# Perimeter Investigation

Extracts from this document...

Introduction

I will be investigating the shape, or shapes, that could be used to fence a plot of land, which contains the maximum area, using exactly 1000 metres. To start I will be investigating the rectangle family as shown below:

From these results, the maximum area using exactly 1000 metres of fencing is the rectangle that measures 250 by 250. Its area is 62500m², which is the biggest area and it is square in shape, which proves that the square is the best to use, when investigating four sided shapes.

I have plotted a graph from these results obtained:

Middle

Base/m | Sides/m | Perpendicular Height/m | Area/m² |

50 | 475 | 474.3 | 11857.5 |

100 | 450 | 447.2 | 22360 |

150 | 425 | 418.3 | 31372.5 |

200 | 400 | 387.3 | 38730 |

250 | 375 | 353.6 | 44200 |

300 | 350 | 316.2 | 47430 |

350 | 325 | 273.9 | 47932.5 |

400 | 300 | 223.6 | 44720 |

450 | 275 | 158.1 | 35572.5 |

From these results, the maximum area using exactly 1000 metres of fencing is the triangle that measures, base 350m, sides 325m and an area of 47932.5m². This triangle is the best to use when investigating 3 sided shapes, because it has the biggest area and a perimeter of 1000 metres.

I decided to use regular shapes through-out my investigation because only regular shapes, i.e. shapes with equal sides, give the maximum area. So this is why I decided to use equilateral triangles.

In my triangle investigation, I decided to start with a base of 50 metres and investigate the area using that base. I did not investigate triangles with a base above 450 metres because the area kept on decreasing. For example, the triangle with base 500 metres and sides 250 metres gave an area of 0 metres.

(Here will be a graph for Base/Area and another graph for sides/perpendicular height)

From the triangle exercise, I have determined that the shape with equal sides gives the maximum area.

Conclusion

From these results, it appears that if we make a polygon of infinite sides, it would give me the maximum area. The polygon with maximum sides can only be a circle because each point on the circumference could be a side and the height will be the radius. If we make a polygon with 1000 sides of 1m each, the length of a side would become a dot and the shape would become a circle. Therefore it can be concluded that the circle would have the maximum area for a given perimeter.

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

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