Embed Size (px)
Citation preview
Area Of Shapes.
8cm
2cm
5cm
3cmA1
A2
16m
12m
10m
12cm
7cm A1 A2
What Is Area ?
Area is the amount of space inside a shape:
Area Area Area Area Area
Area Area Area Area Area
Area Area Area Area Area
Area Area Area Area Area
Area is measured in square centimetres.
1cm
1cm
A square centimetre is a square measuring one centimetre in each direction.
It is written as : 1cm2
1cm2
Estimating The Area.Look at the four shapes below and use your judgement to order them from smallest to largest area:
AB
C
D
AB
C
D
To decide the order of areas consider the four shapes again:
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
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.
Length
Breadth
We can now calculate the area of each rectangle very quickly:
(1)
A= L x B
A = 6 x 6 =36cm2
(2)
A= L x B
A = 12 x 3 =36cm2
(3)
A= L x B
A = 9 x 4 =36cm2
(4) A= L x B
A = 18 x 2 =36cm2
Example 1
Calculate the area of the rectangle below:
Solution
A = LB
L = 7 B = 4
A = 7 x 4
A = 28cm2
7cm
4cm(1) (2)
3m
5m
This area is in square metres:
1m
1m
Solution
A = LB
L = 3 B = 5
A = 3 x 5
A = 15m2
Example 3.
Calculate the area of the shape above:
8cm
2cm
5cm
3cm
Solution.
Split the shape up into two rectangles:
A1A2
Calculate the area of A1 and A2.
A1
2
5A2 3
6
Area = A1 + A2
Area = ( 2 x 5) + (6 x 3)
Area = 10 + 18
Area = 28cm2
What Goes In The Box ?Find the area of the shapes below :
(1)
8cm
6cm
4.2m
2.7m
(2)
(3)
17cm
8cm
12cm
5cm
48cm2
11.34m2
141cm2
The Area Of A Triangle.Consider the right angled triangle below:
What shape is the triangle half of ?
Rectangle
8 cm
5cm
What is the area of the rectangle?
Area = 8 x 5 = 40 cm2
What is the area of the triangle ?
Area = ½ x 40 = 20cm2
Base
Height
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 ?
Base (B)
Height (H)
Can we make this triangle into a rectangle ?
Yes
The triangle is half the area of this rectangle:
B
HA1
A1
The areas marked A1 are equal.
A2
A2 The areas marked A2 are equal.
For all triangles:
Area = ½ BH
Calculate the areas of the triangles below:
Example 1
10cm
6cm
Solution.
Area = ½ x base x height
base = 10 cm height = 6cm
Area = ½ x 10 x 6
Area = ½ x 60 = 30cm2
Example 2
6.4m
3.2m
Solution.
Area = ½ x base x height
base = 6.4m height = 3.2m
Area = ½ x 6.4 x 3.2
Area = ½ x 20.48 = 10.24m2
Example 3.
16m
12m
10m
Calculate the area of the shape below: Solution.
Divide the shape into parts:
A1A2
Area = A1 + A2
A1A2
12
10
16-12 =4
10
Area = LB + 1/2 BH
Area = 10 x 12 + ½ x 4 x 10
Area = 120 + 20
Area = 140m2
What Goes In The Box ? 2Find the area of the shapes below :
(1)
8cm
10cm
(2)
10.2 m
6.3m
(3)
25m
18m
12m
40cm2
32.13m2
258m2
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:
a
b
h
Divide the shape into parts:
A1A2
A3
Work out the dimensions of the shapes:
A1
b
h A2
a – b
h
Area = A1 + ( A2 + A3 )
Area = b x h + ½ x (a - b) x h
Area = bh + ½ h(a - b)
Area = bh + ½ ah – ½ bh
Area = ½ ah + ½ bh
Area = ½ h ( a + b )
A3
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 :
16cm
11cm
13cm
Solution ( Using the formula).
Area = ½ h ( a + b )
a = 16 b =11 h = 13
Area = ½ x 13 x ( 16 + 11 )
Area = ½ x 13 x 27
Area = 175.5cm2
16cm
11cm
13cm
Solution ( Using composite shapes).
Divide the shape into parts:
11
13 13
16 – 11 = 5
Area = rectangle + triangle
Area = LB + ½ BH
Area = (11x 13) + ( ½ x 5 x 13 )
Area = 143 + 32.5
Area = 175.5cm2
Decide for yourself if you prefer the formula or composite shapes.
Example 2
8m14m
10m
Divide the shape into parts:
10
8
10
14 – 8 = 6
Area = rectangle + triangle
Area = LB + ½ B H
A = ( 10 x 8 ) + ( ½ x 6 x 10 )
A = 80 + 30
A = 110 m 2
What Goes In The Box ? 3Find the area of the shapes below :
(1)
20cm
13cm
10cm
2.7m5.4m4.9m
(2)
165cm2
19.85m2 (to 2 d.p)
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:
This time the shapes fit the rectangle more closely:
L
B
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 = r x r
A = r 2
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
A = r 2
r = 10
A = 3.14 x 10 x 10
A = 314 cm2
Example 2
2.7m
A = r 2
r = 1.35m
A = 3.14 x 1.35 x 1.35
A = 5.72m2 ( to 2 d.p)
A = r 2
2
Example 3
7cm
Find half the area of a circle:
A = 3.14 x 7 x 7
2
A = 76.93cm2
Example 4
12cm
7cm
Split the shape into two areas.
A1 A2
Area = A1 + A2
Area = LB + ½ r 2.
L = 12 B = 7 r = 3.5
A = 12 x 7 + ½ x 3.14 x 3.5 x 3.5
A = 84 + 19.23
A = 103.2cm 2. (to 1 d.p)
What Goes In The Box ? 4Find the area of the shapes below :
(1)
7cm
(2)6.3m
6.7cm
4.2cm
(3)
153.86cm2
31.16m2 ( 2 d.p)
35.1cm 2 ( 1 d.p)
