MEASUREMENT
Area
Area
• The area of a figure or shape is the amount of 2-dimensional space taken up by
that figure
• Area is measured in square units
• mm2
• cm2
• m2
• km2
• or just units squared
• Hectare (ha) is also used with
1ha = 10 000 m2
• Area can be defined as the number of square units contained within a shape’s
boundary.
• 2-d shapes are sometimes called ‘plane’ shapes
The Unit Square
If this represents 1 square unit
4 square
units
This shape is then made up of 8
of these single units.
2 square
units
Equivalent to
4 ⨯ 2 = 8 square units
To work out the area of any shape, you can reduce the problem into this
simplified format.
Area Conversions
To convert between square units, the base conversion factor must be squared.
10mm
1cm
A = 1cm2
or
A = 100mm2
100cm
10mm
1cm
Conversion factor
1 cm = 10mm
1 cm2 = 102 mm2 = 100 mm2
A = 1m2
or
1m
A = 10000cm2
100cm
1m
Conversion factor
1 m = 100cm
1 m2 = 1002 cm2 = 10000 cm2
Area Conversions
Action
Conversion
mm2  cm2
100 mm2 = 1 cm2
mm2  m2
1000000 mm2 = 1 m2
cm2  m2
10000 cm2 = 1m2
m2  km2
1000000 m2 = 1 km2
m2  hectares
10000 = 1 hectare
hectares  km2
100 hectares = 1 km2
15 minute activity
Each group will be allocated a different 2-d shape
- Square, rectangle, parallelogram, rhombus, triangle, circle
In your groups I want you to investigate all the properties of your shape
(ensure some are mathematical), and anything else interesting about your
shape.
You will be given an A4 sized piece of card to decorate your card. Feel free
to make use of coloured pencils and pens.
Prize tomorrow for the best looking and most informative poster.
Area - Square
A square has the following properties.
•
•
•
•
4 sided shape
Each side is the same length
Each internal angle is 90°
Opposite sides are parallel
Area = base ⨯height
=a⨯a
= a2
a
a
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Area - Rectangle
A rectangle has the following properties.
• 4 sided shape
• Each internal angle is 90°
• Opposite sides are parallel and are of equal length
Area = base ⨯ height
=a⨯b
= ab
b
a
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Area - Parallelogram
A parallelogram has the following properties.
•
•
•
•
4 sided shape
Opposite sides are parallel and are of equal length
Opposite internal angles are equal
Opposite angles are supplementary i.e. add to 180°
Area = base ⨯ height
=a⨯b
= ab
b
a
Note: height is perpendicular height, not length of side
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Area - Rhombus
A rhombus has the following properties.
•
•
•
•
•
b
4 sided shape with all sides of equal length
Opposite sides are parallel
Opposite internal angles are equal
Adjacent angles are complementary i.e. add to 180°
Diagonals bisect each other at right angles
Area = base ⨯ height
=a⨯b
= ab
a
Note: height is perpendicular height, not length of side
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Area -Trapezium
A trapezium has the following properties.
• 4 sided shape
• 1 pair of opposite sides parallel
b
Area = average (parallel sides) ⨯ height
h
=
(𝑎+𝑏)
2
×ℎ
a
Note: height is perpendicular height, not length of side
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Area -Triangle
A triangle has the following properties.
• 3 sided shape
• 3 internal angles
• The internal angles always add to 180°
Area =
h
a
(𝑏𝑎𝑠𝑒)
2
𝑎
=
2
×ℎ
×ℎ
Note: height is perpendicular height, not length of side
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Area –Circle
A circle is
• A set of points of equal distance from the centre
d
r
Area = π × 𝑟2 or
=π×
𝑑2
4
Note: a semi-circle (half a circle) is also known as a hemisphere
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Area –Sector of a Circle
If a sector has an angle at the centre equal to 𝞱, then what would the
area of the sector be?
r
Area sector
𝞱
r
𝜃
=
360
× π × r2
Summary
Practice Problems
Calculate the area of the following shapes:
7.5 cm
What % is the
blue area?
Find grey area
Some Harder Practice Problems
Calculate the area of the following shapes:
A chocolate bar is wrapped in
a rectangular piece of foil
measuring 14cm by 20cm.
Calculate the area of the piece
of foil.
How many pieces could be
cut out from a larger sheet of
foil measuring 2.4m by 1.4m?
What is the total area of the shaded part of rectangle ACDB?
Areas of Composite Shapes
A composite shape (also called a compound shape) is made up of various
parts.
To find the area of a composite shape, find the areas of each individual
shape, and either add or subtract as you need to.
These are examples of composite shapes:
Example: find the area of this shape
Area of compound shape:
2 cm
= area of Rectangle + area of Triangle
+ area of semi-circle
5 cm
Area = b × h + ½ b × h + ½ π × r2
4 cm
= 4 × 5 + ½ 4 × 2 + 0.5 π × (2)2
= 30.3 cm2 (1 dp)
Example:
Think of a typical running track.
What is the perimeter?
How long are the straight sections?
Calculate the area enclosed by the track.
Acircle + Arectangle = Atotal
3183.1 + 6366.2 = 9549.3 m2 (1dp)
100 m
d=?
Example:
A glass porthole on a ship has a diameter of 28
cm. It is completely surrounded by a wooden ring
that is 3 cm wide.
a.) Calculate the area of glass in the porthole
A = πr2
r = 14 cm
A = π (14)2
A = 616 cm2
b.) Calculate the area of the wooden ring
Area of porthole = πr2 , r = 17 cm
(including frame) = 908 cm2
Area of frame = 908-616
= 292 cm2
2d shapes:
Composite Shapes :
Homework
Exercise F: Pages 170-171
Exercise G: Pages 172-174