# The Fencing Problem

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Introduction

GCSE Maths Coursework

### The Fencing Problem –

To investigate shapes with a perimeter of 1000m and find the largest area this project will be divided into three main sectors.

The first will be investigating shapes with different number of vertices, Triangles, Squares, and different polygons.

The Second will be introducing algebra to make calculating the areas easier. This may only work for regular shapes, I predict that regular shapes will have biggest areas, I hope to prove this in my investigation.

Finally displaying graphs and tables to verify patterns in areas and possible coherency’s.

Middle

Octagon (Regular, 8 sides)

Triangle is a section of the octagon.

Octagon = 75444.17382m2

Dodecagon (Regular, 10 sides)

Triangle is a section of the dodecagon.

Dodecagon = 76942.08842m2

## Circle

Circumference = 1000m

Diameter = 318.3098m

Radius = 159.1549m

Area = Πr2

Area = 3.14 X 159.15492 = 79577.42845m2

## Circle = 79577.42845m2

Π | times | r2 | = | Area |

3.1415926454… | x | 25330.28219 | = | 79577.42845m2 |

So far I have been using basic algebra to calculate the areas of the shapes.

My results so far have proved my prediction that the higher the number of sides, the larger the area, with the circle being the largest.

I feel that I can change my formula so that there is only one variable, the number of vertices.

(2(250/n)2)

Conclusion

Area (m2)

n

(2(500/n)2) X TAN (90-(180/n)) n

3

48112.52243

4

62500

5

68819.09602

6

72168.78365

7

74161.47845

8

75444.17382

9

76318.81721

10

76942.08843

11

77401.9827

12

77751.05849

16

78552.17956

18

78767.80305

20

78921.89393

25

79158.15088

50

79472.72422

100

79551.28988

Circle

79577.42845

This table shows that a hundred sided shape still has a smaller area than a circle.

As there are more vertices in a shape, the difference between the previous one becomes less, so eventually, a million sided shape would not only look very much like a circle, but would also have a very similar area.

This investigation has proved that the largest areas come from shapes with the most sides, the largest being a circle with an infinite number of sides.

The shape of the fence that the farmer should build would be a circle to get the largest area.

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

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