# The Fencing Problem

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Introduction

The Fencing Problem

Aim:

The aim of this project is to find the largest possible area of containment by a fence of 1000 meters. This is achieved by experimenting with different shapes.

Prediction:

I predict that the largest area of containment made by the fence possible would be in the shape of a circle. This is because there are no edges:

Area of circle

Area of Hexagon

The area shaded is the possible area that can be contained by a circle. As you can see the edges cut possible areas of containment, which the circular shape can hold. Therefore I predict that the maximum area from a shape with a perimeter of 1000 would be produced in a circle.

Method:

To carry out this investigation I started with a polygon with the least amount of side’s possible that could contain an area. This was a triangle.

To go about investigating the possible areas for an Isosceles triangle I made a table containing all the possible side lengths adding to 1000 and proceeded to work out each of their areas. I did this on a computer programme called excel. It allowed me to produce formulas to save me working out every individual triangle area.

The first column in my table determined which triangle I was finding the area of. I numbered these to save confusion from 1-50.

Middle

Area of Quadrilateral

Area in Triangle

As shown by this diagram the area for the Quadrilateral is larger than the triangle. This is the first sign of a trend that will be apparent as I proceed with the investigation. This is the fact that the more sides the polygon has the larger the area will be.

I found the maximum area of the quadrilateral to be a square, with lengths of 250 on each side creating an area of 62500 meters squared as shown on the table.

Pentagon:

Once I found out that there appeared to be a pattern the next test would be to do the same experiment on the next polygon. A pentagon. A regular pentagon however cannot have a series of different combinations for the side lengths, as there is only one combination where all sides (the perimeter) add up to 1000 meters or else the pentagon would be irregular. This is 1000 divided by 5, which is 200.

200m 200m Perimeter = 1000

200m 200m

200m

There for the area found is the maximum area for a regular pentagon with a perimeter of 1000.

To find this, the pentagon has to be divided up into 5 triangles each exactly the same. The area of one of these triangles is found and then multiplied by 5 to get the overall area of the pentagon.

Conclusion

Base, which is 1000/the n’th term and height of the triangles within the polygons, which is found using the same method as before. Dropping the perpendicular and calculating the height using Tan. All this information is enough to find the area for every triangle within the polygons using the same methods as before. Finally to find the maximum areas in the last column a simple formula is needed which is the triangle area multiplied by the n’th term this will give you the maximum area for every polygon up to infinity.

I conclude that this experiment agrees with firstly my prediction, which was that the maximum area would increase as the polygon sides increased. The results table and graphs show this clearly. Also I conclude that the best shape to use to contain this plot of land is a circle, as it will give the largest area with a perimeter of 1000. I think that this investigation went well and I managed to prove my theory and discover that although perimeters in a shape can stay the same increasing the shapes sides alters the area dramatically.

I could expand on this experiment by finding the formula in a shortened version, as one fraction so it would be faster and more practical to discover maximum areas.

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

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