# Maths investigation - The Fencing Problem

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Introduction

Maths investigation – The Fencing Problem

Aim – to investigate which geometrical enclosed shape would give the largest area when given a set perimeter.

In the following shapes I will use a perimeter of 1000m. I will start with the simplest polygon, a triangle.

Since in a triangle there are 3 variables i.e. three sides which can be different. There is no way in linking all three together, by this I mean if one side is 200m then the other sides can be a range of things. I am going to fix a base and then draw numerous triangles off this base. I can tell that all the triangles will have the same perimeter because using a setsquare and two points can draw the same shape. If the setsquare had to touch these two points and a point was drawn at the 90 angle then a circle would be its locus. Since the size of the set square never changes the perimeter must remain the same.

Middle

From the diagram the area must equal –X² +500X

Unlike in the previous example, this turns out to be a quadratic equation so I can plot it on a graph.

As you can see from the graph the maximum point is when X = 250. When this number is plugged into the formula the rectangle is really a square.

What do a square and an equilateral triangle have in common? They are both regular shapes i.e. all angles equal, all sides equal.

Why is this?

Lets take the triangle example first. When you make one side longer you will make the other shorter. This will decrease the height, which means the area will be smaller. When both sides are the same length they extend the height to its highest possible. Why does an equilateral triangle have a larger area than an icosoles triangle? you could think of it like this.

Lets take the square example. Obviously the longer the sides the bigger the area.

Conclusion

2πr = 1000

πr = 500

r = 159.2

A=πr²

A = π159.2²

A = 79622.53

Lets add this to our table of results.

Number of sides | Maximum area with perimeter of 1000M |

3 | 48112.5 |

4 | 62500.0 |

5 | 68794.7 |

6 | 72168.8 |

7 | 74288.7 |

8 | 75425.4 |

∞ | 79622.5 |

The circle has the biggest area with a 1000M perimeter out of all the polygons.

Why is this?

When a shape is split up into triangles, the more sides it has, the more triangles there will be yet these triangles will become smaller as the number of sides increase. The amount at which the area of the triangle decreases is not as great as the amount the side increases by. When you split the shape into triangle, the more sides the shape has the smaller the angle gets in between the two equal sides but the perimeter of these triangles increase as the shape has more sides. The higher the perimeter, the larger area you can make providing the perimeter is well used i.e. the triangle is in the form of an icosoles triangle. A circle would have infinite sides and its angles are bigger since bigger angles can encompass more.

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

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