# Fencing Problem

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Introduction

Page of Mustafa Saai , 10 P

Fencing Problem

In this investigation I am trying to find the largest area possible to cover a field with, using a 1000 metres of fencing (Perimeter) for a farmer who wants to fence a certain area off. In order for me to find the largest area possible I am going to have to test a variety of different shapes and values to find out their properties and to find me the largest area possible with a perimeter of 1000 metres.

I hypothesize that the shape with the largest area possible with a perimeter of 1000 metres will have an infinite amount of sides. I predict this as the more sides a shape has the less area restriction is caused by sides.

Also throughout the investigation I will be testing the sides of shapes in ascending order so that my results are in a logical manner, which will make it easier to assess and evaluate my results later on. For example I will test the value of shapes with sides of - 3,4,5,6,7,8,9,10,14,22,57 etc

NOTE - None of my drawn diagrams will be to scale.

## Triangles

I will be testing the values of three different types of triangles –Equilateral, Isosceles and Scalene triangles.

In order for me to be able to work out the area of triangles by using only the lengths of 3 sides I will be using Hero’s Formula, which is;

S = P / 2

A = S(S-a)(S-b)(S-c)

## For example if I wanted to figure out the area of this triangle (or any triangle) I would follow this method (and change the values for other triangles);

- I would divide the Perimeter by 2 to give me the Semi perimeter which is 500 m
- Then I would follow Hero’s formula to figure out the area.
- I would do the calculation (S-a) first, which would be, 500m – 300m = 200m
- Then I woulddo the calculation (S-b), which would be 500m – 300m = 200m
- After that I would do the calculation (S-c), which would be 500m – 400m = 100m
- Following that I would multiply the product of (S-a) , (S-b) and (S-c) by S (500) to give me the result which will : 500 x 200 x 200 x 100 = 2000000000 m²
- Then I would square root () the answer of S(S-a)(S-b)(S-c) to give me the area of the triangle which will be : 2000000000 m² = 44721.35955 m²

PERIMETER = 1000 m AREA = 44721.35955 m²

## Equilateral

I will start investigating triangles with Equilateral triangles. As an equilateral triangle has 3 equal sides and my perimeter is 1000 m I will divide 1000m (Perimeter) by 3 (Sides), which equals 333.3333333· m (recurring) per side.

I will follow Hero’s formula in order for me to find the area :

- 1000 / 3 = 333.3333333· m (Lengths of Each Side)

- 1000 / 2 = 500 m (Perimeter / 2 = Semi perimeter)

- 500 x (500 – 333.3333333·) x (500 – 333.3333333·) x (500 – 333.3333333·) =

- 500 x 166.6666667 x 166.6666667 x 166.6666667 = 2314814815 m

- 2314814815 = 48112.52243 m²

PERIMETER = 1000 m AREA = 48112.52243m²

Side A | Side B | Side C | Perimeter | S | Area (m²) |

333.3333333 | 333.3333333 | 333.3333333 | 1000 | 500 | 48112.52243 |

### Isosceles

I am now going to find the largest possible area possible for an Isosceles triangle. As an Isosceles triangle has 2 equal sides, and 1 odd side I think that the area will not be larger than a Equilateral triangle, as the shape will be long and thin, or short and fat due to the angles.

I am also going to follow Hero’s formula in order for me to find the largest possible area.

- 1000 / 2 = 500 m
- 500 x (500 – 475) x (500 – 475) x (500 – 50) = 140625000
- 140625000 = 11858.54123

PERIMETER = 1000 m AREA = 11858.54123 m²

- 1000 / 2 = 500 m
- 500 x (500 – 450) x (500 – 450) x (500 – 100) = 500000000
- 500000000 = 22360.67978 m²

- PERIMETER = 1000 m AREA = 22360.67978 m²

- 1000 / 2 = 500 m
- 500 x (500 – 425) x (500 – 425) x (500 – 150) = 9843750000 m
- 9843750000 m = 31374.751 m²

PERIMETER = 1000 m AREA = 31374.751m²

- 1000 / 2 = 500 m
- 500 x (500 – 400) x (500 – 400) x (500 – 200) = 1500000000 m
- 1500000000 m = 38729.83346 m²

PERIMETER = 1000 m AREA = 38729.83346 m²

- 1000 / 2 = 500 m
- 500 x (500 – 375) x (500 – 375) x (500 – 250) = 1953123000 m
- 1953123000 m = 44194.17382 m²

PERIMETER = 1000 m AREA = 44194.17382 m²

- 1000 / 2 = 500 m
- 500 x (500 – 350) x (500 – 350) x (500 – 300) = 2250000000 m
- 2250000000 m = 47434.1649 m²

PERIMETER = 1000 m AREA = 47434.1649 m²

- 1000 / 2 = 500 m
- 500 x (500 – 325) x (500 – 325) x (500 – 350) = 2296875000 m
- 2296875000 m = 47925.72378 m²

PERIMETER = 1000 m AREA = 47925.72378 m²

- 1000 / 2 = 500 m
- 500 x (500 – 300) x (500 – 300) x (500 – 400) = 2000000000 m
- 2000000000 m = 44721.35955 m²

Middle

48112.23

334.5

332.75

332.75

1000

500

48111.64

335

332.5

332.5

1000

500

48110.71

After testing my fourth table of values I noticed the exact same pattern, and I also notice the area had increased quite fairly since I began narrowing down my results from my first table. I am now going to further investigate the values with the base of 332 m – 335 m. to find the area of the largest possible Isosceles triangle with a perimeter of a 1000 m.

Base | Side A | Side B | Perimeter | S | ## Area (m²) |

333 | 333.5 | 333.5 | 1000 | 500 | 48112.45 |

333.2 | 333.4 | 333.4 | 1000 | 500 | 48112.51 |

333.4 | 333.3 | 333.3 | 1000 | 500 | 48112.51955 |

333.6 | 333.2 | 333.2 | 1000 | 500 | 48112.48 |

333.8 | 333.1 | 333.1 | 1000 | 500 | 48112.38 |

334 | 333 | 333 | 1000 | 500 | 48112.23 |

After analysing and evaluating my fifth and final set of results, I have concluded that the closer the dimensions of the Isosceles triangles are to an Equilateral triangle the larger the area will be. The largest Isosceles triangle I have found with a perimeter of 1000 m has a base of 333.4 metres and sides of 333.3 metres, which is extremely close to the dimensions of an Equilateral triangle, which has sides of 333.3333333.

## Scalene

Finally I will test the values of scalene triangles. I hypothesize that the area of the triangle will not be as large as an isosceles or equilateral triangle as the dimensions cannot possibly get closer to an Isosceles triangle, which has 2 equal sides. I will be using Hero’s formula yet again to figure out the area of scalene triangles.

- 1000 / 2 = 500 m
- 500 x (500 – 350) x (500 – 305) x (500 – 345) = 2266875000 m
- 2266875000 m = 47611.71074 m²

PERIMETER = 1000 m AREA = 47611.71074 m ²

- 1000 / 2 = 500 m
- 500 x (500 – 350) x (500 – 310) x (500 – 340) = 2280000000 m
- 2280000000 m = 47749.34555 m²

PERIMETER = 1000 m AREA = 47749.34555 m²

- 1000 / 2 = 500 m
- 500 x (500 – 350) x (500 – 315) x (500 – 335) = 2289375000 m
- 2289375000 m = 47847.41372 m²

PERIMETER = 1000 m AREA = 47847.41372 m²

- 1000 / 2 = 500 m
- 500 x (500 – 350) x (500 – 320) x (500 – 330) = 2295000000 m
- 2295000000 m = 47906.15827m²

PERIMETER = 1000 m AREA = 47906.15827 m²

- 1000 / 2 = 500 m
- 500 x (500 – 350) x (500 – 325.1) x (500 – 324.9) = 2296874250 m
- 2296874250 m = 47925.71596 m²

PERIMETER = 1000 m AREA = 47925.71596 m²

I have conclude after testing the values of Scalene triangles that the closer the dimensions get to an Isosceles triangle the larger the area becomes, That means the closer the dimension are to an equilateral triangle the larger the area becomes.

I have also formulated a spreadsheet on Microsoft Excel including Hero’s Formula to display my results so analysing them is more efficient.

Side A | Side B | Side C | Perimeter | S | ## Area (m²) |

350 | 305 | 345 | 1000 | 500 | 47611.71 |

350 | 310 | 340 | 1000 | 500 | 47749.35 |

350 | 315 | 335 | 1000 | 500 | 47847.41 |

350 | 320 | 330 | 1000 | 500 | 47906.16 |

350 | 325.1 | 324.9 | 1000 | 500 | 47925.71596 |

From looking at these results and referring to my past results for my Isosceles triangles I will now try and test the values that are the closest to Equilateral triangles as they have shown the largest area possible with a perimeter of 1000 m.

- 1000 / 2 = 500 m
- 500 x (500 – 333.3) x (500 – 333.2) x (500 – 333.5) = 2314812870 m
- 2314812870 m = 48112.50222 m²

PERIMETER = 1000 m AREA = 48112.50222 m²

Now that I have investigated the scalene triangle with sides of 333.3 x 333.2 x 333.5 which equals 48112.50222 m² I have concluded that the closer the dimensions are to an Equilateral triangle are the larger the area will be.

Side A | Side B | Side C | Perimeter | S | ## Area (m²) |

350 | 305 | 345 | 1000 | 500 | 47611.71 |

350 | 310 | 340 | 1000 | 500 | 47749.35 |

350 | 315 | 335 | 1000 | 500 | 47847.41 |

350 | 320 | 330 | 1000 | 500 | 47906.16 |

350 | 325.1 | 324.9 | 1000 | 500 | 47925.71596 |

333.3 | 333.2 | 333.5 | 1000 | 500 | 48112.50222 |

After analysing my second table of results for scalene triangles, the table has strengthened my assumption that the closer the dimensions are to an equilateral the larger the area is.

I am now going to draw a table to show the largest areas I have found for each triangle. I am then going to analyse my results one final time.

Triangle | Area | Perimeter | Dimensions | Comment |

Equilateral | 48112.52243 | 1000 metres | 333.3333333· x 333.3333333· x 333.3333333· | The largest triangle of all of them. |

Isosceles | 48112.51955 | 1000 metres | 333.4 x 333.3 x 333.3 | 2nd largest triangle. The more narrowed down the closer dimension were to an equilateral |

Scalene | 48112.50222 | 1000 metres | 333.2 x 333.3 x 333.5 | The smallest triangle amongst the other 2. The more narrowed down the closer the dimensions were to an Isosceles and then an Equilateral |

After testing, analysing and evaluating the values for Equilateral, Isosceles and Scalene triangles I have come to a conclusion that a shape with all sides that are equal will have a larger area than a shape which has only 2 or more sides equal. This is shown in my investigation to find the largest area of a triangle with a perimeter of 1000 metres.

For example;

Quadrilaterals

I am now going to investigate quadrilaterals. I will investigate; Squares, Rectangles, Parallelograms and Trapeziums. I hypothesize that the largest area for a quadrilateral shape with a perimeter of 1000 metres will be a square, as all of the sides are equal.

Square

I am going to start of investigating Quadrilaterals with a square. As a square has 4 equal sides I will only have one example as there is only one possible square with a perimeter of 1000 m. I will divide 1000 m (perimeter) by 4 (sides), to give me the length of each of the sides – 250 m. Then I will multiply the Length by Length to give me the area.

- 1000 m / 4 = 250 m (Sides)

- 250 x 250 = 62500 m²

PERIMETER = 1000 m AREA = 62500 m²

The largest area possible for a Square

Side A | Side B | Side C | Side D | Perimeter | Area (m²) |

250 | 250 | 250 | 250 | 1000 m | 62500 m² |

This table presents the largest possible area using 1000 metres of fence for a square.

Rectangles

I am now going to investigate the dimensions of Rectangles. I think that the area of a rectangle will not be as large as the area of a square as it has 2 equal sides, and another 2 equal sides whereas as a square has 4 equal sides.

- 499 x 1 = 499 m²

PERIMETER = 1000 m AREA = 499 m²

- 450 x 50 = 22500 m²

PERIMETER = 1000 m AREA = 22500 m²

- 400 x 100 = 40000 m²

PERIMETER = 1000 m AREA = 40000 m²

- 350 x 150 = 52500 m²

PERIMETER = 1000 m AREA = 52500 m²

- 300 x 200 = 60000 m²

I am not going to draw the next rectangle because if I increase a.1 and a.2 and decrease b.1 and b.2 by 50 again the shape created will be a square (250 x 250). Instead I am going to construct a table with the aid of Microsoft Excel in order to investigate the area’s I have for the rectangles I have created so far.

I am going to create a formula in Microsoft Excel yet again to make my investigation into rectangles more accurate and efficient.

Length of a.1 / a.2 | Length of b.1 / b.2 | Perimeter | Area (m²) |

1 | 499 | 1000 | 499 |

50 | 450 | 1000 | 22500 |

100 | 400 | 1000 | 40000 |

150 | 350 | 1000 | 52500 |

200 | 300 | 1000 | 60000 |

250 | 250 | 1000 | 62500 |

300 | 200 | 1000 | 60000 |

350 | 150 | 1000 | 52500 |

400 | 100 | 1000 | 40000 |

450 | 50 | 1000 | 22500 |

499 | 1 | 1000 | 499 |

An observation I have made from analysing my table is that the lengths of sides surrounding the dimensions > 250 x 250 are the same dimensions except that sides b.1 / b.2 have taken the values of sides a.1 / a.2 and vice versa.

I have also learnt that the closer the dimension are to a square the larger the area is which in some ways similar to my investigation into triangles as the closer the dimensions for my isosceles triangle were to the equilateral the larger the area was.

I am now going to further investigate the dimensions surrounding the ones of a square by narrowing the range of results I am going to investigate to try and find the largest possible area for a rectangle with a perimeter of 1000 metres. In order to do this I am going to construct a table so I can test the values around the dimensions of a square.

Length of a.1 / a.2 | Length of b.1 / b.2 | Perimeter | Area (m²) |

249.8 | 250.2 | 1000 | 62499.96 |

249.9 | 250.1 | 1000 | 62499.99 |

250.1 | 249.9 | 1000 | 62499.99 |

250.2 | 249.8 | 1000 | 62499.96 |

Conclusion

- I am going to use this formula to calculate the area of the Circle

- Circumference = x Diameter

- I know my perimeter must be 1000 m so I will replace circumference by 1000 m

- 1000 m = x Diameter

- Now I will change the subject of the formula

- Diameter = 1000 m

- Diameter = 318.3098862 m

- Now that I have my diameter I will use this formula to work out the area of the circle.

- Area = x Radius²

- The radius is half of the diameter so I will divide the diameter by 2 .

- 318.3098862 = 159.1549431 m

- Now that I have my radius I can follow the rest of the formula.
- Area = x 159.1549431²

- Area = 79577.47155 m²

Radius | Circumference | Area (m²) |

159.1549431 | 1000 m | 79577.47155 |

Conclusion

Now that I have found the shape with the largest area with a fixed perimeter of 1000 metres I can draw a table to show the largest version of each shape that I have discovered.

Shape | Sides | Area (m²) |

Parallelogram | 4 | 39998.48 |

Scalene Triangle | 3 | 48112.50222 |

Isosceles Triangle | 3 | 48112.51955 |

Equilateral Triangle | 3 – Equal Sides | 48112.52243 |

Rectangle | 4 | 62499.99 |

Square | 4 - Equal Sides | 62500 |

Pentagon | 5 - Equal Sides | 68819.096 |

Hexagon | 6 - Equal Sides | 72349.2467 |

Heptagon | 7 - Equal Sides | 74161.47852 |

Octagon | 8 - Equal Sides | 75444.1738 |

Nonagon | 9 - Equal Sides | 76468.81721 |

Decagon | 10 - Equal Sides | 76942.08846 |

20 Sided Shape | 20 - Equal Sides | 78921.89395 |

50 Sided Shape | 50 - Equal Sides | 79472.72422 |

100 Sided Shape | 100 - Equal Sides | 79551.28988 |

Circle | Infinite | 79577.47155 |

By analysing this table and my graph I can conclude that the more equal sides there are in any shape the larger the area is, except for circles because circles have an infinite amount of sides and that is the reason that my formula did not work with the circle. Also the reason the area of the parallelograms I investigated are quite small compared to the other quadrilaterals is because I decided to stop searching for the largest area of a parallelogram as I knew it would not be larger than a rectangle or square.

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

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