# the fencing problem

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Introduction

The Fencing Problem

Introduction

In this piece of maths coursework I am going to find out which shapes give the biggest area with a perimeter of 100m so that a farmer can fence of his field. I am going to find the area of several different shapes like triangles, rectangles, polygons and circles to see which shape give the biggest area to fence off his field. I will prove which shape gives the biggest area by putting my data into a line graph and a table of results.

## Rectangles

First I will be looking at different rectangles with the same perimeter to find out which rectangle will give me the maximum area.

450m 350m

50m 50m 150m 150m

450m 350m

Perimeter = 1000m perimeter = 1000m

Middle

H = 83316.67

H = 83316.67

### H = 288.64

333.3m

## Base x height

2

333.3 x 288.64

2

Area = 48101.85

Base (m) | Height (m) | Area (m²) |

333.3 | 288.64 | 48101.85 |

## Graph including equilateral triangle

Circle

I am now going to find out the area of a circle with a perimeter of 1000m to see if it gives the maximum area.

1000m

159.2m

The formula for the diameter is

C

D = R = D

2

1000

D = 318.5m R = 318.5

2

D = 318.5m

Radius = 159.2m

Pentagon

Now I am going to find the area of regular polygons. I will start off with a pentagon cause it has the least amount of sides in my tables and graphs

Pentagons have 5 sides and the perimeter is 1000m so each side must be 200 because 1000 ÷ 5 = 200

This is equivalent

Conclusion

111.11

20

H

55.555m

152.63

111.11

## Decagon

A decagon has 10 sides and has a perimeter of 1000m so I have to do 1000 ÷ 10 = 100m

18

H

50m 50m

153.88

100m

Number of sides | Base | Area |

3 | 333.3 | 48101.85 |

4 | 250 | 62500 |

5 | 200 | 68800 |

6 | 166.66 | 72162.1134 |

7 | 142.857 | 74174.87 |

8 | 125 | 75440 |

9 | 111.11 | 76314.24 |

10 | 100 | 76940 |

Circle | Infinite | 79582.2 |

## Conclusion

After finding the area of the following shapes – rectangle, square, triangle, circle and the polygons, I found out that the shape with the maximum area is the circle. You can clearly see in the graph and table above that no other shape reaches the area of the circle, which makes it the shape with the maximum area. Looking back at the graph you can see that none of the other shapes even if you carry on will ever reach the area of the circle. So the shape that gives the biggest area, for the farmer to fence off his field is the circle.

By Joseph Marlow

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

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