Transcript Area Of Shapes

Area Of Shapes.
2cm
5cm
12m
10m
A1
A2
3cm
16m
8cm
7cm
A1
12cm
A2
What Is Area ?
Area is the amount of space inside a shape:
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Area is measured in square centimetres.
1cm2
A square centimetre is a square measuring
one centimetre in each direction.
1cm
It is written as : 1cm2
1cm
Estimating The Area.
Look at the four shapes below and use your judgement to order
them from smallest to largest area:
B
A
C
D
To decide the order of areas consider the four shapes again:
B
A
C
D
To measure the area we must determine how many square centimetres
are in each shape:
Each shape is covered by 36 squares measuring a centimetre by a
centimetre .We can now see that all the areas are equal at 36cm2 each.
Area Of A Rectangle.
Look again at one of the shapes whose area we estimated:
C
Breadth
Length
What was the length of the rectangle ?
9cm
How many rows of 9 squares can the breadth hold ? 4
We can now see that the area of the rectangle is given by 9 x 4.
The formula for the area of a rectangle is:
Area = Length x Breadth
or
A = LB
for short.
We can now calculate the area of each rectangle very quickly:
(1)
(2)
A= L x B
A = 12 x 3 =36cm2
A= L x B
(3)
A = 6 x 6 =36cm2
(4)
A= L x B
A= L x B
A = 18 x 2 =36cm2
A = 9 x 4 =36cm2
Example 1
Calculate the area of the rectangle below:
(1)
4cm
7cm
(2)
Solution
3m
This area is in square metres:
A = LB
Solution
L=7
B=4
A=7x4
A=3x5
A = 28cm2
A = 15m2
1m
1m
A = LB
L=3
5m
B=5
Example 3.
2cm
Solution.
Split the shape up into two rectangles:
Calculate the area of A1 and A2.
5cm
A1
2
A2
3cm
A2
8cm
5
A1
6
Calculate the area of the shape above:
Area = A1 + A2
Area = ( 2 x 5) + (6 x 3)
Area = 10 + 18
Area = 28cm2
3
What Goes In The Box ?
Find the area of the shapes below :
(1)
(2)
6cm
2.7m
8cm
4.2m
48cm2
17cm
11.34m2
(3)
5cm
12cm
141cm2
8cm
The Area Of A Triangle.
Consider the right angled triangle below:
What is the area of the triangle ?
Area = ½ x 40 = 20cm2
5cm
Height
8 cm
What shape is the triangle half of ?
Rectangle
What is the area of the rectangle?
Area = 8 x 5 = 40 cm2
Base
The formula for the area
of a triangle is:
Area = ½ x Base x Height
A = ½ BH
Does the formula apply to all triangles ?
Height (H)
Base (B)
Can we make this triangle into a rectangle ?
Yes
The triangle is half the area of this rectangle:
The areas marked A1 are equal.
A1
A2
A1
B
A2
H
The areas marked A2 are equal.
For all triangles:
Area = ½ BH
Calculate the areas of the triangles below:
Example 1
Example 2
6cm
3.2m
10cm
Solution.
Solution.
Area = ½ x base x height
base = 10 cm
6.4m
height = 6cm
Area = ½ x base x height
base = 6.4m
height = 3.2m
Area = ½ x 10 x 6
Area = ½ x 6.4 x 3.2
Area = ½ x 60 = 30cm2
Area = ½ x 20.48 = 10.24m2
Example 3.
Calculate the area of the shape below:
12m
10m
Solution.
Divide the shape into parts:
Area = A1 + A2
A1
A2
16m
10
A1
10
12
A2
16-12 =4
Area = LB + 1/2 BH
Area = 10 x 12 + ½ x 4 x 10
Area = 120 + 20
Area = 140m2
What Goes In The Box ? 2
Find the area of the shapes below :
40cm2
(1)
(2)
6.3m
10cm
10.2 m
8cm
32.13m2
18m
(3)
12m
258m2
25m
The Area Of A Trapezium.
A Trapezium is any closed shape which has two sides that are parallel
and two sides that are not parallel.
We are now going to find a formula for the area of the trapezium:
b
Area = A1 + ( A2 + A3 )
h
Area = b x h + ½ x (a - b) x h
a
Divide the shape into parts:
Area = bh + ½ h(a - b)
Area = bh + ½ ah – ½ bh
A2
A1
A3
Area = ½ ah + ½ bh
Work out the dimensions of the shapes:
b
h
h
A2
A3
A1
a–b
Area = ½ h ( a + b )
Often common sense is as
good as the formula to work
out the area of a trapezium.
Example 1
Calculate the area of the trapezium below :
11cm
Solution ( Using the formula).
13cm
Area = ½ h ( a + b )
a = 16
16cm
b =11
h = 13
Area = ½ x 13 x ( 16 + 11 )
Area = ½ x 13 x 27
Area = 175.5cm2
11cm
Solution ( Using composite shapes).
Divide the shape into parts:
13cm
Area = rectangle + triangle
Area = LB + ½ BH
16cm
Area = (11x 13) + ( ½ x 5 x 13 )
Area = 143 + 32.5
Area = 175.5cm2
11
13
13
16 – 11 = 5
Decide for yourself if you
prefer the formula or composite
shapes.
Example 2
Divide the shape into parts:
Area = rectangle + triangle
8m
14m
Area = LB + ½ B H
A = ( 10 x 8 ) + ( ½ x 6 x 10 )
10m
A = 80 + 30
A = 110 m 2
10
10
8
14 – 8 = 6
What Goes In The Box ? 3
Find the area of the shapes below :
13cm
(1)
165cm2
10cm
(2)
20cm
2.7m
19.85m2 (to 2 d.p)
4.9m
5.4m
The Area Of A Circle.
Consider the circle below divided into quarters:
We are going to place the quarters
as shown to make the shape below
We can fit a rectangle around this shape:
At the moment it is hard to see why this
should tell us how to calculate the area of
a circle.
Now consider the same circle split into eight
parts:
The eight parts are arranged into the same
pattern as last time:
L
B
This time the shapes fit the rectangle
more closely:
L
B
This time the shapes fit the rectangle
more closely:
What length must the breadth B be close to ?
B=r
What length must the length L be close to ?
Half of the circumference of the circle.
If C = 2  r then L =  r .
We now have an approximate length and breadth of our rectangle.
r
.
r
What is the area of the rectangle ?
A=rxr
A=r2
If the circle was split into more and more smaller segments and the
segments arranged in the same pattern, then the parts would become
the rectangle shown above.
See “Autograph Extras”, “New”, “Area Of Circle” for further info’.
r
Conclusion.
The area of a circle of radius r is given by the formula
A =  r 2.
Find the area of the circles below:
Example 1.
20 cm
Example 2
2.7m
A=r2
A=r2
r = 10
r = 1.35m
A = 3.14 x 10 x 10
A = 3.14 x 1.35 x 1.35
A = 314 cm2
A = 5.72m2 ( to 2 d.p)
Example 4
Example 3
7cm
7cm
A1
A2
12cm
Split the shape into two areas.
Find half the area of a circle:
Area = A1 + A2
A=r2
2
Area = LB + ½  r 2.
L = 12
B=7
r = 3.5
A = 3.14 x 7 x 7
2
A = 12 x 7 + ½ x 3.14 x 3.5 x 3.5
A = 84 + 19.23
A = 76.93cm2
A = 103.2cm 2. (to 1 d.p)
What Goes In The Box ? 4
Find the area of the shapes below :
(1)
(2)
6.3m
7cm
153.86cm2
31.16m2 ( 2 d.p)
(3)
4.2cm
35.1cm 2 ( 1 d.p)
6.7cm