2535 18923329.nzl h area and perimeter nzl
TRANSCRIPT
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PASSPO
RT
PASSPO
RT
AreaandPerimeter
AREA ANDAREA AND
PERIMETERPERIMETER
www.mathlecs.co.nz
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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 dierent from what you expected.
This booklet shows how to calculate the area and perimeter of common plane shapes.
Football elds use rectangles, circles, quadrants and minor segments with specic areas and perimeters
to mark out the playing eld.
Write down the name of another sport that uses a playing eld 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:
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Area and Perimeter
Mathletics Passport 3P Learning
2 9HSERIES TOPIC
Howdoes it work?
Area using unit squares
Calculate the area of these shapes
Area and Perimeter
Area is just the amount of at 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 counng 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
22
1
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
Lile dashes on each
side mean they are all
the same length.
1unit
1 cm
1 unit
1unit
Area ( ) whole squares 2 half squares quarter squares
square cm square cm square cm
square centimetres
cm
A
.
.
2 2
2 22
12
4
1
2 1 0 5
3 52
# #
= + +
= + +
= + +
=
^ h
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Area and Perimeter
Mathletics Passport 3P Learning
39HSERIES TOPIC
How does it work? Area and PerimeterYour Turn
Area using unit squares
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
+ 41
# units2
= units2
Area = whole + half+ quarter squares
= units2+2
1# units2 +
4
1# units2
= units2
Area = whole + quarter squares
= cm2
+ 41
# cm2
= cm2
Area = whole + half squares
= units2+2
1# units2
= units2
Area = whole squares
= mm2
1unit
1unit
1m
1unit
1 unit 1mm
1cm
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Area and Perimeter
Mathletics Passport 3P Learning
4 9HSERIES TOPIC
How does it work? Area and PerimeterYour Turn
Calculate the area of these shaded shapes, using the correct short version for the units:
Area using unit squares
2
Area = Area =
Area = Area =
Area = Area =
Area = Area =
a
c
e
g
d
f
h
b1cm
1 m
1 unit
1km
1cm
1mm
1mm
1unit
...../...../20....
AREAUSINGUNITS
QUARES*AR
EAU
SING
UNIT
SQUA
RES*
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Area and Perimeter
Mathletics Passport 3P Learning
59HSERIES TOPIC
How does it work? Area and PerimeterYour Turn
Shade shapes on these square grids to match the area wrien in square brackets.3
4
Number of dierent designs you found =
a
c
b
d
Area using unit squares
units8 26 @, using whole squares only.
mm326 @, include quarter squares in
your shape.
units5 26 @, include half squares in your shape.
cm4.526 @, include halves and quarters.
An arst has eight, 1m2, square-shaped panels which he can use to make a paern.
The rules for the design are:
- the shape formed cannot have any gaps/holes.
i.e. or
- it must t enrely inside the display panel shown,
- all the eight panels must be used in each design.
How many dierent designs can you come up with?
Sketch the main shapes to help you remember your count.
1unit
1mm 1cm
1 m
1 unit
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Area and Perimeter
Mathletics Passport 3P Learning
6 9HSERIES TOPIC
How does it work? Area and Perimeter
Perimeter using unit squares
The word perimeter is a combinaon of two Greek words peri (around) and meter (measure).
So nding the perimeter (P) means measuring the distance around the outside!
Start/end of path around the outside
1unit
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, lile 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 maer where you start/nish, but it is usually easiest to start from one corner.
Start/end of path around the outside
Start/end of path around the outside
These examples shows that we only count all the outside edges.
Sides of unit squares inside the shape not included
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Area and Perimeter
Mathletics Passport 3P Learning
79HSERIES TOPIC
How does it work? Area and PerimeterYour Turn
Perimeter using unit squares
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 of6 units2.
Use your results in queson 2 to help you explain briey whether or not all shapes with the
same area have the same perimeter.
...../...../20....
PERIMETERU
SING
UNIT
S
QUARES*PERIMETERUSING
UNIT
SQUAR
ES*
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Area and Perimeter
Mathletics Passport 3P Learning
8 9HSERIES TOPIC
How does it work? Area and PerimeterYour Turn
Perimeter using unit squares
4 Draw six paerns on the grid below which:
Draw another ve paerns on the grid below which:
a
b
all have an area of5units2 and,
have a dierent perimeter from each other.
all have an area of5units2 and,
have a dierent perimeter than the shapes formed in part a .
All squares used for each paern must share at least one common side or corner point .
All squares used for each paern must share at least halfof a common side
ora corner
point .
1unit
1unit
1unit
1unit
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Area and Perimeter
Mathletics Passport 3P Learning
99HSERIES TOPIC
Area and PerimeterWhere does it work?
Area: Squares and rectangles
Calculate the area of these shaded shapes
A simple mulplicaon will let you calculate the area of squares and rectangles.
For squares and rectangles, just mulply 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
92
#
#
=
=
=
Area length width
m cm
cm cm
cm
2 60
200 60
12 0002
#
#
#
=
=
=
=
(i)
(ii)
(iii)
All measurements (or dimensions) must be wrien in the same units before calculang 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
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Area and Perimeter
Mathletics Passport 3P Learning
10 9HSERIES TOPIC
Area and PerimeterWhere does it work? Your Turn
Area: Squares and rectangles
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 of121m2?
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 calculang the square root.
1.4 km
2 units
2 units
3
units
3.2 mm
7 cm
43 mm
5
mm
0.6 m
* AREA: SQUARESAND
RE
CTAN
GLE
S
*AREA:SQUARESAND
RE
CTAN
GLE
S
...../...../20....
4 unitsArea =28units
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Area and Perimeter
Mathletics Passport 3P Learning
119HSERIES TOPIC
Area and PerimeterWhere does it work?
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 h2
1 2# #
This rule works to nd 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
2
13 4
62
# #
# #
=
=
=
Area base height
mm mm
mm
. 4
.
2
1
21 5 4
10 82
# #
# #
=
=
=
Area base height
units units
units
.
.
2
1
2
11 5 2
1 52
# #
# #
=
=
=
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 sll mulply 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.
6m
1.5units
2units
5.4mm
5m4 m
4 mm
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Area and Perimeter
Mathletics Passport 3P Learning
12 9HSERIES TOPIC
Area and PerimeterWhere does it work? Your Turn
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 =2
1 # # units2
= units2
Area = # # mm2
= mm2
Area = # # units2
= units2
Area = # # m2
= m2
Area = # # cm2
= cm2
Area = # # m2
= m2
Area =2
1 # # units2
= units2
base baseheight height
a
a
c
e
b
b
d
2 units
4 units8 units
12mm
600cm
4 m5m
14cm
8mm
7.5units 4.5m
10units
12cm
Remember:
same units
needed.
*AREA:TR
IANGLES
*A
REA:TRIAN
GLES
*AR
EA:TRIANG
LES
...../...../20....
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Area and Perimeter
Mathletics Passport 3P Learning
139HSERIES TOPIC
Area and PerimeterWhere does it work?
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 distancebetween a pair of
parallel sides is called
the perpendicular height
We can make them look like a rectangle by cung the triangle o one end and moving it to the other.
height
Perpendicular
height (h)
` Area of a parallelogram = Area of the rectangle formed aer moving triangle
= length # perpendicular height units2
=l#h units2
Parallelogram Rectanglemove triangle cut o
Calculate the area of these parallelograms
A parallelogram can also be formed joining together two idencal triangles.
Area length height
mm mm
mm
30 15
4502
#
#
=
=
=
Area length perpendicular height
m m
m
5 12
602
#
#
=
=
=
Area area of the triangle
m m
m
2 5 12
2
2
1
602
#
# # #
=
=
=
(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 ip both
vercally and horizontallyBring them together Parallelogram
20 mm
30 mm
15 mm
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Area and Perimeter
Mathletics Passport 3P Learning
14 9HSERIES TOPIC
Area and PerimeterWhere does it work? Your Turn
Area: Parallelogram
1
2
3
Complete the area calculaons for these parallelograms:
Calculate the area of the parallelograms formed using these triangles.
Fill the grid below with as many diferent parallelograms as you can which have an area of4 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 cm2.2cm
26 m1.6 mm10 m 2 mm
24 m1.2 mm
*AREA:PARAL
LELO
GR
AM*AREA:P
ARALLELOGRA
M
...../...../20....
1unit
1unit
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Area and Perimeter
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159HSERIES TOPIC
Area and PerimeterWhere does it work?
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 nd 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 =2
1 #2 cm #8cm =8cm2
Area 1 =4.5m #3.5m =15.75m2
Area 1 =8m #7m =56m2
Area 2 =8cm #8cm =64cm2
Area 2 =3.5m #7cm =24.5m2
Area 2 =3.5m #4.5m =15.75m2
` Total area = Area 1 + Area 2
=8cm2 +64cm2
=72cm2
` Total area =15.75m2 +24.5m2
=40.25m2
` Total area =56m2 -15.75m2
=40.25m2
This next one shows how you can use addion or subtracon 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
1
2
8 m
7 m
3.5 m
4.5m
4.5m
3.5m
8m
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 pung
together separate parts
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Area and Perimeter
Mathletics Passport 3P Learning
16 9HSERIES TOPIC
Area and PerimeterWhere does it work? Your Turn
Area of composite shapes
1 Complete the area calculaons for these shaded shapes:
a
b
c
d
1
2
6mm
4m
m
2m
m
5m
5m
4m
m
6m
6m
3m
3m
11m
11m
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
2
2c
m
2.5cm
6.5cm
2c
m
2.5cm
6.5cm
3 m
3 m
5m
4cm
5m
2m
2m
1
2
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Area and Perimeter
Mathletics Passport 3P Learning
179HSERIES TOPIC
Area and PerimeterWhere does it work? Your Turn
Area of composite shapes
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
200cm
4.5m
300cm
2 mm
5 units10u
nits
6units
psst: this one needs three area calculaons
AREA OF COMPOSITESHAPE
S
*
AREAOFCOMPOSITESHAPE
S
*
...../...../20....
psst: change all the units to metres rst.
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Area and Perimeter
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18 9HSERIES TOPIC
Area and PerimeterWhere does it work?
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
(zunits)
side 1 (x units)
RectangleSquareTriangle
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
# #
= + + +
= + + +
= +
= +
^
^ ^
h
h 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 calculang 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 1040
# #= +
= +
=
15mm15mm
5mm 5mm0.5 cm
15mm
Sum of all the side lengths
Four lots of the same side length
All side lengths in mm
Opposite sides in pairs
Start/nish
Start/nish
Start/nish
Start/nish
Start/nish
Start/nish
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Area and Perimeter
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199HSERIES TOPIC
Area and PerimeterWhere does it work? Your Turn
Perimeter of simple shapes
15 units8 units
17 units
1
2
a
a
b
b
c
c
d
d
Complete the perimeter calculaons 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
9mm
6mm
11cm
5 m
2.4mm
5.8cm
15m
1.6mm 5m3m
5 m
1.6m 2.4m
3.4 m
PERIMETEROF SIMPLE SHAPES *
.....
/.....
/20....
PERIMETEROFSIMPLESHAPES*
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Area and Perimeter
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219HSERIES TOPIC
Area and PerimeterWhere does it work?
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
3.5m 3.5m
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-posioned to makethe calculaon easier
Start/nish
The lengths of the unlabelled sides must be found in composite shapes before calculang their perimeter.
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Area and Perimeter
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22 9HSERIES TOPIC
Area and PerimeterWhere does it work? Your Turn
Perimeter of composite shapes
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
13m
8cm
2cm
2.2m
m
1mm
b
b
ba
a
a
1.6mm
3.4 mm
18m
15m
b
a
5cm
14cm
15cm
8cm
4.8cm
9.8mm
4.1cm
38mm
2m
1.2cm
16mm
Perimeter = # cm
= cm
Perimeter =2#am +2# m
= m + m
= m
Perimeter =3# mm + mm + mm
= mm
Perimeter = #4.1cm + # cm
= cm
4 m
20mm
am
Be careful with the units for these next two
PERIMETEROFCOMP
OSITE
SHAP
ES
*PERIME
TERO
FC
OMPOSITE
SHAPES*
312
...../...../20....
Calculate the value of the sides labelled a andb in each of these composite shapes:
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Mathletics Passport 3P Learning
239HSERIES TOPIC
Area and PerimeterWhere does it work? Your Turn
Perimeter of composite shapes
3
4
Calculate the perimeter of these composite shapes in the units given in square brackets.
Show all working.
4 mm
48 mm
3.6c
m1200m
1.5km
a
c
b
d
mm6 @
cm6 @
m6 @
km6 @
2.2m
Perimeter = mm
Perimeter = cm
Perimeter = m
Perimeter = km
Total perimeter of completed path = m
22m
m
Earn an awesome passport stamp for this one!
The incomplete geometric path shown below is being constructed using a combinaon of
the following shaped pavers:
The gap in between each part of the spiral path is always 1m wide.
Calculate what the total perimeter of this path will be when nished.
1m
1.41m1m
2m
2m
2m
1.41m
1m
1m 1mCompleted path
AWESOMEAWESOMEAWESOME
AWES
OMEA
WESO
MEAWE
SOME
psst: 1km =1000 m
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Area and Perimeter
Mathletics Passport 3P Learning
24 9HSERIES TOPIC
Area and PerimeterWhere does it work? Your Turn
Perimeter of composite shapes
5 The four composite shapes below have been formed using ve, unit squares.
a
b
Using your knowledge of perimeter and the grid below, combine all four pieces to create two
dierent 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.
1unit
1unit
Briey describe the strategy you used to achieve each outcome below:
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Area and Perimeter
Mathletics Passport 3P Learning
259HSERIES TOPIC
Area and PerimeterWhere does it work?
Simple word problems involving area and perimeter
Somemes we can only communicate ideas or problems through words.
So it is important to be able to take wrien/spoken informaon and turn it into something useful.
For example,
Miguel wants to paint a square. He has just enough paint to create a line 240cm 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 240cm.
So each side length cm
cm
240 4
60
'=
=
` The longest length each side of the square painted by Miguel can be is 60cm.
This is useful for Miguel to know because if he painted the rst side too long, he would run out of paint!
Here are some more examples
(i)
(ii)
A rectangular park is four mes longer than it is wide. If the park is 90m long, how much area
does this park cover?
At a fun run, competors run straight for 0.9km before turning le 90 degrees to run straight for a
further 1.2 km. The course has one nal corner which leads back to the start along a straight 1.5km
long street. How many laps of this course do competors complete if they run a total of18km?
90m
m m22.590 4' =^ h
Area length width
m m
m
90 22.5
20252
#
#=
=
=
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
1.5km 0.9km
1.2km
Start/nish
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 3.6km
` Competors must complete 5 laps of the course to nish
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Area and Perimeter
Mathletics Passport 3P Learning
26 9HSERIES TOPIC
Area and PerimeterWhere does it work? Your Turn
Simple word problems involving area and perimeter
1
2
3
Three equilateral triangles, each with sides of length 3cm 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 h of its height. If the base of this triangle is
4.2m, 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 relaonship between
area and perimeter.
You have been employed by a fabric design company called Double Geometrics. Your rst task
as a paern maker is to design the following using all seven idencal squares:
Closed shapes for a new paern in which the value of their perimeter is twice the value of theirarea. Draw ve possible dierent paerns that match this design request.
SIMPLE WORD
PROBLEMS
INVOLVING
AREA AND
PERIMETER
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Area and Perimeter
Mathletics Passport 3P Learning
279HSERIES TOPIC
Area and PerimeterWhere does it work? Your Turn
a
b
c
d
Simple word problems involving area and perimeter
4 An architect is asked to design an art gallery building. One of the design rules is that the oor must
be a rectangle shape with an area of64m2.
Another design rule is to try ensure a large perimeter so there is more space to hang painngs
from. Use calculaons to show which oor plan will have the largest perimeter.
Would the design with the largest possible perimeter be a good choice?
Explain briey 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 12.5cm wide and 24.5cm long if they all t perfectly without any edges overlapping?
3.5cm
2.5cm
12.5cm
24.5cm
If only whole metre measurements can be used, sketch all the dierent possible oor dimensions.
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Area and Perimeter
Mathletics Passport 3P Learning
28 9HSERIES TOPIC
Area and PerimeterWhere does it work? Your Turn
Simple word problems involving area and perimeter
5
6
A fence used to close o a parallelogram-shaped area is being rearranged to create a square area
with the same perimeter. The short side of the area is 34m 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 aer using the whole length of this fence?
A 500mL n of white paint has been purchased to paint the wall. The instrucons on the
paint n say this is enough to cover an area of11 500cm2.
If the distance between the longer sides of the original area was 30m and the length did not
change, use calculaons to show which fencing arrangement surrounded the largest area.
Use calculaons to show that there is enough paint in the n to cover side of the wall.
28cm
16cmEach brick =
34 m
If a beetle walked all around the outside of the wall (including along the ground), how many
metres did it walk?
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Area and Perimeter
Mathletics Passport 3P Learning
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 9352
#
'
'
=
=
=
^
^
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
nd the area:
Here are some examples:
(i)
(ii)
For this rhombus, WY=12cm andXZ=16cm.
For the kite ABCD shown below,AC=4.7m andBD=2.1m.
length
height
B
X
C
Y
A
W
D
Z
B
B
C
C
A
A
D
D
10cm
3.7m
1.5m
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Area and Perimeter
Mathletics Passport 3P Learning
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=18cm and QS=52cm BD=1.8mm andAC=2.4mm
Rhombus and Kite shapes
Q
S
P R
Area = # ' cm2
= cm2
Area = # #
2
1
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:
3.4cm
5.1cm
3.6mm
41 cm
15cm
6.5cm
HL=30m,IK=IM=16m and JL=21m
14m
9m
J
K
LM
HI
22
1#' =
RHOMBUSANDKITE
SHAP
ES
RHOMBUSAN
D
KITE
SHAPES
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Area and Perimeter
Mathletics Passport 3P Learning
319HSERIES TOPIC
Area and PerimeterWhat else can you do?
10.1m
22mm10mm
20mmArea sum of theparallel sides height
mm mm mm
mm mm
mm
16 2
32 16 2
2
22 10
2562
#
#
#
'
'
'
=
= +
=
=
^
^
h
h
Area sum of theparallel sides height
m m m
m m
m
. .
.
.
2
6 7 14 5 2 2
21 2 2 2
21 22
#
#
#
'
'
'
=
= +
=
=
^
^
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)
16mm
6.7m
14.5m
2.9m2m
Trapeziums
B Baa
bbD D
A A
C C
A trapezium is a quadrilateral which has at least one pair of parallel sides.
Squares, rectangles, parallelograms and rhombi are all just special types of trapeziums.
So the area formula for a trapezium would also work on all of those shapes.
Two common trapezium 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 sum of the parallelsides height 2
2a b h#
: '
'
#=
= +
^
^
h
h
Perimeter AB BD CD AC : = + + +
height (h) height (h)
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Area and Perimeter
Mathletics Passport 3P Learning
32 9HSERIES TOPIC
Area and PerimeterYour TurnWhat else can you do?
a
a b
b
1
2
3
41km8.2m
15km1.8m
1km
12.5m
9km
53km 4.5m
Area = + # '2km2
= km2
Area = cm2
Area = m2
Area = mm2
Perimeter = m
Area = + # '2m2
= m2
Perimeter = km Perimeter = m
5cm
15cm
16.7 m
240cm
33.4 m
8cm
1mm 2.4mm 14.3 mm
170 cm
1 2
Trapeziums
Calculate the area and perimeter of these trapeziums:
Use the trapezium method to calculate the area of these composite plane shapes.
Use the trapezium method to calculate the area of this composite plane shapes.
TRAPEZIUMS*
TRAPE
ZIUMS*TRA
PEZIUMS
*TRAPEZ
IUMS*
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Area and Perimeter
Mathletics Passport 3P Learning
339HSERIES TOPIC
Area and PerimeterYour TurnWhat else can you do?
Area challenge
1unit
1unit
Fill the grid below with as many dierent squares, triangles, rectangles, parallelograms, rhombi, kites
and trapeziums as you can which all have the same area of8 units2.
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Area and Perimeter
Mathletics Passport 3P Learning
34 9HSERIES TOPIC
Area and PerimeterYour TurnWhat else can you do?
Reflection Time
Reecng 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 calculaons to a real
life situaon.
If you discovered or learnt about any shortcuts to help with calculang area and perimeter or some
other cool facts/conversions, jot them down here:
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Area and Perimeter
Mathletics Passport 3P Learning
359HSERIES TOPIC
Cheat Sheet Area and Perimeter
Here is what you need to remember from this topic on Area and perimeter
Area using unit squares
Perimeter using unit squares
Area: Squares and rectangles
Area: Triangles
Area: Parallelograms
Area is just the amount of at 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 fracons of squares to calculate the area.
The perimeter with unit squares means count the number of edges around the outside of the shape.
Just mulply 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
unitsxy2
#=
=
height height
base base` Area of the triangle = (half the base mulplied by the perpendicular height) units2
= unitsb h2
1 2# #
Perpendicular
height (h)
` Area of a parallelogram =length # perpendicular height units2=l#h units2
height
base
side (x)
side (x)
Area length width
unitsx2 2
#=
=
Length (l)
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Area and Perimeter
Mathletics Passport 3P Learning
36 9HSERIES TOPIC
Cheat Sheet Area and Perimeter
Area of composite shapes
Perimeter of simple shapes
Perimeter of composite shapes
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)
RectangleSquareTriangle
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 3m
7 m 7 m
? m (3 + 3.5) m =6.5 m
(7 - 2) m =5 m? m
3.5m 3.5m
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 AreaPerimeter
( ) 2AC BD
AB AD2 2
#
# #
'=
= +
Area
Perimeter
( ) 2a b h
AB BD CD AC
# '= +
= + + +
Start/nish
Start/nish
Start/nish
Start/nish
perpendicular
height (h)
perpendicular
height (h)
The lengths of all unlabelled sides must be found in composite shapes before calculang their perimeter.
It is easier to add them together if the lengths are all in the same units.
Rhombus, Kites and Trapeziums
Trapezium
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...
PERIM
ETERUSINGU
NITSQUARES
*PERIMETER
USIN
GUN
ITSQ
UARES*
*ARE
A:PARALLELOGRA
LLEL
OGRAM
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AREAOFCOMPOSITE
SHAPES
*AR
EAO
FCO
MPOS
ITESHA
PES
*
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SIMPLE WORD
PROBLEMS
INVOLVING
AREA AND
PERIMETER
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PERIMETEROF SIMPLE SHAPES *
.....
/.....
/20....
PERIMETEROFSIMPLESHAPES*
* AREA: SQUARESAND
RE
CTAN
GLE
S
*AREA:SQUARESAND
RE
CTAN
GLE
S
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