Calculate the Area of a Shape

                                           2

If you look at my table this correct.

Now I’m correct with my formula I can find area of a shape of any size as long as there is 1 dot inside.

E.G    

             N= 100       A=100          =50cm2

                                       2

Now I have investigated shapes with 2 dot inside I will put my answers into a table.

• After creating a table 3, I have noticed a pattern in the area column.
• The pattern is going up in ½ cm2 each time a dot is added to the edge.

By looking at the pattern I have created formula in both words and algebra.

                         Area = number of dots +1        A= N +1

2. 2

To show I’m right:

    When =N=6            A= 6+1             = 4cm2

                                           2                            

If you look at my table it is correct.

Now I’m correct with my formula I can find area of a shape of any size as long as there are 2 dots inside.

E.G

    When =N=80            A=80 +1         =40+1

2. =41cm2

After completing all the diagrams, tables and formulas I have noticed a pattern between the three formulas:

I have put all the formulas into a table:

A=N-1                0 dots inside

     2

A=N+0               1 dot inside   (I have added the 0 just to show the pattern works)

     2

A=N+1              2 dots inside    

     2

As you can see the formulas are going up in 1, every time the dots inside the shape are increased.

By doing this I have created 1 formula which is the three formulas combined as 1

This is:

Area= number of dots + dots inside -1

                                                              2

                                      A=N + D -1

                                            2

Now I’m correct with my formula I can find area of a shape of any size with any number of dots inside and joint.

EG:

           When= N= 12             D= 5            A=12+5-1         = 6+5-1=10cm2

                                                                          2

Calculating the area of a shape

In this project I have investigated the area of shapes; I have investigated shapes with 3,4,5,6 and 7 dots joined and I will look at building up the dots inside the shapes later on.

After drawing the shapes I have noticed that, shapes with the equal number of dots have the same area, this can be seen on the previous diagram pages. So from now on I will only draw 1 of each type of diagram.

I will now put all of my results for no dots inside into a table.

Table 1

                   

• After creating a table 1, I have noticed a pattern in the area column.
• The pattern is going up in ½ cm2 each time a dot is added to the edge.
• By looking at the pattern I have created formula in both words and algebra:

Area = the number of dots   - 1          A=N – 1          

A= Area                                                             2                                      2  

N= Number of dots                                          

To prove I’m right

When N = 3    A=3 - 1 = 1 ½ -1     When N=4 A=4 -1 =2-1      

                              2      = ½ cm2                             2     =1 cm2                    

When N=5      A=5 -1 = 2 ½ -1      When N=6 A=6 -1 =3-1      When N=7 A=7 – 1 =3 ½ -1

                              2     = 1 ½ cm2                          2      =2cm2                           2       =2 ½ cm2

If you look at my table you will see this correct.

Now I know I am definitely correct with my formula I can find area of a shape of any size as long as there are no dots inside and this is shown on back of the diagram sheet.

E.G

                 N=10             A=10 – 1            = 5 - 1

                                             2                   = 4cm2

Now I have investigated shapes with 1 dot inside I will put my answers into a table.

Table 2

 

 

•  After creating a table 2, I have noticed a pattern in the area column.
• The pattern is going up in ½ cm2 each time a dot is added to the edge.

By looking at the pattern I have created formula in both words and algebra.

A= area                         area= the number of dots         A=N

N=number of dots                                 2                               2

To show I’m right:

   

When N=3      A= 3 =1 ½cm2   When N=4 A=4 =2cm2   When N=5 A=5 =2 ½cm2

                            2                                            2                                       2

When N=6   A=6 =3cm2         When N=7 A=7=3 ½cm2

                           2                                              2

If you look at my table this correct.

Now I’m correct with my formula I can find area of a shape of any size as long as there is 1 dot inside and this is shown on back of the diagram sheet.

E.G    

             N= 10       A=10        =5cm2

                                     2

Now I have investigated shapes with 2 dot inside I will put my answers into a table.

Table 3

• After creating a table 3, I have noticed a pattern in the area column.
• The pattern is going up in ½ cm2 each time a dot is added to the edge.

By looking at the pattern I have created formula in both words and algebra.

                         Area = number of dots +1        A= N +1

3. 2

To show I’m right:

     

  When N=3      A=3+1 =2 ½cm2   When N=4 A=4 +1 =3cm2   When N=5 A=5 +1 =3 ½ cm2

                             2                                                2                                              2

When N=6   A=6+1=4cm2         When N=7 A=7+1=4 ½cm2

                           2                                                 2

If you look at my table it is correct.

Now I’m correct with my formula I can find area of a shape of any size as long as there are 2 dots inside and this is shown on the diagram sheet.

E.G

    When =N=10            A=10 +1         =5+1

3. =6cm2

After completing all the diagrams, tables and formulas I have noticed a pattern between the three formulas:

I have put all the formulas into a table:

Table 4

A=N-1                0 dots inside

     2

A=N+0               1 dot inside   (I have added the 0 just to show the pattern works)

     2

A=N+1              2 dots inside    

     2

As you can see the formulas are going up in 1, every time the dots inside the shape are increased.

By doing this I have created 1 formula which is the three formulas combined as 1

This is:

A=Area    D=Dots inside N=number of dots joined

                          Area= number of dots + dots inside -1

                                                 2

                                      A=N + D -1

                                            2

Now I’m correct with my formula I can find area of a shape of any size with any number of dots inside.

EG:

           When= N= 10             D=2             A=10+2-1         = 5+2-1=6cm2

                                                                          2

Now I will show some examples for the final formula for 0 dots, 1 dot and 2 dots in side on some of the past diagrams I have drew , but with the final formula not the individual formulas.

   

EG 1:

When N=5 D=0      A=5 +0 - 1 =1 ½ cm2

                                      2

EG 2:

When N=7 D=1      A=7 +1-1 = 3 ½ cm2

                                      2

EG 3:

When N=3 D=2    A=3 +2 -1 =2 ½ cm2

                                    2

     

Mathematics

Coursework

Area of a shape

Leigh Bevan