# Calculate the Area of a Shape

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Introduction

## Calculating the area of a shape

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

My prediction is that shapes with the same amount of dots joined and number of dots inside, will have the same area.

After drawing the shapes I have notice 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

Number of dots N | Area A |

3 | ½ cm2 |

4 | 1 |

5 | 1 ½ cm2 |

6 | 2 |

7 | 2 ½ cm2 |

- 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 = 6 A= 6 - 1 = 3-1

2 = 2cm2

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

Now I know I am

Middle

Table 2

A=N+0

2

Table 3

A=N +1

2

Final formula

A=N+D -1

2

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

Number of dots N | Area A |

3 | ½ cm2 |

4 | 1 cm2 |

5 | 1 ½ cm2 |

6 | 2 cm2 |

7 | 2 ½ cm2 |

- 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

Number of dots N | Area A |

3 | 1 ½cm2 |

4 | 2cm2 |

5 | 2 ½cm2 |

6 | 3cm2 |

7 | 3 ½cm2 |

Conclusion

E.G

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

- =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

Table | Formula |

Table 1 | A=N -1 2 |

Table 2 | A=N+0 2 |

Table 3 | A=N +1 2 |

Final formula | A=N+D -1 2 |

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

This student written piece of work is one of many that can be found in our GCSE Fencing Problem section.

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