# Fencing Problem

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Introduction

John Smith Thomas Alleynes High School Maths Coursework

GCSE Maths Coursework-

The Fencing Problem

Introduction

I am going to investigate different a range of different sized shapes made out of exactly 1000 meters of fencing. I am investigating these to see which one has the biggest area so a Farmer can fence her plot of land. The farmer isn’t concerned about the shape of the plot, but it must have a perimeter of 1000 meters, however she wishes to fence off the plot of land in the shape with the maximum area.

Rectangles

I am going to look at different size rectangles to find which one has the biggest area.

Formula:Length x Width

Table Of results

Length | Width | Area |

0 | 500 | 0 |

50 | 450 | 22500 |

100 | 400 | 40000 |

150 | 350 | 52500 |

200 | 300 | 60000 |

250 | 250 | 62500 |

300 | 200 | 60000 |

350 | 150 | 52500 |

400 | 100 | 40000 |

450 | 50 | 22500 |

500 | 0 | 0 |

Conclusion

I have found that the four sided shape that had the biggest area when using 1000 meters of fencing, was a square with the measurement of 250m x 250m and the area=62500m²

Isosceles Triangles

I am now going to look at different size Isosceles triangles to find which one has the biggest area. I am going to use Pythagoras Theorem to find the height of the triangle.

Pythagoras Theorem:a²=b²+c²

Formula To Find A Triangles Area:½ x base x height

1. Base=100m Sides=450m

Area: ½ x b x h

½ x 100 x 477

=23850m²

2. Base=200m Sides=400m

Area: ½ x b x h

½ x 200 x 387

Middle

Tan = opp

adj

Tan 64.3 = h

71.4275

2.08 x 71.4275 = h

148.6 = h

½ x 142.855 x 148.6 = 10614.1

10614.1 x 7 = 74298.7

Area = 74298.7m²

Octagons

I am now going to look at an Octagon to find out its area when using 1000meters of fencing.

Formula To Find the Area Of A Octagon:

Exterior Angle = 360°

No. Of Sides

Interior Angle = 180°-Exterior Angle

I will then use tangent of an angle, to work out the area of one of the eight segments of the octagon. I will multiply this number by 8 because the 8 segments are all the same size. Therefore I will get the overall area of the octagon.

Octagon Area

Area: Ex. Angle = 360° = 45°

8

Int. Angle = 180°-45°=135°

= 135°÷2=67.5°

Tan = opp

adj

Tan 67.5 = h

62.5

2.41 x 62.5 = h

150.625 = h

½ x 125 x 150.625 = 9414.0625

9414.0625 x 8 = 75312.5

Area = 75312.5m²

Nonagons

I am now going to look at a Nonagon to find out its area when using 1000meters of fencing.

Formula To Find the Area Of A Nonagon:

Exterior Angle = 360°

No. Of Sides

Interior Angle = 180°-Exterior Angle

I will then use tangent of an angle, to work out the area of one of the nine segments of the nonagon. I will multiply this number by 9 because the 9 segments are all the same size. Therefore I will get the overall area of the nonagon.

Nonagon Area

Area: Ex. Angle = 360° = 40°

9

Int. Angle = 180°-40°=140°

= 140°÷2=70°

Tan = opp

adj

Tan 70 = h

55.55

2.75 x 55.55 = h

152.8(1d.p) = h

½ x 111.1 x 152.8 = 8488.04

8488.04 x 9 = 76392.36

Area = 76392.36m²

Decagons

Conclusion

Conclusion

I was set the task of finding a shape with the biggest area when using 1000meters of fencing, so that a farmer could fence a plot of her land. The shape didn’t matter, however it had to have the maximum area.

I investigated regular shapes because it is only when all the sides are equal I will get the highest area of that shape.

From investigating different shapes I overall found that a circle had the maximum area. A circles area when using 1000meters of fencing is 79577.52m².

I noticed that the area didn’t change much when I started investigating the area of a Nine sided shapes and onwards. Also when I reached a 40-sided shape the area change in a very small amount. As you can see from my graph (on the previous page) the line goes up then curves round and becomes a constant line until we reach the area of a circle, where it stops as I found out because a circle has the maximum area.

In conclusion I have found that the more sides a shape has the bigger the area will be until you get to a circle, which as I found out has the maximum 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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