Chapter 13
Area
Imagine having a sheet of paper in front of you. A person who can’t see the piece of paper wants to know how
big it is. One way of describing the size of the sheet of paper is to meausre its dimensions (its length and its
width) and then tell the other person. However this requires telling the other person multiple measurements,
and could become very tricky if the piece of paper isn’t a simple shape such as a rectangle.
An easy way of describing the size of a two-dimensional shape is area. In the case of a piece of paper, area
tells us how much room there is for writing on. The area of a room can tell us how much furniture can fit into
it and the area of a block of land can tell us how big we can build our house.
Area is measured in square units. For example, mm2 , cm2 , m2 or km2 . The little ‘2’ tells us that it is an
area that is being measured and not a length. A square that has a side length of 1m has an area of 1m2 . Large
areas can also be measured in hectares. 1 hectare (ha) = 10 000m2 . Notice that hectare does not have a little
‘2’. When we talk about hectares, we automatically know that it must be an area.
Area describes the size of a two-dimensional shape.
Area is measured in square units, such as m2 .
In this chapter we will learn how to calculate the area of simple two-dimensional shapes.
13.01
Area of Square
Because all the sides of a square are the same length, we only need to know the side length of a square to be
able to work out its area.
Area of Square = Side2
Example 1 Find the area of a square with a side length of 4cm
4cm
Area of Square
= side2
= 42
= 16cm2
Example 2 Find the area of a square with a side length of 2.5m
2.5m
103
104
CHAPTER 13. AREA
Area of Square
= side2
= 2.52
= 2.5 × 2.5
= 6.25m2
If we know the area of a square, we can find its side length.
Side of Square =
√
Area
Example 3 Find the side length of a square with an area of 64mm2
area=64mm2
Side of Square
√
= area
√
= 64
= 8mm
Example 4 Find the side length of a square with an area of 9m2
area=9m2
Side of Square
√
= area
√
= 9
= 3m
13.01.1
Finding Square Roots by Factorisation
Once numbers go past our tables (12x12=144), we need another way of finding a number’s square root. Assuming
that a number’s square root is a whole number, we can find it using factorisation.
To find the square root of a number, first write the number as the product of its prime factors. Its square
root can then be found by halving all the powers in its prime factorisation (and then multiplying the
remaining factors together).
√
Example 5 Find the square root of 196 ( 196)
196
49
4
2
2
7
7
105
13.02. EXERCISE 13.02
196 = 2 × 2 × 7 × 7
√
= 22 × 72
196 = 21 × 71
= 14
√
Example 6 Find the square root of 324 ( 324)
324
81
4
2
2
9
3
9
3
3
3
324 = 2 × 2 × 3 × 3 × 3 × 3
= 22 × 34
√
324 = 21 × 32
=2×9
= 18
13.02
Exercise 13.02
1. Find the area of each of the following squares with side lengths of:
a) 2mm
b) 8cm
c) 3mm
d) 5km
e) 11cm
f) 7mm
g) 1km
h) 19cm
i) 14m
j) 27cm
k) 35km
l) 68mm
m) 2.4cm
n) 1.8mm
o) 4.5km
2. Find the side lengths of the following squares with areas of:
a) 16cm2
b) 9km2
c) 4mm2
d) 1cm2
e) 64m2
13.03
f) 121mm2
g) 81m2
h) 25cm2
i) 144km2
j) 225mm2
k) 400cm2
l) 441m2
m) 625mm2
n) 729cm2
o) 1ha
Area of Rectangle
To calculate the area of a rectangle, we need two measurements, the length and the width. The length is the
longer side of the rectangle and the width is the shorter side.
Area of Rectangle = Length × Width
Length of Rectangle = Area ÷ Width
Width of Rectangle = Area ÷ Length
106
CHAPTER 13. AREA
Example 7 Find the area of a rectangle with a length of 8cm and a width of 4cm
8cm
4cm
Area of Rectangle
= L×W
= 8×4
= 32cm2
Example 8 Find the length of a rectangle with an area of 77m2 and a width of 7m
area=77m2
7m
Length of Rectangle
= A÷W
= 77 ÷ 7
= 11m
Example 9 Find the width of a rectangle with an area of 10mm2 and a length of 5mm
5mm
area=10mm2
Width of Rectangle
= A÷L
= 10 ÷ 5
= 2mm
13.04
Exercise 13.04
1. Find the area of the following rectangles
a) length=2m, width=1m
b) length=10cm, width=3cm
c) length=12mm, width=5mm
d) length=8km, width=7km
e) length=5m, width=2m
f) length=22cm, width=5cm
g) length=19mm, width=8mm
h) length=34km, width=4km
i) length=48m, width=15m
j) length=61cm, width=23cm
k) length=29mm, width=17mm
l) length=37km, width=26km
m) length=2.8m, width=1.5m
n) length=5.4cm, width=3.2cm
2. Find the length of the following rectangles
a) area=4m2 , width=1m
b) area=12cm2 , width=2cm
c) area=15mm2 , width=3mm
d) area=50km2 , width=5km
e) area=20m2 , width=4m
f) area=48cm2 , width=4cm
g) area=35mm2 , width=5mm
h) area=56km2 , width=7km
i) area=288m2 , width=9m
j) area=138cm2 , width=6cm
3. Find the width of the following rectangles
k) area=279mm2 , width=9mm
l) area=336km2 , width=7km
m) area=39.36m2 , width=4m
n) area=64.05cm2 , width=5cm
107
13.05. AREA OF TRIANGLE
f) area=56cm2 , length=8cm
g) area=11mm2 , length=11mm
h) area=40km2 , length=8km
i) area=120m2 , length=12m
j) area=112cm2 , length=16cm
a) area=8m2 , length=4m
b) area=18cm2 , length=6cm
c) area=6mm2 , length=3mm
d) area=21km2 , length=7km
e) area=27m2 , length=9m
13.05
k) area=231mm2 , length=21mm
l) area=119km2 , length=17km
m) area=451m2 , length=41m
n) area=189cm2 , length=27cm
Area of Triangle
To calculate the area of a triangle, we don’t actually need all three side lengths. We only need two measurements,
the base of the triangle and its perpendicular height. The base is one side of the triangle, and the perpendicular
height is the height of the triangle taken at a right angle to the base. Perpendicular means to be at right angles.
Perpendicular
Height
Base
Perpendicular
Height
Base
By multiplying the base and the perpendicular height together, we will find the area of the rectangle that the
triangle fits into. However every triangle is simply half the size of the rectangle that it fits into. Therefore the
area of the triangle can be found by multiplying the base and perpendicular height together and then dividing
by two.
You can check that the area of the triangle is half that of the rectangle by drawing a triangle onto a sheet
of paper as shown below. Cut out the triangle, and the left over pieces will exactly cover the triangle.
Perpendicular
Height
Base
Perpendicular
Height
Base
Area of Triangle = Base × Perpendicular Height ÷ 2
Base of Triangle = 2 × Area ÷ Perpendicular Height
Perpendicular Height of Triangle = 2 × Area ÷ Base
Example 10
Find the area of a triangle with a base of 8cm and a perpendicular height of 4cm
PH = 4cm
8cm
Area of Triangle
= B × PH ÷ 2
= 8×4÷2
= 32 ÷ 2
= 16cm2
Example 11
Find the base of a triangle with an area of 40km2 and a perpendicular height of 5km
Area = 40km2
PH = 5km
108
CHAPTER 13. AREA
Base of Triangle
= 2 × Area ÷ PH
= 2 × 40 ÷ 5
= 80 ÷ 5
= 16km
Example 12
Find the perpendicular height of a triangle with an area of 60m2 and a base of 10m
Area = 60m2
10m
Perpendicular Height of Triangle
= 2 × Area ÷ Base
= 2 × 60 ÷ 10
= 120 ÷ 10
= 12m
13.06
Exercise 13.06
1. Find the area of the following triangles
a) base=3cm, PH=2cm
b) base=6m, PH=3m
c) base=4km, PH=4km
d) base=11mm, PH=6mm
e) base=10cm, PH=1cm
f) base=12m, PH=2m
g) base=10km, PH=9km
h) base=5mm, PH=8mm
i) base=7cm, PH=12cm
j) base=9mm, PH=6mm
k) base=13km, PH=18km
l) base=30mm, PH=25mm
m) base=36cm, PH=45cm
n) base=55m, PH=18m
2. Find the base of the following triangles
a) area=8cm2 , PH=4cm
b) area=3m2 , PH=1m
c) area=5km2 , PH=5km
d) area=27mm2 , PH=9mm
e) area=24cm2 , PH=4cm
f) area=33m2 , PH=6m
g) area=9km2 , PH=2km
h) area=20mm2 , PH=5mm
i) area=60cm2 , PH=10cm
j) area=55mm2 , PH=11mm
k) area=50km2 , PH=4km
l) area=162mm2 , PH=9mm
m) area=128cm2 , PH=8cm
n) area=112m2 , PH=7m
3. Find the perpendicular height of the following triangles
a) area=30cm2 , base=5cm
b) area=20m2 , base=4m
c) area=36km2 , base=9km
d) area=15mm2 , base=3mm
e) area=60cm2 , base=10cm
13.07
f) area=30m2 , base=5m
g) area=7km2 , base=7km
h) area=33mm2 , base=11mm
i) area=42cm2 , base=7cm
j) area=36mm2 , base=12mm
k) area=143km2 , base=26km
l) area=198mm2 , base=22mm
m) area=512cm2 , base=32cm
n) area=540m2 , base=40m
Area of Circle
To find the area of a circle, we only need one measurement and that is the radius. We also need a constant,
π (pronouced pi), which is approximately equal to 3.14. The constant π is irrational which means that there
are an infinite number of numbers after the decimal place. However when using π, we only need the first 1 or
2 decimal places to be able to calculate a fairly accurate answer. Therefore we will round π to 3.14.
Area of Circle = 3.14 × radius2
Area of Circle = πr2
109
13.08. EXERCISE 13.08
Example 13
Find the area of a circle with a radius of 5cm
radius=5cm
Area of Circle
=πr2
=3.14 × 52
=3.14 × 5 × 5
=3.14 × 25
=78.5cm2
Example 14 Find the area of a circle with a diameter of 14mm
In this example we have been given the diameter of the circle. However the formula for area of a circle
requires the radius. Therefore we must half the diameter to obtain the radius of the circle before doing any
other step.
diameter=14mm
radius=7mm
Area of Circle
=πr2
=3.14 × 72
=3.14 × 7 × 7
=3.14 × 49
=153.86mm2
13.08
Exercise 13.08
1. Find the area of the following circles with radii of:
a) 3cm
b) 4km
c) 9mm
d) 6m
e) 2cm
f) 8mm
g) 5km
h) 1cm
i) 7mm
j) 12km
k) 28cm
2. Find the area of the following circles with diameters of:
a) 8cm
b) 4m
c) 6mm
d) 10km
e) 12cm
f) 20m
g) 16mm
h) 2cm
i) 18m
j) 46mm
k) 28cm
110
13.09
CHAPTER 13. AREA
Exercise 13.09
1. Find the area of a square that has a side length of 6m.
2. If the area of a square is 121cm2 , what is its side length?
3. If a triangle has an area of 80cm2 and a base of 10cm, what is its perpendicular height?
4. A rectangular room has a length of 9m and a width of 4m. What is the area of the room?
5. A new classroom is being designed. The room must have an area of 108m2 and a length of 12m. What
will the width of the room be?
6. A soccer club is considering using a triangular flag for their club instead of the standard rectangular flag.
If the flag has a base of 1m and a perpendicular height of 2m, what is the area of the flag?
7. On the wall of a new building a triangle is drawn with a base of 3m and a perpendicular height of 4m.
What is the area of the triangle?
8. In the centre of a rectangular soccer field, a circle with a radius of 4m is drawn. What is the area of the
circle?
9. What is the area of the largest triangle that can be cut from a rectangular sheet of paper that is 20cm
long and 15cm wide?
10. A rectangle’s length is 3cm longer than its width. If its width is 5cm, what is the area of the rectangle?
11. A landscaper is building a square garden in a backyard. If the garden needs to be 16m2 , what will be the
length of one side?
12. What is the area of the largest circle that will fit into a square of side length 6cm?
13. From a rectangular sheet of paper that has a length of 11cm and a width of 7cm, a smaller rectangle of
length 8cm and width 5cm is cut. What is the area of paper that is wasted?
14. Five identical squares are drawn on a sheet of paper with none of them overlapping. Each square has a
side length of 9cm. What is the total area of all the squares?
15. A triangular section of concrete needs to be painted. The triangle has a base of 5m and a perpendicular
height of 6m. If the painter charges $5 per square metre, how much will the painter charge to paint the
whole triangle?
111
13.09. EXERCISE 13.09
Answers
Exercise 13.02
1a) 4mm2
b) 64cm2
c) 9mm2
d) 25km2
e) 121cm2
f) 49mm2
g) 1km2
h) 361cm2
i) 196m2
j) 729cm2
k) 1225km2
l) 4624mm2
m) 5.76cm2
n) 3.24mm2
o) 20.25km2
2a) 4cm
b) 3km
c) 2mm
d) 1cm
e) 8m
f) 11mm
g) 9m
h) 5cm
i) 12km
j) 15mm
k) 20cm
l) 21m
m) 25mm
n) 27cm
o) 100m
l) 962km2
m) 4.2m2
n) 17.28cm2
2a) 4m
b) 6cm
c) 5mm
d) 10km
e) 5m
f) 12cm
g) 7mm
h) 8km
i) 32m
j) 23cm
k) 31mm
l) 48km
m) 9.84m
n) 12.81cm
3a) 2m
b) 3cm
c) 2mm
d) 3km
e) 3m
f) 7cm
g) 1mm
h) 5km
i) 10m
j) 7cm
k) 11mm
l) 7km
m) 11m
n) 7cm
l) 375mm2
m) 810cm2
n) 495m2
2a) 4cm
b) 6m
c) 2km
d) 6mm
e) 12cm
f) 11m
g) 9km
h) 8mm
i) 12cm
j) 10mm
k) 25km
l) 36mm
m) 32cm
n) 32m
3a) 12cm
b) 10m
c) 8km
d) 10mm
e) 12cm
f) 12m
g) 2km
h) 6mm
i) 12cm
j) 6mm
k) 11km
l) 18mm
m) 32cm
n) 27m
g) 78.5km2
h) 3.14cm2
i) 153.86mm2
j) 452.16km2
k) 2461.76cm2
2a) 50.24cm2
b) 12.56m2
c) 28.26mm2
d) 78.5km2
e) 113.04cm2
f) 314m2
g) 200.96mm2
h) 3.14cm2
i) 254.34m2
j) 1661.06mm2
k) 615.44cm2
5)
6)
7)
8)
9) 150cm2
10) 40cm2
11) 4m
12) 28.26cm2
13) 37cm2
14) 405cm2
15) $75
Exercise 13.04
1a) 2m2
b) 30cm2
c) 60mm2
d) 56km2
e) 10m2
f) 110cm2
g) 152mm2
h) 136km2
i) 720m2
j) 1403cm2
k) 493mm2
Exercise 13.06
1a) 3cm2
b) 9m2
c) 8km2
d) 33mm2
e) 5cm2
f) 12m2
g) 45km2
h) 20mm2
i) 42cm2
j) 27mm2
k) 117km2
Exercise 13.08
1a) 28.26cm2
b) 50.24km2
c) 254.34mm2
d) 113.04m2
e) 12.56cm2
f) 200.96mm2
Exercise 13.09
1)
2)
3)
4)
36m2
11cm
16cm
36m2
9m
1m2
6m2
50.24m2