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Perimeter and Area

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Perimeter and Area

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Perimeter and Area

1SERIES TOPIC

J 12

PERIMETER AND AREA

Answer these questions, before working through the chapter.

Answer these questions, after working through the chapter.

The perimeter of a shape is the total length of its edges. The area of a shape is how much space it takes up on a 2D surface. These shapes can be joined together to form "composite shapes" with larger areas and perimeters.

But now I think:

What do I know now that I didn’t know before?

I used to think:

What does "circumference" mean?

What does "circumference" mean?

What is a sector?

What is a sector?

A quadrilateral is a shape with four sides. Do different quadrilaterals have different perimeters and areas?

A quadrilateral is a shape with four sides. Do different quadrilaterals have different perimeters and areas?


Perimeter and Area

2SERIES TOPIC

J 12

Basics

Circumference

cm

cm

. ...

.

r2

2 3

2 3 14 3

18 8

#

# #

r

r

=

=

=

=

^ h

Perimeter of Shapes

The perimeter of a shape is found by adding the lengths of all its sides.

The only shape which has a tricky method to find the perimeter, is a circle. This is because it has no corners. The perimeter of a circle is called the "circumference."

The perimeter of a circle with a radius r is given by:

where 3.14fr =

(1 decimal place)

Square Rectangle Rhombus

cm5

cm4cm6

cm7

Perimeter

cm

5 5 5 5

4 5

20

= + + +

=

=

^ h

Perimeter

cm

4 4 7 7

2 4 2 7

22

= + + +

= +

=

^ ^h h

Perimeter

cm

6 6 6 6

4 6

24

= + + +

=

=

^ h

r

r r

3 cm 4 cm

Diameter

Circumference r2r=

Perimeter of semicircle Circumference Diameter

cm cm

cm.

r r

21

21 2 2

21 2 4 2 4

20 6

#

#

# # # #

r

r

= +

= +

= +

=

`

` ^

^ ^

j

j h

h h

(1 decimal place)


Perimeter and Area

3SERIES TOPIC

J 12

Basics

Composite Shapes

"Composite Shapes" are formed when shapes join together.

Remember: The lines show us which sides have the same length.cm4

cm5

cm8

cm8 cm4

cm4

cm12

cm6 cm?

Find the perimeter of this "composite shape"

This composite shape is made up of a rectangle and two semicircles

Perimeter

cm

4 4 4 4 8 5 5

4 4 8 2 5

34

= + + + + + +

= + +

=

^ ^h h

a

b

c

How long is the diameter of the bottom semicircle?

Find the radius of the top semicircle and the radius of the bottom semicircle.

Find the Perimeter of this composite shape to the nearest cm.

To find P add up the length of all the straight sides and the circumferences of the semicircles

The total length of the rectangle is 12 cm. So, the diameter of the bottom semicircle is cm cm cm12 6 6- = .

The radius is half the diameter.

Radius of the top semicircle cm cm8 2 4'= = Radius of the bottom semicircle cm cm6 2 3'= =

top semicircle bottom semicircle

cm

cm

. ...

P

P

P

4 4 6 4

182

2 42

2 3

39 99

40

# #r r

= + + + + +

= + +

=

=

^

` `

h

j j

(nearest cm)


Perimeter and Area

4SERIES TOPIC

J 12

BasicsQuestions

1. Find the perimeter of these shapes to the nearest cm. (All units in cm)

2. Look at this triangle:

a

a

d

b

e

c

f

3

5

10

12

6

5

.6 2

cm39

6

11

5

4

Use Pythagoras's theorem to find the length of the missing side.

b

cm31

cm1 .0 4

Find the perimeter of the triangle.


Perimeter and Area

5SERIES TOPIC

J 12

Questions Basics

3. Find the perimeter of this composite shape (all measurements in m).

4. An athlete runs around the track below. What distance does he run after 3 laps?

5. A composite shape is made up of a quarter of a circle and a right angled triangle. Find the perimeter.

16

m50

m46

cm12

cm5

18

14


Perimeter and Area

6SERIES TOPIC

J 12

Knowing More

Area of Shapes

The area of a shape, is the amount of space it covers. Each shape has its own formula for finding its area.Here is a summary of formulas for area for common shapes.

Rectangle

Parallelogram

Trapezium

Triangle

Circle

Semicircle(half a circle)

Area length breadth

l b

#

#

=

=

Area base height

bh

#=

=

Area sumof parallel sidesh

h a b

21

21

=

= +

^

^

h

h

Area Area of circle

r

21

21 2r

=

=

^ h

Area r2r=

Area base height

bh

21

21

# #=

=

r

r r

r

Square

Rhombus

Kite

Area product of diagonals

xy

21

21

=

=

^ h

Area product of diagonals

xy

21

21

#=

=

Area length length

l2

#=

=

yb

l b

h

h

b

b

a

h

x

l

x

y


Perimeter and Area

7SERIES TOPIC

J 12

Knowing More

So the square has a larger area.

The trapezium has the smaller area.

Which of the two shapes below has the larger area, the square or the rectangle?

Which of these two shapes below has the smaller area?

The kite below has an area of 35 cm2. How wide is the kite (x)?

cm3.4

cm6

cm.4 8

cm2

Square:

cm

.

.

A l

3 4

11 56

2

2

2

=

=

=

^ h

cm12

cm7

cm16

cm9

Trapezium:

cm

A h a b21

21 7 12 16

98 2

= +

= +

=

^

^ ^

h

h h

Rectangle:

cm

.

.

A l b

4 8 2

9 6 2

#

#

=

=

=

Circle:

cm ( d.p.)113.1 1

A r

6

2

2

2

#

r

r

=

=

=

^ h

Kite: product of diagonals

cm

A

A x

x

x

x

21

21 10

3521 10

5 35

7

#

`

`

`

=

=

=

=

=

^

^

^

h

h

h

cm10

x


Perimeter and Area

8SERIES TOPIC

J 12

Knowing More

Area of Composite Shapes

We can find areas of composite shapes by joining these shapes together.

The following shape needs to be painted on a field. Find its area to the nearest square unit.

Not to scale

Split the composite shape into shapes with area formulas we know:

Area of shape = Area of semicircle + Area of rectangle + Area of triangle

Not to scale

m

m

m nearest unit

.

r

A r

24 2 12

21

21 12

226 19

226

2

2

2

2

'

f

.

r

r

= =

=

=

=

^

^

h

h

m

A l b

20 24

480 2

#

#

=

=

=

m

h

h

A bh

13 12 25

5

21

21 24 5

60

2 2 2

2

`

# #

= - =

=

=

=

=

Area of shape m m m

m

226 480 60

766

2 2 2

2

` = + +

=

m20

m13

m24

m20

m13

m24

Area of Semicircle Area of Rectangle Area of Triangle

Find h usying pythagoras

h


Perimeter and Area

9SERIES TOPIC

J 12

Questions Knowing More

1. Identify the following shapes and find the areas of (all measurements in cm)

2. Use Pythagoras to find the missing length, and then find the area (measurements in cm).

16

.9 6

h

A

7

15

11D

8AC

BD 9

=

=

B

C

a

a

b c

7

13


Perimeter and Area

10SERIES TOPIC

J 12

Knowing MoreQuestions

b 5

h

x

x

y

13

20

3. Find the area of each of these shapes if cm7x = and cm10y = to the nearest cm2 .

Circle Kitea b


Perimeter and Area

11SERIES TOPIC

J 12


x

m40The shaded "D" is a semicircle. Find x.

Find the area of the shaded region (to 1 decimal place).

Find the area of the unshaded region (to 1 decimal place).

4. A sports field has a painted "D" with these measurements:

a

b

c


Perimeter and Area

12SERIES TOPIC

J 12

Knowing MoreQuestions

5. A square table has an area of m9 2 . A tablecloth needs to be designed in the shape below.

What is the side length of the square table?

Square table Tablecloth

The square centre of the tablecloth needs to fit on top of the table exactly. How much material is needed to make this tablecloth (one decimal place)?

a

b

m9 2


Perimeter and Area

13SERIES TOPIC

J 12


6. Marissa wants to paint this heart on a square wall.

What are the diameters of the equal semicircles at the top of the heart?

What is the height of the triangle?

What is the area of the heart (2 decimal places)?

What is the total area of the wall without paint on it (2 decimal places)?

a

b

c

d

m4


Perimeter and Area

14SERIES TOPIC

J 12

Using Our Knowledge

Perimeter and Area of Quadrants

A "Quadrant" is a quarter of a circle.

The perimeter of a quadrant is:

The straight sides are equalsince each is a radius.

r

r

The curved part is called the arc

Since a quadrant is 41 of a circle, the length of the arc must be

41 of the circumference of a circle.

The area of a quadrant is 41 of the area of a circle. So to find the area of a quadrant:

Arc circumference

Arc

Arc

r

r

41

41 2

2

#

# r

r

=

=

=

Area of Quadrant Area of Circle

Area of Quadrant r

41

41 2

#

# r

=

=

radius2P Arc #= + ^ h

2P r r2r= +

Area of Quadrant r4

2r=

arc


Perimeter and Area

15SERIES TOPIC

J 12

Using Our Knowledge

So the arc length of a sector must be 360ci of the circumference of a circle.

Perimeter and Area of Sectors

A sector is a fraction of a circle with an angle i.

For a full circle, 360ci = . So a sector with angle i is 360ci of a full circle.

The straight sides are equalsince each is a radius.

The curved part is called the arc

So the perimeter of a sector is:

The area of a sector is 360ci of the area of a circle. So to find the area of the sector:

Area of Sector Area of Circle360

#i=c

Area of Sector r360

2#i r=c

Arc radiusP 2 #= + ^ h

Arc circumference360

#i=c

Arc 2 r360

#i r=c

2P r r360

2#i r= +c

c m

r

r

arc

i


Perimeter and Area

16SERIES TOPIC

J 12

Using Our Knowledge

Identify the shapes below. Find their perimeters and areas correct to 2 decimal places.

Find the radius of this sector to 1 decimal place if it's area is cm214 2 .

a b

This is a Quadrant

This is a sector with θ = 40c

6 cm

4 cm40c

120c

cm

.

.

.

P r r

P

P

P

22

2

62 6

9 424 12

21 424

21 42

f

f

.

r

r

= +

= +

= +

=

^ ^h h

cm

.

.

P r r

P

P

P

3602 2

36040 2 4 2 4

91 8 8

10 792

10 79

#

#

#

f

.

i r

r

r

= +

= +

= +

=

c

cc

c

^ ^

m

h h

cm

.

.

A r

A

A

4

46

28 274

28 27

2

2

2

f

.

r

r

=

=

=

^ h

cm

cm decimalplace

.

. .

. ( )

A r

r

r

r

r

360120 214

214120360 642

642 204 354

204 354 14 295

14 3 1

2 2

2

2

#

` #

` '

`

`

f

f f

.

r

r

r

= =

= =

= =

= =

cc

cc

cm

.

.

A r

A

A

A

360

36040 4

91 16

5 585

5 59

2

2

2

#

#

#

f

.

i r

r

r

=

=

=

=

c

cc ^ h

r


Perimeter and Area

17SERIES TOPIC

J 12

Using Our Knowledge

Perimeter and Area of Ellipses

An Ellipse is a stretched circle.

The "longer axis" is called the major axis. The "shorter axis" is called the minor axis.

a

b

c

a is the semi-major axis (half of the major axis)

b is the semi-minor axis (half of the minor axis)

The area of an ellipse is given by the formula:

There is no formula for the exact perimeter of an ellipse, but a good approximation is given by the formula:

What are the lengths of the semi-major axis (a) and the semi-minor axis (b)?

Find the area of this ellipse to 2 decimal places.

Approximate the perimeter to 2 decimal places

A abr=

2P a b2

2 2

. r +

cm and cm12a b 6= =

cm decimalplaces

226.194

226.19 (2 )

A ab

A 12 6

2

f

.

r

r

=

=

=

^ ^h h

cm decimalplaces

.

.

P a b22

22

12 6 2 90

59 607

59 61 2

2 2

2 2

f

. r

r r

+

= + =

=

= ^ h

b

a

ab

6 cm

12 cm


Perimeter and Area

18SERIES TOPIC

J 12

Using Our KnowledgeQuestions

360i =c

• Identify i, the angle inside the shaded sector.

• A sector is 360ic

of a full circle. Find the fraction of the full circle represented by the shaded sector.

1. For each of the four following sectors:

2. Find the arc length of these sectors to the nearest cm.

a b c

53 c130c8 cm

10 cm

9 cm

45c

120c

288c

324c

i = i =

i =i =

b d

hf

a c

ge

360i =c

360i =c360

i =c


Perimeter and Area

19SERIES TOPIC

J 12

Questions Using Our Knowledge

3. Use the arc lengths to find the perimeter of the above sectors.

4. Find the area of these sectors to 1 decimal place.

a

a

b

b

c

c

12 cm

150c

15 cm

11 cm

55c


Perimeter and Area

20SERIES TOPIC

J 12

Using Our KnowledgeQuestions

5. Use this ellipse to answer the following questions

a

b

c

Find a and b, the lengths of the semi-major axis and semi-minor axis respectively.

Find the area of the ellipse to the nearest cm.

Find the perimeter of the ellipse to the nearest cm.

20 cm

50 cm

ab


Perimeter and Area

21SERIES TOPIC

J 12

Thinking More

12 cm

Sometimes we need to find the angle or the length of the radius of a sector based on the area.

Working Backwards

Find i (nearest degree) if the area of this sector is cm100 2

i

Write the formula for Area:

Make i the subject of the formula.

Solve for i using the given values.

A r360

2#i r=c

r

A3602#ir

=c

(nearest degree)

. ...

12

360 100

79 577

80

2#

c

c.

ir

=

=

c^ h

A sector has 60i = c, and area m32 2 . Find the radius of the sector to the nearest metre.

Identify what has been given:

The formula for the perimeter of a sector is:

Make r the subject of the formula and substitute the given values.

m60 32A 2i = =c

A r360

2#i r=c

Find the area of an ellipse with semiminor axis 14 cm and semimajor axis 22 cm to 1 decimal place.

Always identify what has been given. Determine the needed formula, and substitute the given values in. You can always draw a rough sketch of the shape in the question to help you.

Identify what has been given:

Word Problems

cm cm22 14a b= =

The formula for area of an Ellipse is A abr=

cm

.

.

A 22 14 967 61

967 6 2

f

.

r= =^ ^h h

m

7.81764...

r A 3606032 360

8

##

##c

.

i r r= =

=

c c


Perimeter and Area

22SERIES TOPIC

J 12

Thinking More

Composite Area

Sectors and ellipses can join with other shapes to make composite shapes.

A special key is made up of a triangle and a sector. The lock for this key is shown below. (All measurements are in cm)

Find the area of the shaded lock to the nearest cm.

Area of lock = Area of Ellipse - Area of Triangle - Area of Sector

Area of lock = Area of Ellipse - Area of Triangle - Area of Sector

65

16

2

40c

Area of Ellipse

semiminor axis

semimajor axis

cm.

b

a

A ab

26 3

216 8

8 3

75 398 2f

r

r

= = =

= = =

=

=

=

^ ^h h

Area of Triangle

base height

cm

A21

21 5 2

5 2

# #

# #

=

=

=

Area of Sector

cm.

A r360

36040 2

91 4

1 396

2

2

2

#

#

#

f

i r

r

r

=

=

=

=

c

cc ^ h

cm nearest cm

. .

.

( )

75 398 5 1 396

69 001

69 2

f f

f

.

= - -

=


Perimeter and Area

23SERIES TOPIC

J 12

Questions Thinking More

1. Find all these answers to the closest unit.

2. The perimeter of an ellipse is approximately given by:

a

a

b

b

c

Find the area of a sector if the radius is 6 cm and 180i = c.

Make b the subject of the formula.

An ellipse has a perimeter of 402 cm. Find the length of the semimajor axis if the semiminor axis is 4 cm (1 decimal place).

Find i if the area is 70 cm2 and the radius is 10 cm.

Find the length of the radius is the area is 85 cm2 and 135i = c.

2P a b2

2 2

. r +


Perimeter and Area

24SERIES TOPIC

J 12

Thinking MoreQuestions

3. A semicircle is really just a sector with 180i = c.

180c

r r

a

b

c

Write the formula for the area of a sector.

Use this formula to find the area of a semicircle 180i = c^ h.

Compare this formula to the one given at the beginning of the chapter? Does this make sense?


Perimeter and Area

25SERIES TOPIC

J 12

Questions Thinking More

4. A pizza is cut into 8 equal slices. 3 Slices are eaten.

a

b

What area of the plate is covered by pizza after the 3 slices are eaten? Find to 2 decimal places.

What area of the plate is uncovered after the 3 slices are eaten? Find to 2 decimal places.

3 slices eaten

20 cm


Perimeter and Area

26SERIES TOPIC

J 12

Thinking MoreQuestions

5. A new elliptical engine is attached to a jet plane on a trapezium connector.

The striped sections represent the parts of the engine for which the area is to be calculated. Find the area of the striped regions of the engine.

50 cm

15 cm

80 cm

50 cm

60 cm


Perimeter and Area

27SERIES TOPIC

J 12

Answers

Basics:

Knowing More:

Knowing More:

Using Our Knowledge:

1.

1.

2.

3.

4.

5.

1.

2.

3.

4.

5.

6.

2.

3.

4.

5.

a

a

a

a

a

a

a

a

d

b

b

b

b

e

c

c

f

cm16

cm28

cm39 cm24

cm30

cm28

cm.7 8 cm.31 2

Perimeter m182=

Distance m.733 54=

Perimeter cm3 .857=

cm91 2 cm36 2

cm91 2

cm12.8h =

Area cm.61 44 2=

cmh 12=

Area cm240 2=

x = 20 m

c

Area shaded cm.628 3 2.

Areaunshaded cm.171 7 2.

b

b

b

b

Area cm154 2. Area cm35 2=

ml 3= Area m.23 1 2.

c

d

Areaheart m.9 14 2.

Areaunpainted m.6 86 2=

Diameter = 2m

Height = 3m

i = 45c81b

b

b

b

a

a

a

a

a

i = 288cc

c

c

c

i = 120ce

d54

h61

f31

g

cml 5. cml 23.

cm42l .

i = 36c

cm21 cm43

cm60

Area cm.58 1 2. Area cm.176 7 2.

Area cm.188 5 2.

b

c

a = 25cmb = 10cm

Area cm785 2.

Perimeter cm120.


Perimeter and Area

28SERIES TOPIC

J 12

Answers

Thinking More:

c cmr 8.

a

b

Area of sector cm57 2.1.

2.

3.

4.

5.

a

b

b P a2 2

22.

r-

cm90.b 4.

a Area of sector r360

2r i=c

80.i c

b Area of sector r2

2r=

b Area cm.117 81 2.

a Area cm.196 35 2.

Area striped cm3981.2 2.

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Perimeter and Area

perimeter and area · pdf fileperimeter and area curriculum ready perimeter ... b l b h h a b...

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