# To discover the largest obtainable area within a fenced area 1000m I was given the project of finding what shape with circumference of 1000m has the greatest area. How I am going to set about this task

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Introduction

To discover the largest obtainable area within a fenced area 1000m I was given the project of finding what shape with circumference of 1000m has the greatest area. How I am going to set about this task First I am going to start with 3-sided shapes as these have the least amount of sides, which enclose an area. From there I am going to work out the areas of a selection of 3-sided shapes, 4-sided shapes and then I am going to look at which ones has the best area. I am then going to see if there is anything particular to each shape and whether the 3 or 4-sided shapes had the greatest area. I am then going to do the same for 5, 6, 7 and 8-sided shapes. I am going to find if the 8-sided shape has a better area in any instance than the 3 and 4-sided shapes. My Prediction I think that as the number of sides increases then the area at the equilateral will be greater. I think this because if u put a square inside a circle then there is a great amount of wasted space visible and as the number of sides increases the hemispheres off the chords of the circle decrease in area. ...read more.

Middle

I also know that each side is 200m long, because 1000 divided by 5 is 200, so the base of the triangle is 100m. Because of the law SOHCAHTOA I can see I need to us Using SOHCAHTOA Tangent. H=100 Tan36 H=137.638 This is the value of the height so I can now work out the area of the right-angled triangle. Area = 1/2*b*h Area = 1/2*100*137.638=6881.91 This is the area of one right-angled triangle. Area = 6881.91*2 Is the area of one Isosceles triangle. Area = 13763.82*5 Is the area of the regular Pentagon. Area = 68819.1m2 Now I am going to work out the areas of 6, 7-sided shapes using the same method: 6-sided H = 83.3 Tan30 H = 144.38 Area = 1/2*b*h Area = 1/2*166.6*144.38 Area = 6014.065*2 Area = 12024.13*6 Area = 72144.78 7-sided H = 71.4 Tan25.7 H = 148.323 Area = 1/2*b*h Area = 1/2*71.4*148.323 Area = 5295.131 Area = 5294.131*2*7 Area = 74161.644m2 Analysing the results As you can see form my table of results as the number of sides increases the area increases. The best shape is shows is the circle where thought the area is not very much higher when the number of sides is very high, never the less it is still higher. ...read more.

Conclusion

Tan(180/N) Area of a Circle =p*Radius� Circumference =p*Diameter Diameter = Circumference p Diameter = 1000 p Radius = Diameter 2 Area of a Circle: A=p*(1/2(1000)�) ( ( p ) ) A=p((500)�) (( p ) ) A=p(250000) ( p*p ) A=250000 p Area of a 10000000-sided polygon: A=(250000/N) Tan(180/N) A=250000/10000000 Tan(180/10000000) A=250000/10000000 Tan0.000018 A= 250000 10000000*(Tan0.000018) A= 250000 10000000*(Tan0.000018) Tan0.000018=0.0000003141592654 10000000*0.0000003141592654=3.141592654 p=3.141592654 A=250000 p This shows that algebraically the area of a 10000000-sided polygon is the same as a circle. This is because of the lack of precision with the calculating equipment I used. If I was to have an infinitely accurate calculator then even with a shape of 10trillion sides it would show a minute area difference. Why is it that the more sides you have the larger the area? If you look at shapes with small numbers of shapes, for example to triangle with 3 sides and put an equilateral inside a circle you can see the wasted space. As you increase the amount of sides the hemisphere from each corner to the next is decreased even when you get to shapes with large amounts of sides where the area is only increased slightly there is still a change. ...read more.

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