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Measuring Area | Area
measurement facts |
| Summary of area formulae of some shapes|
Quick quiz |
Measuring Area
Area
is the amount of surface of an object. The area of regular, irregular
and complex plane figures can be measured. Both flat and curved surfaces
have area.
Example
1: Work out the area of the coloured region if the coloured square
is the unit.
If
we slide the two blue triangles in the bottom row of the grid up to
the second row of the grid we can see that in total four units on
the grid are coloured blue, so the area is 4 times the area of the
coloured square.
Area
measurement facts
| AREA |
SI
unit: square metre (m2) |
| 100
square millimetres (mm2) |
=
1 square centimetre (cm2) |
|
1
cm2
|
=
1 cm x 1cm |
|
|
= 10 mm x 10 mm
= 100 mm2 |
NOTE:
The illustration has been enlarged,
1 cm is actually smaller than pictured above
|
| 10
000 square centimetres (cm2) |
=
1 square metre (m2) |
| 10
000 square metres (m2) |
=
1 hectare (ha) |
| 100
hectares (ha) |
=
1 square kilometre (km2) |
| Odd
spot: An are is the basic unit for measuring area. A hectare,
our common unit for measurement of area of land, is 100 ares.
It is the area of a square 100m by 100m, but of course it can
be any shape. |
Summary
of area formulae of some shapes:
|
Area
of a square = l x l
|
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Area
of a rectangle =
l x w
|
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Example
2: A
rectangular block of land is measured 300 m long and 120 m wide. Calculate
the area of the block and express your answer in hectares.
| Setting
Out |
Thinking |
|
Area
= Length x Width
= 300 m x 120 m
= 36000 m2
=
3.6 hectares
|
The
length is 300 m and the width is 200 m, so I need to multiply
these measurements together.
Metres
x metres is square metres, so the units of my answer are square
metres.
I
know that 10000
m2 = 1 hectare and hectares are larger than square
metres so I divide by 10000. So, 36000
m2 = 3.6 hectares.
|
Example
3: Movie - intermediate explanation of calculation of the area of
a rectangular object
Area
of a triangle = 1/2 x (b x h) where b is length
of base and h is the height measured perpendicular to the base.
We
can show justification for the formula of the area of a triangle by
looking at the area of a square.
|
Justification
for the formula of the area of a triangle
|
Thinking
Text |
Area
of a square = l x l
|
We
know the area of a square can be found by multiplying the length
of 2 sides (l).
Area
of a square = l x l
|
Area
of a square = b x h |
If
we divide a square in half diagonally we can see that the square
has the same area as the 2 triangles. We can rename the sides
of the square as base (b) and height (h) and the area of the
square can also be expressed as, area = b x h. All we have done
is use different labels for the sides of the square.
|
Area
of a triangle = 1/2 x b x h
|
We
can now see that the area of each equal sized triangle must
be half of b x h because the area of the 2 triangles is b x
h. That is, 1/2 x b x h.
NOTE:
We are referring to the height perpendicular to the base. This
is called the perpendicular height. As these triangles are 'right-angled'
triangles, the perpendicular height can easily be seen. It is
not always as obvious as is shown in the following examples.
|
|
The
orientation of the triangle does not determine how the height
is measured. The height of a triangle is always perpendicular
to the base. The base can be any side. The perpendicular height
connects from a vertex to the opposite side (which can be the
base). It is measured at right angles to the chosen base.
|
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Example
4: Movie - full
explanation of calculation of the area of a triangular shape.
What is the area of a triangular sheet of metal which has a perpendicular
height of 1.2 m and a base of 3.7 m?
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The
perpendicular height of a parallelogram is always measured at
right angles to the chosen base,
irrespective of the orientation of the parallelogram.
The area formula works because the translation (a slide) preserves
size and shape. |
Example
5: Movie - finding the area of a parallelogram with a base of 8 cm
and a perpendicular height of 4cm.
|
Area
of a trapezium =
1/2 x (a
+ b)
x h
= half
sum of parallel side lengths x perpendicular height
|
| The
diagram below shows that 2 congruent trapeziums have been put
together by flipping and sliding to form a parallelogram with
a base of (a+b). So we can see that the the area
of a trapezium can be found by finding the area of the resulting
parallelogram, using (a+b) x h, and
dividing it in half. |
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|
| We
can say, in other words, the area of a trapezium is half the sum
of the lengths of the parallel sides multiplied by the perpendicular
distance between them. |
Area
of a circle 
|
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|
|
 ,
pronounced as 'pi', is a
special number which is approximately equal to 3.142. As pi
is used regularly in calculations of circular measurement most
calculators will have a button.
'r'
is the radius of the circle,
which is the distance from the centre of the circle to the edge
of the circle.
The
diameter of a circle, 'D',
is the distance across the circle through the centre. The diameter
is twice the length of the radius. Can you express the area
of a circle in terms of pi and its diameter, D?
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Justification
for the area of a circle formula
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|
By
dividing a circle into segments and then lining up each segment
to form a parallelogram we can see an informal justification
for the area formula for a circle.
We
know that the area of a parallelogram = base x perpendicular
height, so using the picture above we can see that
if we had taken even smaller segments of the circle and lined
them up to form a parallelogram, the radius would eventually
approximate the perpendicular height. The base of the parallelogram
is formed by the outer edge of half the segments of the circle
which is half the circumference of the circle.
So
the area of the 'parallelogram' can be approximated by x
r2
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Example
7: Movie - expert performance.
Calculation
of the approximate area of a compound shape.
|
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Do
you notice anything wrong with this movie?
The
narrator refers to the units of the shape as 'metres squared'.
The correct expression is 'square metres'.
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Quick
quiz
| 1. |
Find
the area of the following shapes: |
| |
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| 2. |
Find
the area covered by a circle with a radius of 4.5 cm.
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| 3. |
Find
the area of a parallelogram if its base is 12 cm and its perpendicular
height is 3.4 cm.
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| 4. |
I
need to find the area of a kitchen bench which is rectangular
in shape but which also has a circular breakfast area at one end.
The actual shape of the bench is similar to a rectangle with a
circle at one end. If the distance from one edge of the circular
area to the centre of the circular area is 80 cm and the rectangular
section of the bench is 2.3m long and 1 m wide. What is the approximate
total area of the bench in metres?
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| 5. |
Stanley
needs to move some trapezoidal desks (which have a table top
of the following dimensions) into a rectangular room which is
8m2 in area.
Will
he be able to fit the 12 of these desks comfortably into the
room?
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| 6. |
A
pattern which is made up of parallelograms has been incorporated
into the pavement paving and now needs to be covered with a
non-slip coating. If the perpendicular height of each of the
3 parallelograms is the same, 1.8 metres, what area in square
metres will need to be covered with the non-slip coating?
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To
view the quiz answers, click here.
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©
University of Melbourne
2003 |
