# The Fencing Problem.

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Introduction

Year 10 Coursework

Jas Singh 10D

## The Fencing Problem

The task set to us was to solve a tricky problem. A farmer had exactly 1000 metres of fencing, and she wanted to fence off a plot of level land. She was not concerned about the shape of the plot, but she said that it must have a perimeter of 1000 m. It could be any shape, square, rectangle, trapezium, circle, pentagon, anything, but it must have a perimeter (or circumference) of 1000 m. She also wanted to have the shape that would give her the maximum possible area. So our task was to investigate the shape, or shapes, that could be used to fence in the maximum area using exactly 1000 metres of fencing each time.

Once I had discovered what the task was, I had to figure out a method to go through the different shapes systematically and eventually discover which shape would give me the largest area. To do this I had an idea. Firstly, I would start off by investigating rectangles and squares, as it is easy to find the area of them. After this, I would investigate trapeziums and other quadrilaterals, and compare those results to the results I obtained with the rectangles and squares. After doing this, I would begin to investigate triangles and I would see what kind of triangle would give the largest area. Then after this, I would finally go to investigate polygons and have a look at different sided shapes, and investigate which shapes give the largest area and investigating whether the number of sides affects the area of the shape.

To further my investigation, I needed to be able to calculate the area of various shapes in order to find out the total area.

To figure out the area of rectangle

Middle

So far with the isosceles triangles, the maximum area obtained is around 48112.41 m2. So now I had to find out the area of an equilateral triangle. There is only one equilateral triangle available, and this is when all of its sides are 333.333 metres in length. By following the same method above, I was able to calculate the area of the equilateral triangle. However, using Excel was not necessary as there was only one set of results that I required. Here are the exact calculations I used to calculate the area of the equilateral triangle:

(333.3332) – (0.5 X 333.333)2 = 83333.1666 (this number was the height squared)

√83333.1666 = 288.6748459 (so now this number was the height of the triangle)

288.6748459 X 333.333 = 96224.85242

96224.85242/(divided)2 = 48112.4262 (this number was the total area of the triangle)

Success! My total area for the equilateral triangle was 48112.

Conclusion

So now I have to investigate a circle. The use of Excel was not required for this part however, as there is only one circle there can be - a circle with a circumference of 1000 metres. I needed to investigate this circle to find out what the area of this circle is.

So if we know the circumference of this circle is 1000 metres, and we want to find out the area, first we must somehow find out the radius of this circle.

If circumference is 2 X Radius X Pi (3.141592654),

Then to find the radius, you must do:

1000/(divided by) 2 = 500

500/3.141592654 = 159.1549431

If we know that to find the area of a circle is Pi X Radius2 (Radius Squared), then we have to do the following to find the area of a circle with circumference of 1000 metres:

159.15494312 X Pi =

75977.47155

So in conclusion, my task was to find a shape that could be used to fence in the maximum area using exactly 1000 metres of fencing with the shape. In this investigation, I have look at squares, rectangles, trapeziums, circles, triangles and shapes with up to 1000 sides! But I have discovered that if the farmer wants to have a fence in the shape that will give the largest area, she should use a circle with a circumference of 1000 metres. This will provide the farmer with

75977.47155 metres of land.

I can back this up by saying that I have investigated quadrilaterals, all of which provide me with a lower area that that of a circle. I have also investigated triangles, with gave an area much less than that of a circle. I proved in my polygon section that theoretically the shape with most sides would have the largest area. And so in theory a shape with infinite sides would have the largest area – and a circle is a shape that has infinite sides. Thus, I can successfully say fully backed up with evidence, that a circle with a circumference of 1000 metres will give 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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