2 4 4 6 6

26 4

# #= + +

=

^ h

mm cm

mm cm

48 4.8

.22 2 2

=

=

m km1200 1.2=

m

1 1 2 3 4 5 6 7 8 9 9 1 10

11 10 9 8 7 6 5 4 3 2 1 132

+ + + + + + + + + + + +

+ + + + + + + + + + + =

Total perimeter of completed path = 132 m

Each new spiral arm increases by 1ĂŌĞƌĞĂĐŚƚƵƌŶŐŽŝŶŐŽƵƚ;ĞdžĐĞƉƚĨŽƌůĂƐƚǀĞƌƟĐĂůĂƌŵͿ^ƚĂƌƟŶŐĂƚƚŚĞĚŽƚĂŶĚǁŽƌŬĚŽǁŶƚŚĞŶĂŶƟͲĐůŽĐŬǁŝƐĞŽŶƚŚĞǁĂLJŽƵƚĂŶĚƚŚĞŶĐůŽĐŬǁŝƐĞŽŶƚŚĞǁĂLJďĂĐŬŝŶƚŽƚŚĞĐĞŶƚƌĞ

9HSERIES TOPIC

11ƌĞĂĂŶĚWĞƌŝŵĞƚĞƌ^ŽůƵƟŽŶƐ

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Where does it work? Solutions Area and Perimeter

Page 24 questions

Page 26 questions

Perimeter of composite shapes

Simple word problems involving area and perimeter

5

1

a

b

c

Perimeter cm

cm

5 3

15

#=

=

^ŵĂůůĞƐƚƉŽƐƐŝďůĞƉĞƌŝŵĞƚĞƌ= 18ƵŶŝƚƐ

>ĂƌŐĞƐƚƉŽƐƐŝďůĞƉĞƌŝŵĞƚĞƌ= 40ƵŶŝƚƐ

dƌLJƚŽŵĂŬĞĂƐŵĂŶLJƐŝĚĞƐĂƐƉŽƐƐŝďůĞƚŽƵĐŚŝŶŐĞĂĐŚŽƚŚĞƌ ǁŚŝĐŚƌĞĚƵĐĞƐƚŚĞŶƵŵďĞƌŽĨƐŝĚĞƐƚŚĂƚĂƌĞĐŽƵŶƚĞĚĨŽƌƚŚĞƉĞƌŝŵĞƚĞƌ

DĂŬŝŶŐĞĂĐŚƐŚĂƉĞũŽŝŶĞĚƚŽƚŚĞŽƚŚĞƌďLJŽŶĞƐŝĚĞŽŶůLJ ǁŚŝĐŚŝŶĐƌĞĂƐĞƐƚŚĞŶƵŵďĞƌŽĨƐŝĚĞƐƚŚĂƚŵƵƐƚĐŽƵŶƚĞĚĨŽƌƚŚĞƉĞƌŝŵĞƚĞƌ

12 ƌĞĂĂŶĚWĞƌŝŵĞƚĞƌ^ŽůƵƟŽŶƐ

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9HSERIES TOPIC

Where does it work? Solutions Area and Perimeter

Page 26 questions

Page 27 questions

6LPSOHZRUGSUREOHPVLQYROYLQJDUHDDQGSHULPHWHU

6LPSOHZRUGSUREOHPVLQYROYLQJDUHDDQGSHULPHWHU

2

3

4

^ŚĂƉĞ1 ^ŚĂƉĞ2

WĞƌŝŵĞƚĞƌ= 10ƵŶŝƚƐ WĞƌŝŵĞƚĞƌ= 8ƵŶŝƚƐ

dŚĞĂƌĞĂŽĨďŽƚŚƐŚĂƉĞƐŝƐƚŚĞƐĂŵĞďƵƚƚŚĞƉĞƌŝŵĞƚĞƌƐĂƌĞĚŝīĞƌĞŶƚdŚŝƐƐŚŽǁƐƚŚĂƚƚŚĞƌĞŝƐŶŽƌĞůĂƟŽŶƐŚŝƉďĞƚǁĞĞŶƚŚĞĂƌĞĂĂŶĚƚŚĞƉĞƌŝŵĞƚĞƌŽĨĂƐŚĂƉĞ

4.2 m

1 m

2 m4 m

8 m

64 m

32 m 16 m 8 m

Height of triangle m

m

5 4.2

21

#=

=

Area of triangle m m

m

4.2 21

.

2

44 1 2

#` '=

=

^ h

a

a

b

b

1

1 3

2 4

23

4

Perimeter m m

m

: 2 1 2 64

130

# #= +

=

Perimeter m m

m

: 2 4 2 16

40

# #= +

=

Perimeter m m

m

: 2 2 2 32

68

# #= +

=

Perimeter m m

m

: 4 8

32

#=

=

`ŇŽŽƌƉůĂŶ 1 ŚĂƐƚŚĞůĂƌŐĞƐƚƉĞƌŝŵĞƚĞƌ

9HSERIES TOPIC

13ƌĞĂĂŶĚWĞƌŝŵĞƚĞƌ^ŽůƵƟŽŶƐ

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Where does it work? Solutions Area and Perimeter

Page 27 questions

Page 28 questions

6LPSOHZRUGSUREOHPVLQYROYLQJDUHDDQGSHULPHWHU

6LPSOHZRUGSUREOHPVLQYROYLQJDUHDDQGSHULPHWHU

4

5

c

d

a

EŽŇŽŽƌƉůĂŶ 1 ǁŽƵůĚŶŽƚďĞĂŐŽŽĚĐŚŽŝĐĞtŚŝůĞŝƚŵĞĞƚƐƚŚĞƌƵůĞŽĨŚĂǀŝŶŐƚŚĞůĂƌŐĞƐƚƉŽƐƐŝďůĞƉĞƌŝŵĞƚĞƌ ŝƚǁŽƵůĚďĞǀĞƌLJůŽŶŐĂŶĚƚŽŽŶĂƌƌŽǁĨŽƌŵŽƌĞƚŚĂŶŽŶĞƉĞƌƐŽŶƚŽǁĂůŬƚŚƌŽƵŐŚĂƚĂƟŵĞ

Number of stamps that fit width cm12.5 .52

5

'=

=

Area of stamp cm

cm

2.5 .

8.75

3 5

2

#=

=

Area of envelope cm

cm

. .

.

12 5 24 5

306 25 2

#=

=

Length of parallelogram m

m

2 34

68 2

#=

=

Therefore length of fence long side short side

m m

m

2 2

2 68 2 34

204

# #

# #

= +

= +

=

Length of each side of the square area m

m

204 4

51

` '=

=

cm cm

stamps were needed.

306.25 8.75 35

35

2 2` '

`

=

Number of stamps that fit length cm. .524 5 3

7

'=

=

Tota number of stamps that fit

stamps

5 7

35

#` =

=

OR

`ĂĐŚƐŝĚĞŽĨƚŚĞŶĞǁƐƋƵĂƌĞĂƌĞĂǁŝůůďĞ51ŵůŽŶŐ

14 ƌĞĂĂŶĚWĞƌŝŵĞƚĞƌ^ŽůƵƟŽŶƐ

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9HSERIES TOPIC

Where does it work? Solutions Area and Perimeter

Page 28 questions

6LPSOHZRUGSUREOHPVLQYROYLQJDUHDDQGSHULPHWHU

5

6

b

b

a

Parallelogram area m m

m

68 30

2040 2

#=

=

Area of each brick cm cm

cm

28 16

448 2

#=

=

Perimeter Brick length Half brick length brick width

cm cm cm

cm cm cm

cm

10 8 10

28 10 14 8 16 10

280 112 160

552

# #

# # #

#= + +

= + +

= + +

=

Square area m m

m

51 51

2601 2

#=

=

Total area cm

cm

25

11200

448

2

#` =

=

`dŚĞŶĞǁƐƋƵĂƌĞƐŚĂƉĞǁŝůůƐƵƌƌŽƵŶĚŵŽƌĞĂƌĞĂ;ĂŶĞdžƚƌĂ m561 2ƚŽďĞĞdžĂĐƚͿ

`dŚĞƌĞǁŝůůďĞũƵƐƚĞŶŽƵŐŚƉĂŝŶƚƚŽĐŽǀĞƌƚŚĞƐŝĚĞŽĨƚŚĞǁĂůů;ǁŝƚŚ cm300 2 ƚŽƐƉĂƌĞͿ

`dŚĞďĞĞƚůĞǁĂůŬĞĚĂƚŽƚĂůĚŝƐƚĂŶĐĞŽĨ5.52ŵĞƚƌĞƐ

9HSERIES TOPIC

15ƌĞĂĂŶĚWĞƌŝŵĞƚĞƌ^ŽůƵƟŽŶƐ

Mathletics Passport © 3P Learning

What else can you do? Solutions Area and Perimeter

bArea = + #

' 2 km2

= km2

Area = + #

' 2 m2

= m2

WĞƌŝŵĞƚĞƌ= km WĞƌŝŵĞƚĞƌ= m

Page 30 questions

Page 32 questions

Rhombus and Kite shapes

1

2

3

a

a

a

b

b

b

Area = # '

Đŵ2

= Đŵ2

Area = # #

21

mm2

= mm2

WĞƌŝŵĞƚĞƌ= 2 # + 2 # Đŵ

= Đŵ

WĞƌŝŵĞƚĞƌ= # mm

= mm

18 1.8

468

53 4.5

243 15.3

110 27

1 12.59 1.8

2.16

112 14.4

52 2.4

41 415 3.6

2

Perimeter m m

m

4 9 2 14

64

# #= +

=

Area cm

cm

15 2

135

5 8 5 2

2

# '= + +

=

^ h

Perimeter cm cm cm

cm

5.1 4 3.4 2 6.5

31.7

# #= + +

=

Area mm

mm

. . . 2.4 2

24.12

2 4 1 2 4 14 3 2

2

# '= + + +

=

^ h

Area Area of rhombus area of kite

m m

m

2 21 16 230 16

240 168

408

2 2

2

# #' '

= +

= +

= +

=

^ ^h h

a1

2

Trapeziums

16 ƌĞĂĂŶĚWĞƌŝŵĞƚĞƌ^ŽůƵƟŽŶƐ

Mathletics Passport © 3P Learning

9HSERIES TOPIC

What else can you do? Solutions Area and Perimeter

3

Page 32 questions

Page 33 questions

170Đŵ= 1.7ŵĂŶĚ240Đŵ= 2.4 m

Area Area Area

m m

m m

m

. . 1.7 2 . . 2.4 2

. .

.

33 4 2 4 16 7 2 4

30 43 22 92

53 35

2 2

2 2

2

# #' '

= +

= + + +

= +

=

^ ^h h1 2

Perimeter m m m

m

33.4 2 16.7 3 2.4

74

# #= + +

=

ƌĞĂĐŚĂůůĞŶŐĞ

Here are 20ƉŽƐƐŝďůĞƐŚĂƉĞƐǁŚŝĐŚĂůůŚĂǀĞĂŶĂƌĞĂŽĨ units8 2 dŚĞƌĞĂƌĞŵĂŶLJŵŽƌĞ

Trapeziums

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Area and Perimeter