# Find out the greatest area that can be enclosed with 1000m of fence

Extracts from this document...

Introduction

Fence

The exercise is to find out the greatest area that can be enclosed with 1000m of fence. My first impression is that the circle has the largest area, but this needs to be investigated.

Starting with rectangles, any 2 different length sides will add up to 500, because each side has an opposite with the same length. So in a rectangle of 100m X 400m, there are two sides opposite each other that are 100m long and 2 sides next to them that are opposite each other that are 400m long. This means that you can work out the area if you only have the length of one side. The equation to work out a rectangle is

1000 = width(500 –width)

Using 10m increments here is a table of areas For a graph of Height/Area see graph 1.

## Height (m) | Width | Area (m2) |

0 | 500 | 0 |

10 | 490 | 4900 |

20 | 480 | 9600 |

30 | 470 | 14100 |

40 | 460 | 18400 |

50 | 450 | 22500 |

60 | 440 | 26400 |

70 | 430 | 30100 |

80 | 420 | 33600 |

90 | 410 | 36900 |

100 | 400 | 40000 |

110 | 390 | 42900 |

120 | 380 | 45600 |

130 | 370 | 48100 |

140 | 360 | 50400 |

150 | 350 | 52500 |

160 | 340 | 54400 |

170 | 330 | 56100 |

180 | 320 | 57600 |

190 | 310 | 58900 |

200 | 300 | 60000 |

210 | 290 | 60900 |

220 | 280 | 61600 |

230 | 270 | 62100 |

240 | 260 | 62400 |

250 | 250 | 62500 |

260 | 240 | 62400 |

270 | 230 | 62100 |

280 | 220 | 61600 |

290 | 210 | 60900 |

300 | 200 | 60000 |

310 | 190 | 58900 |

320 | 180 | 57600 |

330 | 170 | 56100 |

340 | 160 | 54400 |

350 | 150 | 52500 |

360 | 140 | 50400 |

370 | 130 | 48100 |

380 | 120 | 45600 |

390 | 110 | 42900 |

400 | 100 | 40000 |

410 | 90 | 36900 |

420 | 80 | 33600 |

430 | 70 | 30100 |

440 | 60 | 26400 |

450 | 50 | 22500 |

460 | 40 | 18400 |

470 | 30 | 14100 |

480 | 20 | 9600 |

490 | 10 | 4900 |

500 | 0 | 0 |

This shape is a parabola. If you look at the table and the graph, the rectangle with a height of 250m has the biggest area.

Middle

I will try isosceles triangles because I can vary the base easily with this, I believe that the biggest area will belong to the equilateral triangle

The formula for working the isosceles triangle out is simple. Split the base in half giving two right angles. Use Pythagoras to work out the triangles height using the base length divided by two and the other side length.

Conclusion

No. of sides | Area (m2) |

20 | 78921.894 |

50 | 79472.724 |

100 | 79551.290 |

200 | 79570.926 |

500 | 79576.424 |

1000 | 79577.210 |

2000 | 79577.406 |

5000 | 79577.461 |

10000 | 79577.469 |

20000 | 79577.471 |

50000 | 79577.471 |

100000 | 79577.471 |

From the first graph, the line straightens out as the number of side’s increases. Because I am increasing the number of sides by large amounts and they are not changing much, this indicates a graph of this would continue this trend.

I am going to see what the result is for a circle. Circles have an infinite amount of sides, so I cannot find the area by using the equation that I have used to find the other amount of sides out. I can find out the area of a circle by using pi.

- 1000/pi = diameter
- diameter / 2 = radius
- pi X radius squared = area
- which for a 1000m circumference 79577.47155 metres squared

This is larger than any number of sides and would fit on the graph, but this difference can become almost imperceptible.

Therefore, I have proved and conclude that the polygon which will give the largest area, for a perimeter of 1000 metres is the circle. It is also quite safe to conclude that this is the same for any perimeter because the size of perimeter and scale in theory make no difference.

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

## Found what you're looking for?

- Start learning 29% faster today
- 150,000+ documents available
- Just £6.99 a month