Area Of Shapes

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					            Area Of Shapes.
      2cm
                                      12m


5cm   A1                        10m
              A2         3cm
                                           16m
            8cm


                   7cm     A1         A2


                               12cm
              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.
                                                 1cm2   1cm
A square centimetre is a square measuring
one centimetre in each direction.                1cm
It is written as : 1cm2
               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)                                     (2)
                              4cm                                5m


             7cm
                                                     3m
  Solution                      This area is in square metres:         1m
  A = LB                        Solution                          1m
                               A = LB
 L= 7      B=4
                               L= 3          B=5
  A= 7 x 4
                                A= 3 x 5
 A = 28cm2
                                A = 15m2
 Example 3.                       Solution.
        2cm                       Split the shape up into two rectangles:

                                  Calculate the area of A1 and A2.
5cm      A1                                         2
                 A2         3cm
                                                                 A2         3
                                              5    A1
              8cm
Calculate the area of the shape above:                            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)                                      (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
                                                        Base
What shape is the triangle half of ?            The formula for the area
                                                of a triangle is:
Rectangle

What is the area of the rectangle?          Area = ½ x Base x Height
                                                    A = ½ BH
 Area = 8 x 5 = 40 cm2
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           The areas marked A2 are equal.
                                       H
               A1         A2                For all triangles:

                B                              Area = ½ BH
Calculate the areas of the triangles below:
Example 1                                 Example 2


  6cm
                                                       3.2m


                  10cm
                                                      6.4m
  Solution.
                                          Solution.
 Area = ½ x base x height
                                         Area = ½ x base x height
 base = 10 cm        height = 6cm
                                         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:   Solution.
          12m                            Divide the shape into parts:

                                         Area = A1 + A2
10m      A1
                    A2

              16m                  10     A1            10
                                                             A2

                                            12               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 :

(1)                40cm2        (2)
10cm                                    6.3m


                                      10.2 m
             8cm
                   18m                         32.13m2

       (3)
        12m

                                      258m2
                     25m
             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=  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.                        Example 2

   20 cm                              2.7m



 A=  r 2                       A=  r 2

 r = 10                         r = 1.35m

 A = 3.14 x 10 x 10             A = 3.14 x 1.35 x 1.35

                                A = 5.72m2 ( to 2 d.p)
 A = 314 cm2
  Example 3                             Example 4


   7cm                            7cm        A1         A2

                                              12cm
                                    Split the shape into two areas.
Find half the area of a circle:
                                    Area = A1 + A2
  A=  r 2
         2                          Area = LB + ½  r 2.
                                    L = 12        B=7   r = 3.5
  A = 3.14 x 7 x 7
                                   A = 12 x 7 + ½ x 3.14 x 3.5 x 3.5
           2
                                   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

				
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