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  • Level: GCSE
  • Subject: Maths
  • Word count: 1274

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:

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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:

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...read more.

Middle

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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.

...read more.

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.

...read more.

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