Area and Perimeter
Area and
Perimeter
Curriculum Ready
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First edition printed 2009 in Australia.
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This booklet shows how to calculate the area and perimeter of common plane shapes.
Football fields use rectangles, circles, quadrants and minor segments with specific areas and perimeters
to mark out the playing field.
Write down the name of another sport that uses a playing field or court and list all the plane shapes
used to create them below (include a small sketch to help you out):
Sport:
Shapes list:
Q
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 different from what you expected.
Work through the book for a great way to do this
Area and Perimeter
Mathletics
©3P Learning Ltd
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How does it work?
Area and Perimeter
Area using unit squares
Area is the amount of flat space a shape has inside its edges or boundaries.
A unit square is a square with each side exactly one unit of measurement long.
1 unit
Little dashes on each
side mean they are all
the same length.
Area (A) = 1 square unit
= 1 unit2 (in shorter, units form)
So the area of the shaded shape below is found by simply counting the number of unit squares that make it.
1 unit
1
2
3
4
Area (A) = 10 square units
5
6
7
8
= 10 unit2
9
10
Here are some examples including halves and quarters of unit squares:
Calculate the area of these shapes
(i)
Area (A) = 2 whole square units + 2 half square units
= 2 square units + 2 # 1 square units
2
= ^2 + 1h square units
1 unit
= 3 units2
When single units of measurement are given, they are used instead of the word ‘units’.
(ii)
1 cm
2
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Area (A) = 2 whole squares + 2 half squares + 2 quarter squares
1 square cm + 2 # 1 square cm
= 2 square cm + 2 # 2
4
= ^2 + 1 + 0.5h square centimeters
= 3.5 cm 2
Area and Perimeter
Mathletics
©3P Learning Ltd
How does it work?
Your Turn
Area and Perimeter
Area using unit squares
1
Calculate the area of all these shaded shapes:
a
b
1 unit
Area =
whole squares
1 mm
Area =
units2
=
whole squares
mm2
=
1 unit
c
d
1 m
Area =
whole +
m2 +
=
half squares
#
Area =
whole +
half squares
=
units2 +
#
1 m2
2
m2
=
units2
=
e
1 units2
2
f
1 cm
1 unit
Area =
whole +
quarter squares
=
units2 +
#
Area =
1 units2
4
cm2 +
=
units2
=
whole +
quarter squares
#
1 cm2
4
cm2
=
g
1 unit
Area =
whole +
half +
=
units2 +
#
=
1 units2 +
2
#
1 units2
4
units2
Area and Perimeter
Mathletics
quarter squares
©3P Learning Ltd
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How does it work?
Your Turn
Area and Perimeter
Area using unit squares
2
Calculate the area of these shaded shapes, using the correct short version for the units:
a
b
1 cm
1 unit
Area =
Area =
c
d
1 mm
1 m
Area =
Area =
e
f
1 mm
1 unit
Area =
Area =
g
h
1 km
IN G
US
UN
ING IT SQ
US
R E S * AR E A
UA
UNI T
SQ
...../...../20....
R
UA
E S * AR E
A
1 cm
Area =
4
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Area =
Area and Perimeter
Mathletics
©3P Learning Ltd
How does it work?
Your Turn
Area and Perimeter
Area using unit squares
3
Shade shapes on these square grids to match the area written in square brackets.
a
68 units2 @, using whole squares only.
b
1 unit
c
1 unit
63 mm2 @, include quarter squares in
d
your shape.
1 mm
4
65 units2 @, include half squares in your shape.
64.5 cm2 @, include halves and quarters.
1 cm
An artist has eight, 1 m2, square-shaped panels which he can use to make a pattern.
The rules for the design are:
- the shape formed cannot have any gaps/holes.
i.e.
or
1 m
- it must fit entirely inside the display panel shown,
- all the eight panels must be used in each design.
How many different designs can you come up with?
Sketch the main shapes to help you remember your count.
Number of different designs you found =
Area and Perimeter
Mathletics
©3P Learning Ltd
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How does it work?
Area and Perimeter
Perimeter using unit squares
The word perimeter is a combination of two Greek words peri (around) and meter (measure).
Finding the perimeter (P) means measuring the distance around the outside!
Start/end of path around the outside
1 unit
Perimeter (P) = 1 unit + 1 unit + 1 unit + 1 unit
= 4 # 1 unit
Remember, little dashes on
each side mean they are all
the same length.
= 4 units
These examples shows that we only count all the outside edges.
Calculate the perimeter of these shapes formed using unit shapes
(i)
Start/end of path around the outside
2 units
1 unit
1 unit
2 units
Perimeter (P) = 1 + 2 + 1 + 2 units
Sides of unit squares inside the shape not included
= 6 units
It does not matter where you start/finish, but it is usually easiest to start from one corner.
(ii)
3 units
1 unit
1 unit
1 unit
1 unit
1 unit
1 unit
1 unit
1 unit
Start/end of path around the outside
Perimeter (P) = 1 + 1 + 1 + 1 + 3 + 1 + 1 + 1 units
= 10 units
6
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Area and Perimeter
Mathletics
©3P Learning Ltd
Area and Perimeter
a
1 unit
Perimeter =
Perimeter =
+
units
+
+
+
+
units
+
units
=
Write the length of the perimeter (P) for each of these shaded shapes:
a
b
P =
3
+
units
=
1 unit
units
+
1 unit
c
+
...../...../20....
units
Perimeter =
b
+
+
=
2
S I NG U N I
RU
T
TE
Calculate the perimeter of these shaded shapes:
RES * PERI
ME
1
R E S * P E RI
ME
S ING U N I
RU
T
Perimeter using unit squares
UA
TE
SQ
UA
Your Turn
SQ
How does it work?
units
c
P =
units
d
P =
units
P =
units
The shaded shapes in 2 all have the same area of 6 units2.
Use your results in question 2 to help you explain briefly whether or not all shapes with the
same area have the same perimeter.
Area and Perimeter
Mathletics
©3P Learning Ltd
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How does it work?
Your Turn
Area and Perimeter
Perimeter using unit squares
4
a
Draw six patterns on the grid below which:
• all have an area of 5 units2 and,
• have a different perimeter from each other.
All squares used for each pattern must share at least one common side
or corner point
1 unit
1 unit
b
Draw another five patterns on the grid below which:
• all have an area of 5 units2 and,
• have a different perimeter than the shapes formed in part a .
All squares used for each pattern must share at least half of a common side
point
.
1 unit
1 unit
8
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Area and Perimeter
Mathletics
©3P Learning Ltd
or a corner
.
Where does it work?
Area and Perimeter
Area: Squares and rectangles
A simple multiplication will let you calculate the area of squares and rectangles.
For squares and rectangles, just multiply the length of the perpendicular sides (Length and width).
Length
Width
Square
Length
Width
Side (x)
Rectangle
Side (x)
Side (y)
Side (x)
Area = length # width
Area = length # width
= Side ^ xh units # Side^ xh units
= Side ^ xh units # Side^ yh units
= x # x units2
= x # y units2
= x2 units2
= xy units2
Here are some examples involving numerical lengths:
Calculate the area of these shaded shapes
(i)
Area = length # width
= 4 units # 4 units
= 42 units2
4 units
= 16 units2
So why units squared for area?
4 units # 4 units = 4 # 4 units # units
= 42 # units2
= 16 units2
Area = length # width
(ii)
1.5 mm
= 6 mm # 1.5 mm
= 9 mm2
Units of area match units of side length
6 mm
All measurements (or dimensions) must be written in the same units before calculating the area.
Area = length # width
(iii)
60 cm
2m
= 2 m # 60 cm
= 200 cm # 60 cm
Write both lengths using the same unit
= 12 000 cm2
Units of area match units of side length
Area and Perimeter
Mathletics
©3P Learning Ltd
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Where does it work?
Your Turn
Area and Perimeter
Area: Squares and rectangles
Calculate the area of these squares and rectangles, answering using the appropriate units.
a
b
Area =
length
d
width
mm2
3.2 mm
Calculate the area of these squares and rectangles. Round your answers to nearest whole square unit.
a
b
Area =
km2
#
length
.
Area =
width
=
km2 (to nearest whole km2)
.
What is the length of this rectangle?
Area = 28 units
4 units
What are the dimensions of a square with an area of 121 m2?
Psst: remember the opposite of squaring numbers is calculating the square root.
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Area and Perimeter
Mathletics
©3P Learning Ltd
...../...../20....
cm2
#
length
km2
=
4
7 cm
* AREA: SQUARES
1.4 km
AND RECTANGLES
* AREA: SQUARES
43 mm
3
width
=
2 units
2
mm2
#
length
units2
=
m2
Area =
5 mm
length
width
=
0.6 m
units2
#
m2
#
length
units2
Area =
3 units
Area =
width
=
2 units
c
units2
#
AND RECTANGLES
1
width
cm2
cm2 (to nearest whole cm2)
Where does it work?
Area and Perimeter
Area: Triangles
Look at this triangle drawn inside a rectangle.
Height
(Length)
Base (width)
The triangle is exactly half the size of the rectangle
` Area of the triangle = half the area of the rectangle units2
= 1 of width (base (b) for a triangle) # Length (height (h) for a triangle) units2
2
= 1 # b # h units2
2
This rule works to find the area for all triangles!
Here are some examples involving numerical dimensions:
Calculate the area of the shaded triangles below
(i)
4 m
5 m
Area = 1
2
= 1
2
# base # height
#3m #4m
Height = use the perpendicular height
= 6 m2
6 m
The rule also works for this next triangle which is just the halves of two rectangles combined.
Area = 1
2
4 mm
= 1
2
(ii)
# base # height
# 5.4 mm # 4 mm
Here, we say the height
is the perpendicular
distance of the third
vertex from the base.
= 10.8 mm2
5.4 mm
For unusual triangles like this shaded one, we still multiply the base and the perpendicular height
and halve it.
(iii)
Area = 1 # base # height
2
= 1 # 1.5 units # 2 units
2 units
2
1.5 units
= 1.5 units2
Area and Perimeter
Mathletics
©3P Learning Ltd
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Area and Perimeter
* A REA
S
LE
a
b
4 units
...../...../20....
REA: TRIANG
Calculate the area of the triangle that cuts these two shapes in half.
G LE S * A R E
A
:
IANGL
TR
:
Area: Triangles
1
IA N
TR
*A
Your Turn
ES
Where does it work?
8 units
2 units
Area =
1
2
#
base
height
#
units2
#
base
units2
=
2
Area =
1
2
units2
#
height
units2
=
Calculate the area of these shaded triangles:
a
b
14 cm
12 cm
8 mm
12 mm
Area =
#
mm2
#
Area =
mm2
=
#
d
7.5 units
4.5 m
10 units
Area =
#
600 cm
#
units2
Area =
units2
=
#
Area =
4 m
#
#
5 m
m2
=
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Area and Perimeter
Mathletics
©3P Learning Ltd
#
m2
=
e
12
cm2
cm2
=
c
#
m2
Remember:
same units
needed.
m2
Where does it work?
Area and Perimeter
Area: Parallelograms
Parallelograms have opposite sides equal in length and parallel (always the same distance apart).
Perpendicular
height (h)
The shortest distance
between a pair of
parallel sides is called
the perpendicular height
We can make them look like a rectangle by cutting the triangle off one end and moving it to the other.
height
Parallelogram
Rectangle
move triangle cut off
` Area of a parallelogram =
Area of the rectangle formed after moving triangle
= length # perpendicular height units2
= l # h units2
Calculate the area of these parallelograms
(i)
Area = length # height
20 mm
= 30 mm # 15 mm
15 mm
= 450 mm2
30 mm
A parallelogram can also be formed joining together two identical triangles.
13 m
12 m
(ii) Find the area of the parallelogram formed using two of these right angled triangles:
13 m
5m
12 m
12 m
13 m
13 m
Copy and flip both
vertically and horizontally
12 m
12 m
5m
Parallelogram
Area = length # perpendicular height
OR
# 5 m # 12 m
13 m
5m
Bring them together
Area = 2 # area of the triangle
= 2#1
2
5m
5m
5m
= 5 m # 12 m
= 60 m2
= 60 m2
Area and Perimeter
Mathletics
©3P Learning Ltd
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Your Turn
Area and Perimeter
G
RALLELO
Complete the area calculations for these parallelograms:
PA
M * A R EA
:
b
10 units
Area =
units2
Area =
height
cm2
=
b
26 m
10 m
2 mm
24 m
Area =
1.6 mm
1.2 mm
m2
Area =
mm2
Fill the grid below with as many different parallelograms as you can which have an area of 4 units2.
1 unit
1 unit
14
height
Calculate the area of the parallelograms formed using these triangles.
a
3
cm2
#
length
units2
=
2
4.6 cm
#
length
3.9 cm
2.2 cm
4.5 mm
G
a
...../...../20....
RALLELO
1
PA
Area: Parallelograms
M * A R EA
:
RA
RA
Where does it work?
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Area and Perimeter
Mathletics
©3P Learning Ltd
Where does it work?
Area and Perimeter
Area of composite shapes
When common shapes are put together, the new shape made is called a composite shape.
Common shape
(Rectangle)
+
Common shape =
Composite shape
(Isosceles triangle)
(Rectangle + Isosceles triangle)
Composite just means
it is made by putting
together separate parts
Just calculate the area of each shape separately then add (or subtract) to find the total composite area.
Calculate the area of these composite shapes
Split into a triangle
(i)
8 cm
10 cm
2
8 cm
1
and a square
2
.
Area
1
= 1
2
Area
2
= 8 cm # 8 cm = 64 cm2
#
` Total area =
Area
1
2 cm # 8 cm = 8 cm2
1
+ Area
2
= 8 cm2 + 64 cm2
2 cm
Add area 1 and 2 for the composite area
= 72 cm2
8 cm
This next one shows how you can use addition or subtraction to calculate the area of composite shapes.
(ii)
• method 1: Split into two rectangles
3.5 m
8m
7m
3.5 m
4.5 m
3.5 m
and
Area
1
= 4.5 m # 3.5 m = 15.75 m2
Area
2
= 3.5 m # 7 cm = 24.5 m2
2
` Total area =
15.75 m2 + 24.5 m2 Add area
1
1 and area 2 together
= 40.25 m2
2
7m
• method 2: Large rectangle
3.5 m
2
8 m
1
1
7m
4.5 m
1
minus the small 'cut out' rectangle
Area
1
= 8 m # 7 m = 56 m2
Area
2
= 3.5 m # 4.5 m = 15.75 m2
` Total area =
56 m2 - 15.75 m2
2
Subtract area 2 from area 1
= 40.25 m2
Area and Perimeter
Mathletics
©3P Learning Ltd
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Where does it work?
Your Turn
Area and Perimeter
Area of composite shapes
1
Complete the area calculations for these shaded shapes:
6 mm
a
4 mm
Area
1
=
mm #
mm2
2 mm
=
4 mm
2
1
mm Area
` Composite area =
1
2
=
mm #
mm2
=
mm2
2
+
mm
mm2
=
11 m
Area
1
3 m
5 m
b
=
#
Area
2
m2
=
6 m
m2
#
=
#
m2
m2
=
2
` Composite area =
3 m
1
5 m
11 m
6 m
c
2 cm
6.5 cm
Area
1
2.5 cm
=
1
=
m2
#
cm2 Area
2
cm2
=
m2
2
+
=
#
#
cm2
cm2
=
2 cm
6.5 cm
4 cm
` Composite area =
1
2
-
2.5 cm
cm2
cm2
=
d
5 m
Area
3 m
2 m
5 m
1
2
3 m
16
1
=
#
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Area
m2
=
` Composite area =
1
Area and Perimeter
Mathletics
©3P Learning Ltd
2
=
=
2
-
m2
=
2 m
H
m2
#
m2
#
m2
m2
Your Turn
Area and Perimeter
Calculate the area of these composite shapes, showing all working:
13 cm
a
...../...../20....
AREA OF COMPOSITE
5 cm
12 cm
b
Area =
cm2
Area =
m2
Area =
mm2
Area =
units2
psst: change all the units to meters first.
300 cm
200 cm
4.5 m
c
2 mm
d
psst: this one needs three area calculations
6 units
10 units
2
SHAPES *
AREA OF COMPOSITE
Area of composite shapes
SHAPES *
Where does it work?
5 units
Area and Perimeter
Mathletics
©3P Learning Ltd
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Where does it work?
Area and Perimeter
Perimeter of simple shapes
By adding together the lengths of each side, the perimeter of all common shapes can be found.
Start/finish
Start/finish
width
(y units)
Square
side (x units)
side 2
(y units)
Rectangle
side 1 (x units) Start/finish
length (x units)
P = 4 # side length
= 4 # x units
= 4x units
Triangle
side 3
(z units)
P = width + length + width + length
= ^ y + x + y + xh units
= ^2 # xh + ^2 # yh units
= 2x + 2y units
P = side 1 + side 2 + side 3
= x + y + z units
You can start/end at any
vertex of the shape
Here are some examples involving numerical dimensions:
Calculate the perimeter of these common shapes
(i)
11 units
8 units
11 units
Start/finish
10 units
8 units
Sum of all the side lengths
10 units
Perimeter = 11 units + 8 units + 10 units
= 29 units
Start/finish
(ii)
2.3 cm
2.3 cm
2.3 cm
2.3 cm
Four lots of the same side length
2.3 cm
Perimeter = 4 # 2.3 cm
= 9.2 cm
All measurements must be in the same units before calculating perimeter.
(iii) The perimeter for parallelograms is done the same as for rectangles. Calculate this perimeter in mm.
15 mm
Start/finish
0.5 cm
15 mm
5 mm
5 mm
All side lengths in mm
15 mm
Perimeter = 2 # 15 mm + 2 # 5 mm
= 30 mm + 10 mm
= 40 mm
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Area and Perimeter
Mathletics
©3P Learning Ltd
Opposite sides in pairs
Where does it work?
Your Turn
Area and Perimeter
Complete the perimeter calculations for these shapes:
Perimeter =
17 units
8 units
15 units
units +
units +
units
units
=
Perimeter =
2#
b
OF SIMPLE SHAPES *
a
...../...../20....
PERIMETER
Perimeter of simple shapes
1
PERIMETER
OF SIMPLE SHAPES *
mm + 2 #
mm
9 mm
mm
=
6 mm
Perimeter =
c
m
#
m
=
5m
d
Perimeter =
2#
cm +
cm
11 cm
cm
=
5m
2
Calculate the perimeter of the shapes below, using the space to show all working:
a
b
15 m
5.8 cm
Perimeter =
cm
Perimeter =
c
m
d
1.6 mm
3 m
1.6 m
2.4 mm
Perimeter =
5 m
3.4 m
mm
Perimeter =
Area and Perimeter
Mathletics
©3P Learning Ltd
2.4 m
m
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Where does it work?
Your Turn
Area and Perimeter
Perimeter of simple shapes
3
Find the perimeter of each shape written using the smaller units of measurement in each diagram.
a
in cm.
in mm.
b
550 cm
16.5 cm
600 cm
3m
225 mm
Perimeter =
4
cm
Perimeter =
mm
Each shape below has its perimeter written inside and is missing one of the side length values.
Rule a straight line between each shape and the correct missing side length on the right to answer:
How many straight sides does an icosagon have?
P = 24 m
8 m
F
V
5.2 m
4.4 m
d
P = 12 m
a
L
b
E
440 cm
S
v
m
H
P = 18 m
d
a
v
R
c
a
TOPIC
6 m
5 m
P = 12 m
SERIES
650 cm
1.1 m
G
Y
9
7 m
c
T
N
6.5 m
H
2 m
b
E
20
380 cm
c
W
P = 14 m
1.6 m
3.5 m
T
P = 32 m
9 m
2.4 m
N
b
c
d
Area and Perimeter
Mathletics
©3P Learning Ltd
m
v
Where does it work?
Area and Perimeter
Perimeter of composite shapes
The lengths of the unlabeled sides must be found in composite shapes before calculating their perimeter.
(3 + 3.5) m = 6.5 m
? m
2 m
2 m
3.5 m
7 m
7 m
3.5 m
? m
(7 - 2) m = 5 m
Start/finish
3 m
3 m
` Perimeter = 7 m + 6.5 m + 2 m + 3.5 m + 5 m + 3 m
= 27 m
Here are some more examples.
Calculate the perimeter of these composite shapes
(i)
9 cm
12 cm
130 mm
5 cm
9 cm
12 cm
13 cm
Calculate each side length of the shape in the same units
9 cm + 5 cm = 14 cm
` Perimeter = 9 cm + 13 cm + 14 cm + 12 cm
= 48 cm
(ii)
You can also imagine the
sides re-positioned to make
the calculation easier
3 m
^6 m - 3 mh ' 2 = 1.5 m
6 m
3 m
3 m
6 m
1.5 m + 1.5 m = 3 m
6 m
` Perimeter = 6 # 1.5 m + 3 m + 6 m
` Perimeter = 2 # 6 m + 2 # 3 m
= 18 m
= 18 m
Area and Perimeter
Mathletics
©3P Learning Ltd
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21
Where does it work?
Your Turn
Area and Perimeter
Perimeter of composite shapes
Calculate the value of the sides labeled a and b in each of these composite shapes:
2 cm
a
b
a =
cm
b
b =
8 cm
cm
3.4 mm
2.2 mm 1 mm
13 m
a
b
b =
mm
a =
cm
b =
cm
1.6 mm
15 m
d
a =
5 cm
m
4.8 cm
b
b =
18 m
m
a
14 cm
b
8 cm
15 cm
Calculate the perimeter of these composite shapes:
Perimeter =
cm
#
b
Perimeter =
2 # a m + 2 #
POSITE SH
AP
m
2 m
=
m+
2
1 3
...../...../20....
R
a m
C OM
OF
cm
=
POSITE SH
COM
AP
OF
E
9.8 mm
E R I ME T
*P
E
R
S
E
a
ERIMET
*P
E
2
mm
a
c
a
a =
S
1
m
4 m
=
Be careful with the units for these next two
c
Perimeter =
3#
1.2 cm
m
mm +
mm +
mm
20 mm
=
mm
Perimeter =
# 4.1 cm +
16 mm
d
4.1 cm
=
38 mm
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Area and Perimeter
Mathletics Passport
© 3P Learning
cm
#
cm
Where does it work?
Your Turn
Area and Perimeter
Perimeter of composite shapes
Calculate the perimeter of these composite shapes in the units given in square brackets.
Show all working.
6mm @
a
b
4 mm
2.2 m
Perimeter =
3.6 cm
mm
d
48 mm
m
6km @ psst: km m
1.5 km
1200 m
Perimeter =
cm
Perimeter =
km
Earn an awesome passport stamp for this one!
The incomplete geometric path shown below is being constructed using a combination of
the following shaped pavers: 1 m
2 m
2 m
1 m
1.41 m 2 m
1 m 1 m
E
SOM
E A
WES
OME
...../.....
/20....
ME
S
Mathletics Passport
© 3P Learning
E
Area and Perimeter
SOM
Total perimeter of completed path =
AWE
Completed path
AWE
SOM
The gap in between each part of the spiral path is always 1 m wide.
Calculate what the total perimeter of this path will be when finished.
AWE
1.41 m
AWE
1 m
ESO
4
Perimeter =
22 mm
6cm @
c
6m @
OME
AW
3
m
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Where does it work?
Your Turn
Area and Perimeter
Perimeter of composite shapes
5
The four composite shapes below have been formed using five, unit squares.
a
Using your knowledge of perimeter and the grid below, combine all four pieces to create two
different shapes so that:
• One shape has the smallest possible perimeter.
• The other has the largest possible perimeter.
All shapes must be connected by at least one whole side of a unit square.
1 unit
1 unit
b
Briefly describe the strategy you used to achieve each outcome below:
• A shape with the smallest possible perimeter.
• A shape with the largest possible perimeter.
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Area and Perimeter
Mathletics
©3P Learning Ltd
Where does it work?
Area and Perimeter
Simple word problems involving area and perimeter
Sometimes we can only communicate ideas or problems through words.
So it is important to be able to take written/spoken information and turn it into something useful.
For example,
Miguel wants to paint a square. He has just enough paint to create a line 240 cm 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
240 cm.
So each side length = 240 cm ' 4
= 60 cm
` The longest length each side of the square painted by Miguel can be is 60 cm.
This is useful for Miguel to know because if he painted the first side too long, he would run out of paint!
Here are some more examples
(i) A rectangular park is four times longer than it is wide. If the park is 90 m long, how much area
does this park cover?
^90 ' 4h m = 22.5 m
Draw diagram to illustrate problem
90 m
Area = length # width
= 90 m # 22.5 m
= 2025 m2
(ii) At a fun run, competitors run straight for 0.9 km before turning left 90 degrees to run straight for a
further 1.2 km. The course has one final corner which leads back to the start along a straight 1.5 km
long street. How many laps of this course do competitors complete if they run a total of 18 km?
1.2 km
0.9 km
1.5 km
Draw diagram to illustrate problem
Start/finish
Perimeter of course = 0.9 km + 1.2 km + 1.5 km
= 3.6 km
Perimeter will be the length of each lap
` Length of each lap of the course is 3.6 km
` Number of laps = 18 km ' 3.6 km
Race distance divided by the length of each lap
=5
` Competitors must complete 5 laps of the course to finish
Area and Perimeter
Mathletics
©3P Learning Ltd
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Where does it work?
Your Turn
Area and Perimeter
Simple word problems involving area and perimeter
1
2
3
Three equilateral triangles, each with sides of length 3 cm 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.
a
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 relationship between
area and perimeter.
b
You have been employed by a fabric design company called Double Geometrics. Your first task
as a pattern maker is to design the following using all seven identical squares:
“Closed shapes for a new pattern in which the value of their perimeter is twice the value of their
area.” Draw five possible different patterns that match this design request.
The base length of a right-angled triangle is one fifth of its height. If the base of this triangle is 4.2 m,
calculate the area of the triangle.
SIMPLE WORD
PROBLEMS
INVOLVING
AREA AND
PERIMETER
...../...../20....
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Area and Perimeter
Mathletics
©3P Learning Ltd
Where does it work?
Your Turn
Area and Perimeter
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 floor must
be a rectangle shape with an area of 64 m2.
a
If only whole meter measurements can be used, sketch all the different possible floor dimensions.
b
Another design rule is to try ensure a large perimeter so there is more space to hang paintings
from. Use calculations to show which floor plan will have the largest perimeter.
c
Would the design with the largest possible perimeter be a good choice?
Explain briefly why/why not.
d
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 12.5 cm wide and 24.5 cm long if they all fit perfectly without any edges overlapping?
2.5 cm
3.5 cm
12.5 cm
24.5 cm
Area and Perimeter
Mathletics
©3P Learning Ltd
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Where does it work?
Your Turn
Area and Perimeter
Simple word problems involving area and perimeter
5
A fence used to close off a parallelogram-shaped area is being rearranged to create a square area
with the same perimeter. The short side of the area is 34 m long (half the length of the long side).
a
How long will each side of the new square area be after using the whole length of this fence?
34 m
b
6
If the distance between the longer sides of the original area was 30 m and the length did not
change, use calculations to show which fencing arrangement surrounded the largest area.
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.
Each brick =
16 cm
28 cm
28
a
A 500 mL tin of white paint has been purchased to paint the wall. The instructions on the
paint tin say this is enough to cover an area of 11 500 cm2.
Use calculations to show that there is enough paint in the tin to cover side of the wall.
b
If a beetle walked all around the outside of the wall (including along the ground), how many
meters did it walk?
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Area and Perimeter
Mathletics Passport
© 3P Learning
What else can you do?
Area and Perimeter
Rhombus and Kite shapes
The area for both of these shapes can be calculated the same way using the length of their diagonals.
A
B
• A rhombus is like a square parallelogram.
Area = ^diagonal lengths multiplied togetherh ' 2
D
= ^ AC # BDh ' 2
C
A Rhombus is a
parallelogram, so
we can also use
the same rule to
find the area:
height
Perimeter = 4 # length of one side
= 4 # AB
B
length
• A kite has two pairs of equal sides which are adjacent (next to) each other.
A
Area = ^diagonal lengths multiplied togetherh ' 2
C
= ^ AC # BDh ' 2
Perimeter = 2 # short side + 2 # long side
= 2 # AB + 2 # AD
D
Here are some examples:
Calculate the area and perimeter of these shapes
(i) For this rhombus, WY = 12 cm and XZ = 16 cm.
W
Area = ^diagonal lengths multiplied togetherh ' 2
X
= ^12 cm # 16 cmh ' 2
Z
10 cm
= 96 cm2
Y
Perimeter = 4 # length of sides
= 4 # 10 cm
= 40 cm
(ii) For the kite ABCD shown below, AC = 4.7 m and BD = 2.1 m.
1.5 m
A
Area = ^diagonal lengths multiplied togetherh ' 2
B
= ^2.1 m # 4.7 mh ' 2
3.7 m
D
= 4.935 m2
C
Perimeter = 2 # short side + 2 # long side
= 2 # 1.5 m + 2 # 3.7 m
= 10.4 m
Area and Perimeter
Mathletics
©3P Learning Ltd
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Area and Perimeter
PR = 18 cm and QS = 52 cm
KITE SHAP
E
ND
b
BD = 1.8 mm and AC = 2.4 mm
A
a
20....
/
.
.
.
.
.
/
.....
E
Calculate the area and perimeter of these shapes:
1
A
KITE SHAP
Rhombus and Kite shapes
R H O M B US
ND
S
R H O MB U S
Your Turn
S
What else can you do?
Q
A
41 cm
P
R
B
D
15 cm
C
3.6 mm
S
Area =
cm2
'
#
Area =
cm2
=
Perimeter =
2#
#
cm
Perimeter =
cm
1 mm2
2
mm2
=
+2#
=
#
mm
#
mm
=
Calculate the perimeter of these composite shapes:
2
a
b
14 m
6.5 cm
3.4 cm
5.1 cm
9 m
Perimeter =
m
cm
Calculate the area of this composite shape, showing all working when:
3
HL = 30 m, IK = IM = 16 m and JL = 21 m
J
I
H
K
M
30
Perimeter =
L
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Area =
Area and Perimeter
Mathletics
©3P Learning Ltd
m2
'2 = # 1
2
What else can you do?
Area and Perimeter
Trapezoids
A trapezoid is a quadrilateral which has only one pair of parallel sides.
So the area formula for a trapezoid would also work on all of those shapes.
a
A
B
A
height (h)
C
B
height (h)
D
b
a
C
D
b
Two common trapezoid 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 parallel sidesh # height ' 2
= ^ a + bh # h ' 2
: Perimeter = AB + BD + CD + AC
Here are some examples:
Calculate the area and perimeter of these shapes
(i)
Area = ^sum of the parallel sidesh # height ' 2
20 mm
= ^22 mm + 10 mmh # 16 mm ' 2
= 32 mm # 16 mm ' 2
10 mm
= 256 mm2
22 mm
Perimeter = 20 mm + 22 mm + 16 mm + 10 mm
16 mm
(ii)
= 68 mm
Area = ^sum of the parallel sidesh # height ' 2
6.7 m
2.9 m
2 m
10.1 m
= ^6.7 m + 14.5 mh # 2 m ' 2
= 21.2 m # 2 m ' 2
14.5 m
= 21.2 m2
Perimeter = 6.7 m + 10.1 m + 14.5 m + 2.9 m
= 34.2 m
Area and Perimeter
Mathletics
©3P Learning Ltd
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Your Turn
Area and Perimeter
b
1 km
15 km
12.5 m
41 km
9 km
1.8 m
53 km
Area =
8.2 m
4.5 m
+
#
Area =
' 2 km2
km2
=
=
km
Perimeter =
2
+
#
m2
m
Perimeter =
Use the trapezoid method to calculate the area of these composite plane shapes.
a
b
8 cm
5 cm
1 mm 2.4 mm
14.3 mm
15 cm
Area =
3
cm2
Area =
mm2
Use the trapezoid method to calculate the area of this composite plane shapes.
16.7 m
33.4 m
1
2
170 cm
240 cm
Area =
32
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...../...../20....
A
a
PE
Calculate the area and perimeter of these trapezoids:
IDS * TR
A
1
DS * T
R
I DS * T R
ZO
A
Trapezoids
OIDS * TR
A
ZO I
PE
Z
PE
ZO
PE
What else can you do?
m2
Perimeter =
Area and Perimeter
Mathletics
©3P Learning Ltd
m
' 2 m2
What else can you do?
Your Turn
Area and Perimeter
Area challenge
Fill the grid below with as many different squares, triangles, rectangles, parallelograms, rhombi, kites
and trapezoids as you can which all have the same area of 8 units2.
1 unit
1 unit
Area and Perimeter
Mathletics
©3P Learning Ltd
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What else can you do?
Your Turn
Area and Perimeter
Reflection Time
Reflecting 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 calculations to a real
life situation.
• If you discovered or learnt about any shortcuts to help with calculating area and perimeter or some
other cool facts/conversions, jot them down here:
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Area and Perimeter
Mathletics
©3P Learning Ltd
Cheat Sheet
Area and Perimeter
Here is what you need to remember from this topic on Area and perimeter
Area using unit squares
Area is just the amount of flat 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 fractions of squares to calculate the area.
Area (A) = 2 units2
Perimeter using unit squares
The perimeter with unit squares means count the number of edges around the outside of the shape.
Perimeter (P) = 6 units
Area: Squares and rectangles
Just multiply the length of the perpendicular sides (length and width).
length
width
Square
length
side (x)
width
side (y)
Rectangle
side (x)
side (x)
Area = length # width
Area = length # width
= x2 units2
= xy units2
height
Area: Triangles
height
height
base
base
base
` Area of the triangle = (half the base multiplied by the perpendicular height) units2
= 1 # b # h units2
2
Area: Parallelograms
Perpendicular
height (h)
Length (l)
` Area of a parallelogram =
length # perpendicular height units2
= l # h units2
Area and Perimeter
Mathletics
©3P Learning Ltd
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Cheat Sheet
Area and Perimeter
Area of composite shapes
1
2
Area 1
(Rectangle)
+
1 + 2
Area 2
=
(Isosceles triangle)
Composite Area = Area 1 + Area 2
(Rectangle + Isosceles triangle)
Perimeter of simple shapes
Add together the lengths of every side which make the shape.
Start/finish
Start/finish
width
(y units)
Square
side 2
(y units)
Rectangle
side (x units)
side 1 (x units) Start/finish
length (x units)
P = side 1 + side 2 + side 3
= x + y + z units
P = width + length + width + length
= 2x + 2y units
P = 4 # side length
= 4x units
Triangle
side 3
(z units)
Perimeter of composite shapes
The lengths of all unlabeled sides must be found in composite shapes before calculating their perimeter.
It is easier to add them together if the lengths are all in the same units.
(3 + 3.5) m = 6.5 m
? m
2 m
2 m
3.5 m
7 m
3.5 m
7 m
? m
(7 - 2) m = 5 m
Start/finish
3 m
3 m
` Perimeter = 7 m + 6.5 m + 2 m + 3.5 m + 5 m + 3 m
= 27 m
B
Rhombus, Kites and Trapezoids
A
B
A
C
A
a
B
perpendicular
height (h)
D
Rhombus
Area = ^ AC # BDh ' 2
Perimeter = 4 # AB
36
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D
Kite
b
a
A
B
perpendicular
height (h)
D C
b
Trapezoid
Area = (AC # BD) ' 2
Perimeter = 2 # AB + 2 # AD
Area and Perimeter
Mathletics
©3P Learning Ltd
Area = (a + b) # h ' 2
Perimeter = AB + BD + CD + AC
D
Answers
Area and Perimeter
Area using unit squares
Area using unit squares
4. Here are 42 solutions. There are more...
1. a Area =
4 whole squares
= 4 units2
b
Area =
6 whole squares
= 6 mm2
c
Area =
2 whole + 2 half squares
= 2 m2 + 2 # 1 m2
2
= 3 m2
d
Area =
4 whole + 4 half squares
= 4 units2 + 4 # 1 units2
2
= 6 units2
e
Area =
2 whole + 4 quarter squares
= 2 units2 + 4 # 1 units2
4
= 3 units2
f
Area =
4 whole + 2 quarter squares
= 4 cm2 + 2 # 1 cm2
4
= 4.5 cm2
g
Area =
3 whole + 4 half + 4 quarter squares
= 3 units2 + 4 # 1 units2 + 4 # 1 units2
2
4
= 6 units2
2. a Area = 9 cm2
c
Area = 6 m2
e
Area = 5 units
g
Area = 12.5 km
3. a
c
2
2
b
Area = 12 units2
d
Area = 4.5 mm2
f
Area = 8 mm
h
Area = 14 cm
2
Perimeter using unit squares
1.
2
b
2.
d
3.
a
Perimeter =
2 + 2 + 2 + 2 units
= 8 units
b
Perimeter =
1 + 3 + 1 + 3 units
= 8 units
c
Perimeter =
2 + 1 + 1 + 1 + 1 + 2 units
= 8 units
a
P = 10 units
a
P = 12 units
c
P = 14 units
d
P = 13 units
Even though the shapes all have the same area,
they do not all have the same perimeter lengths.
This shows that the shapes with the same area will
not necessarily have the same perimeter. So the
perimeter is not related to the area of the shape.
Area and Perimeter
Mathletics
©3P Learning Ltd
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Answers
Area and Perimeter
Perimeter using unit squares
Area: Triangles
1 # 2 # 4 units2
1. a Area =
2
= 4 units2
4. a
b
P = 10 units
P = 12 units
P = 14 units
Area =
1 # 8 # 8 units2
2
= 32 units2
1
2
# 12 # 8 mm
2. a Area =
2
= 48 mm2
P = 16 units
P = 18 units
P = 20 units
Area =
c
Area =
d
Area =
e
Area =
b
P = 11 units
P = 13 units
P = 17 units
1
2
# 14 # 12 cm
2
= 84 cm2
b
1
2
# 10 # 7.5 units
2
= 37.5 units2
P = 15 units
P = 19 units
1
2
# 6 # 4.5 m
2
= 13.5 m2
1
2
#5#4m
2
= 10 m2
Area: Squares and rectangles
Area: Parallelograms
1. a Area =
2 # 2 units2
= 4 units2
b
Area =
0.6 # 0.6 m2
= 0.36 m2
Area =
3 # 2 units2
= 6 units2
d
Area =
5 # 3.2 mm2
= 16 mm2
c
2. a Area =
1.4 # 1.4 km2
= 1.96 km2
. 2 km2 (to nearest whole km2)
b
1. a Area =
4.5 # 10 units2 b Area =
4.6 # 2.2 cm2
= 450 units2
= 10.12 cm2
2. a Area =
240 m2
3.
Area =
7 # 4.3 cm2
= 30.1 cm2
. 30 cm2 (to nearest whole cm2)
3. The length of the rectangle is 7 units.
4. The length of each side is 11 units
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Area and Perimeter
Mathletics
©3P Learning Ltd
b
Area =
1.92 mm2
Answers
Area and Perimeter
Area of composite shapes
1. a Area
1
= 4 mm # 4 mm
= 16 mm2
Area
2
= 2 mm # 2 mm
= 4 mm2
Perimeter of simple shapes
2. a Perimeter =
23.2 cm
Area
1
= 0.5 # 6 # 8 m2
= 24 m2
Area
2
= 11 # 5 m2
= 55 m2
4.
Area
1
= 6.5 # 2 cm2
= 13 cm2
Area
2
= 0.5 # 4 # 2 cm2
= 4 cm2
W
E
N
T
Y
a
b
c
d
m
v
c
a = 9 m
b = 6 m
b
a = 1.8 mm
b = 2.2 mm
d
a = 5.2 cm
b = 11.2 cm
5 # 9.8 cm
2. a Perimeter =
= 49 cm
b
Perimeter =
2 # a m + 2 # 2 m = 12 m + 4 m
= 16 m
= 2 # 2 m2
= 4 m2
c
Perimeter =
3 # 12 mm + 20 mm + 16 mm
= 72 mm
Composite area = 20 - 4 m2
= 16 m2
d
Perimeter =
4 # 4.1 cm + 2 # 3.8 cm
= 24 cm
Area
1
= 0.5 # 8 # 5 m2
= 20 m2
Area
2
2. a Area =
199 cm2
c
Perimeter =
10.8 m
T
1. a a = 4 cm
b = 5 cm
Composite area = 13 - 4 cm2
= 9 cm2
d
d
Perimeter of composite shapes
Composite area = 24 + 55 m2
= 79 m2
c
Perimeter =
45 m
3. a Perimeter =
1450 cm b Perimeter =
780 mm
Composite area = 16 + 4 mm2
= 20 mm2
b
Perimeter =
8 mm
c
b
Area =
32 mm
2
b
Area =
22.5 m2
d
Area =
90 units
48 mm
3. a Perimeter =
2
Perimeter of simple shapes
c
Perimeter =
21.2 cm
b
Perimeter =
26.4 m
d
Perimeter =
6.6 km
4. Total perimeter of completed path = 132 m
1. a Perimeter =
15 units + 17 units + 8 units
= 40 units
b
Perimeter =
2 # 9 mm + 2 # 6 mm
= 30 mm
c
Perimeter =
4#5m
= 20 m
d
Perimeter =
2 # 11 cm + 5 cm
= 27 cm
Area and Perimeter
Mathletics
©3P Learning Ltd
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9
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Answers
Area and Perimeter
Simple word problems involving area
and perimeter
Perimeter of composite shapes
5.
a
2. b
Smallest possible perimeter = 18 units
3. Area of a triangle = 44.1 m2
Largest possible perimeter = 40 units
b
c
4. a
1
Try to make as many sides as possible touching
each other, which reduces the number of sides
that are counted for the perimeter.
64 m
2
Making each shape joined to the other by one
side only, which increases the number of sides
that must counted for the perimeter.
4
Perimeter =
15 cm
Shape 2
2
Perimeter =
40 m
3
Perimeter =
68 m
4
Perimeter =
32 m
1
has the largest perimeter
No, floor plan 1 would not be a good choice.
While it meets the rule of having the largest
possible perimeter, it would be very long and
too narrow for more than one person to walk
through at a time.
d
Total number of stamps that fit = 35 stamps
5. a Each side of the new square area = 51 m long
Perimeter = 8 units
b
The area of both shapes is the same, but the
perimeters are different. This shows that there
is no relationship between the area and the
perimeter of a shape.
The new square shape will surround more area.
6. a There will be just enough paint to cover the side
of the wall (with 300 cm2 to spare)
b
TOPIC
Perimeter =
130 m
c
Perimeter = 10 units
SERIES
16 m
1
Floor plan
Shape 1
9
4 m
8 m
1.
H
3
8 m
b
40
2 m
32 m
Simple word problems involving area
and perimeter
2. a
1 m
The beetle walked a total distance of 5.52 m
Area and Perimeter
Mathletics
©3P Learning Ltd
Answers
Area and Perimeter
Rhombus and Kite shapes
Trapeziums
1. a Area =
18 # 52 ' 2 cm2
= 468 cm2
b
c
d
1. a Area =
(53 + 1) # 9 ' 2 km2
= 243 km2
Perimeter =
110 km
Area =
1.8 # 2.4 # 1 mm2
2
= 2.16 mm2
b
Perimeter =
2 # 41 + 2 # 15 cm
= 112 cm
Perimeter =
27 m
Perimeter =
4 # 3.6 mm
= 14.4 mm
64 m
2. a Perimeter =
b
Area =
(4.5 + 12.5) # 1.8 ' 2 m2
= 15.3 m2
135 cm2
2. a Area =
Perimeter =
31.7 cm
53.35 m2
3. Area =
b
Area =
24.12 mm2
Perimeter =
74 m
408 m2
3. Area =
Area challenge
Here are 20 possible shapes which all have an area of 8 units2 . There are many more.
Area and Perimeter
Mathletics
©3P Learning Ltd
H
9
SERIES
TOPIC
41
Notes
42
Area and Perimeter
H
9
SERIES
TOPIC
Area and Perimeter
Mathletics
©3P Learning Ltd
Area and Perimeter
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