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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:

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.

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