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The Fencing Problem.

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Charles Watson

The Fencing Problem


I have been given 1000 meters of fencing and my aim is to find out the maximum area inside.


I would predict that the more sides the shape has, then possibly the bigger the area it will have, although I have nothing to base this on, it will be what I am about to investigate.


I am going to start with the rectangle, I think this is a good starting block because I am able to vary the widths and lengths to see which has the bigger area. If I discover that the rectangles with equal sides i.e. square bring me the best result, then I will try to direct my investigation into furthering that particular theory.


                                Area = 40 000 m2

                                Area = 60 000 m2

Area = 62 500 m2

It appears that the square

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In order to find out the area of the triangle, I need to multiply half base and height, half base = 400 ÷ 2 = 200,

 200 x 170 = 34 000m

The Equilateral Triangle

I will work out the area by using half base multiplied by height, 333.3 ÷2 = 166.65,

∴166.65 x 270  = 44995.5

I will now try using a trapezium, however my investigation is going according to my prediction.

Trapezium _        

I am using a trapezium because it is a four sided shape that I have not used yet.

I will split it up into 3 sections and find out the middle section and then do the calculations for the outer sections using half base times height:

                                                        200 x 200 = 40 000m                                                                   50 x 200 = 10 000m

                                                                          50 x 200 = 10 000m

Total  = 60 000m

I will now carry on by looking at shapes with more sides, I will now look at a 6 sided shape:

The Hexagon

I will work out the area in the same way that I went about finding out the area of the pentagon.

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                                                        = 155.4

∴41.65 x 155.4 = 6474.1   x 12 = 77 688.9

I can now put all the shapes in a table and I notice a pattern forming,

Equilateral Triangle (3)                        44995.5        

Triangle (3)                                

Square (4)                                        625, 00

Trapezium (4)

Pentagon (5)                                        68, 562.50

Hexagon (6)                                        72, 168.79

Octagon (8)                                        75, 444.20

Dodecagon (12)                                 77, 688.90

I will now use a shape with infinite sides because I can see a trend growing, although I am not sure that I will get any where just using more shapes, so I am going to use:

The Circle

                                                Perimeter = 1000m

                                                Diameter = 1000 ÷

                                                =318.31 ÷ 2= 159.15

∴   r   = 79, 577. 47

This proves my theory correct, as areas increased we grew more closer towards the circle. I have no real solid evidence to say why this is so, although I could suggest that because circles have no corners or straight edges, this increases the overall area of usable space.

This strong positive correlation on my graph shows the pattern which I have seen and predicted since the start, this proves that my results are correct and I do not think I should any anomalies as I worked out each shape using my own calculations.

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