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Perimeter and Area

11

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?

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J 112 100% Perimeter & Area


Perimeter and Area Basics

Circumference

cm

cm

r

r

2

. ...

.

r

2 3

2 3 14 3

18 8

#

# #

=

=

=

=

^ 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 r = 3.14 ...

(1 decimal place)(1 decimal place)(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 cm4 cm

Diameter

Circumference = 2rr

Perimeter of semicircle Circumference Diameter

cm cm

cm

r

r

.

r r

21

21 2 2

21 2 4 2 4

20 6

#

#

# # # #

= +

= +

= +

=

`

` ^

^ ^

j

j h

h h

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Perimeter and Area

11

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

r r2 4 2 3

. ...

P

P

P

4 4 6 4

182 2

39 99 40

# #

= + + + + +

= + +

= =

^

` `

h

j j

(nearest cm)

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Perimeter and Area BasicsQuestions

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

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.

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Perimeter and Area

11

Questions Basics

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

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

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

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Perimeter and Area 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

rr

21

21 2

=

=

^ h

Area rr2=

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

#=

=

y

x

y

b

l b

h

h

b

b

a

x

l

h

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Perimeter and Area

11

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 long is 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

r

r

.

A r

6

113 09

2

2

2

#

f

=

=

=

^ 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

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Perimeter and Area 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.

m20

m13

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

m20

m13

Not to scale

h

m

m

m nearest unit

Area of Semicircle

r

r

.

r

A r

24 2 12

21

21 12

226 19

226

2

2

2

2

'

f

.

= =

=

=

=

^

^

h

h

m

Area of Rectangle

A l b

20 24

480 2

#

#

=

=

=

Find using Pythagoras

m

Area of Triangle

h

h

h

A bh

13 12 25

5

21

21 24 5

60

2 2 2

2

`

# #

= - =

=

=

=

=

Area of shape m m m

m

452 480 60

992

2 2 2

2

` = + +

=

m24

m24

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Perimeter and Area

11

Questions Knowing More

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

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

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Perimeter and Area Knowing MoreQuestions

b 5

h

x

x

y

13

20

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

Circle Kitea b

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Perimeter and Area

11


x

m40

The 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)

A sports field has a painted "D" with these measurements

a

b

c

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Perimeter and Area Knowing MoreQuestions

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

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Perimeter and Area

11


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

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Perimeter and Area 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

r

r

41

41 2

2

#

#

=

=

=

Area of Quadrant Area of Circle

Area of Quadrant rr

41

41 2

#

#

=

=

radius2P Arc #= + ^ h

rP r r2

2= +

Area of Quadrant rr4

2

=

arc

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Perimeter and Area

11

Using Our Knowledge

So the arc length of a sector must be 360c

of the circumference of a circle.

Perimeter and Area of Sectors

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

For a full circle, θ = 360c. So a sector with angle θ is 360c

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 360c

of the area of a circle. So to find the area of the sector:

Area of Sector Area of Circle360

#=c

Area of Sector rr360

2#=

c

Arc radiusP 2 #= + ^ h

θ

Arc circumference360

#=c

θ

Arc r2 r360

#=c

θ

r360

2P r r2#= +c

c mθ

θ

θ

θ

θ

θ

r

r

arc

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Perimeter and Area 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 bThis is a Quadrant

This is a sector with θ = 40c

6 cm

4 cm40c

120cr

cm

r

r

.

. .

P r r

P

P

P

22

2

62 6

9 424 12

21 424 21 42

f

f .

= +

= +

= +

=

^^

hh

cm

r

r

r

3602

4 4

. .

P r r

P

P

P

2

36040 2 2

91 8 8

10 792 10 79

#

#

f .

#= +

= +

= +

=

c

cc

c

^ ^

m

h h

cm

r

r

. .

A r

A

A

4

46

28 274 28 27

2

2

2f .

=

=

=

^ h

decimalplace

r

r

r .

. .

. ( )cm

A r

r

r

r

r

360120 214

214120360 642

642 204 354

204 354 14 295

14 3 1

2

2

2

#

` #

` '

`

`

f

f f

.

= =

= =

= =

= =

cc

cc

cm

r

r

r

4

16

. .

A r

A

A

A

360

36040

91

5 585 5 59

2

2

2f .

#

#

#

=

=

=

=

c

cc ^ h

θ

θ

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Perimeter and Area

11

Using Our Knowledge

a

b

c

Perimeter and Area of Ellipses

An Ellipse is a stretched circle.

The "longer axis" is called the semimajor axis. The "shorter axis" is called the semiminor axis.

b is the semiminor axis

a is the semimajor 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 semimajor axis (b) and the semiminor axis (a)?

Find the area of this ellipse to 2 decimal places.

Approximate the perimeter to 2 decimal places

rA ab=

r2P a b2

2 2

. +

cm and cm7 12b a= =

cm decimalplaces

r

r

. . ( )

A ab

A 12 6

226 194 226 19 22f .

=

=

=

^ ^h h

cm decimalplaces

r

r r

. .

P a b22

22

12 6 2 90

59 607 59 61 2

2 2

2 2

f

. +

= + =

= = ^ h

6 cm

12 cm

b

a

ab

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Perimeter and Area Using Our KnowledgeQuestions

360=

c

• Identify θ, the angle inside the shaded sector.

• A sector is 360c

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

For each of the four following sectors

Find the arc length of these sectors to the nearest cm

a b c

θ

53 c130c8 cm

10 cm

9 cm

45c

120c

288c

324c

θ = θ =

θ =θ =

b d

hf

a c

ge

360=

c

360=

c360=

c

θ θ

θθ

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Perimeter and Area

11

Questions Using Our Knowledge

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

Find the area of these sectors to 1 decimal place

a

a

b

b

c

c

11 cm

15 cm

12 cm

55c150c

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Perimeter and Area Using Our KnowledgeQuestions

20 cm

50 cm

a b

Use this ellipse to answer the following questions

a

b

c

Find a and b, the lengths of the semiminor axis and semimajor axis respectively

Find the area of the ellipse to the nearest cm

Find the perimeter of the ellipse to the nearest cm

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Perimeter and Area

11

Thinking More

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

A sector has θ 60= c, and area 32 m2. Find the radius of the sector to the nearest metre.

12 cm

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

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:

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

Write the formula for Area:

Make θ the subject of the formula

Solve θ for using the given values

Working Backwards

Word Problems

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

θ

nearest degree79.577.. 80 ( )= =c c

rA r360

2#=

cθ

r360

r

A2#=

cθ

θ

r 12

360 1002

#=c^ h

θ

cm cm22 14a b= =

θ mA60 32= =c

rA r360

2#=

c

The formula for area of an Ellipse is rA ab=

cmr 22 14 967.61 967.6A 2f .= =^ ^h h

m

r r60

61.115 61

r A 360 32 360# #

f .

#= =

=

cc

c

SERIES TOPIC



Perimeter and Area 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

r

r

.

a

b

A ab

26 3

216 8

3 8

75 398 2f

= = =

= = =

=

=

=

^ ^h h

Area of Triangle

base height

cm

A21

21 5 2

5 2

# #

# #

=

=

=

Area of Sector

cm

r

r

r

2

4

1.396

A r360

36040

91

2

2

2f

#

#

#

=

=

=

=

c

cc ^ h

cm nearest cm

. .

. ( )

75 398 5 1 396

69 001 69 2

f f

f .

= - -

=

θ

SERIES TOPIC



Perimeter and Area

11

Questions Thinking More

Find all these answers to the closest unit

The perimeter of an ellipse is given by

a

a

b

b

c

Find the area of a sector if the radius is 6 cm and θ 180= 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 θ 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 θ 135= c

r2P a b2

2 2

= +

SERIES TOPIC



Perimeter and Area Thinking MoreQuestions

A semicircle is really just a sector with θ 180= 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 180= c^ h

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

θ

SERIES TOPIC



Perimeter and Area

11

Questions Thinking More

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

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Perimeter and Area Thinking MoreQuestions

An engine is attached to an aeroplane on a trapezium connector

The unshaded area represents the holes in the engine. Find the area of the shaded region

40 cm

25 cm

80 cm

25 cm

40 cm

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Perimeter and Area

11

Notes

SERIES TOPIC



Perimeter and Area Notes

perimeter - wikispacesmathematicsyear8.wikispaces.com/file/view/j... · ^ h perimeter cm 4 4 7 7 2...

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