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Peri
met
er &
Are
a PERIMETER& AREA
www.mathletics.co.nz
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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?
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2SERIES TOPIC
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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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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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4SERIES TOPIC
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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.
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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
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6SERIES TOPIC
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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
h
x
l
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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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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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9SERIES TOPIC
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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
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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 cm.
Circle Kitea b
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Questions Knowing More
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)
4. A sports field has a painted "D" with these measurements
a
b
c
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12SERIES TOPIC
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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
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13SERIES TOPIC
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Questions Knowing More
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
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14SERIES TOPIC
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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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15SERIES TOPIC
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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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16SERIES TOPIC
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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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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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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.
1. For each of the four following sectors
2. 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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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
11 cm
15 cm
12 cm
55c150c
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Using Our KnowledgeQuestions
20 cm
50 cm
a b
5. 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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21SERIES TOPIC
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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
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22SERIES TOPIC
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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
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Questions Thinking More
1. Find all these answers to the closest unit
2. 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
= +
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Thinking MoreQuestions
3. 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
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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
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Thinking MoreQuestions
5. 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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27SERIES TOPIC
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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
cm.38 96 cm24
cm30
cm28
cm.7 8 cm.31 2
Perimeter m182=
Distance m.733 54=
Perimeter cm32.85=
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
θ = 45c81b
b
b
b
a
a
a
a
a
θ = 288cc
c
c
c
θ = 120ce
d54
h61
f31
g
cml 5. cml 23.
cm42l .
θ = 36c
cm21 cm43
cm60
Area cm.58 1 2. Area cm.176 7 2.
Area cm.188 5 2.
b
c
a = 50cmb = 20cm
Area cm.3141 6 2.
Perimeter cm.239 3.
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Answers
Thinking More:
c cmr 8.
a
b
Area of sector cm57 2.1.
2.
3.
4.
5.
a
b
rb P a
4
22
22
= -
cm.b 90 5.
a Area of sector rr360
2=
cθ
80. cθ
b Area of sector rr2
2
=
b Area cm.117 81 2.
a Area cm.196 35 2.
Area shaded m.0 75 2.