A farmer has 1000 metres of fencing. She wants to use it to fence off a field. The fencing has to enclose the maximum area possible; my task is to find the shape and dimension that will give the largest area

Of a triangle:

Base x Perpendicular Height

2

If the base is fixed and the Perpendicular height is at its upper bound the highest result will be at its highest. Since I discovered that isosceles is larger than any other shape I will continue my investigation with isosceles triangles.

Triangle: Isosceles

I will keep 2 sides of the triangle equal making the triangle having one variable making it easier to notice patterns and similarities.

Answer = 11858.54cm2

Answer = 22360.68cm2

Answer = 31374.75cm2

Answer = 38729.83cm2

Answer =44194.17cm2

Answer =47434.16cm2

Answer =47925.72cm2

Answer =44721.36cm2

Answer =35575.62cm2

Table

per

000

a

b

base

475

475

50

1858.54

450

450

00

22360.68

425

425

50

31374.75

400

400

200

38729.83

375

375

250

44194.17

350

350

300

47434.16

325

325

350

47925.72

300

300

400

44721.36

275

275

450

35575.62

Largest area

47925.72

Graph

Analyse

I have noticed as the triangles sides became more equal to each other the area became larger so it is only logical to test and equilateral triangle and compare its area to the other triangles. However the area length of one side of a triangle with a perimeter of 1000m is an irrational number so I have to express this number as 1000/3

Answer =48112.52cm2

The area of the equilateral triangle is the largest out of all the other triangles I tested. The property of this type of triangle is that it is a regular shape. I think that this triangle is the largest area triangle with a perimeter of 1000m.

Quadrilaterals

In this section in my work I will test the area of 4 sided shapes. There will be no point for me to work with parallelograms that does not have right-angled triangles because the perpendicular height will not be at its upper bound. I will prove this theory again for conformation.

Answer= 40000m2

The parallelograms I have just tested both have the same side length but different areas because the perpendicular height has changed. The rectangle has the upper bound of perpendicular height because the sides' area all right angled to each other. Now I will continue my investigation with different rectangles. I will use the formula Base x Height

0 x 500 = 0m2

50 x 450 = 22500m2

100 x 400 = 40000m2

150 x 350 = 52500m2

200 x 300 = 60000m2

250 x 250 = 62500m2

300 x 200 = 60000m2

350 x 150 = 52500m2

400 x 100 = 40000m2

450 x 50 =22500m2

500 x 0 = 0m2

Table

Per

000

0

500

0

50

450

22500

00

400

40000

50

350

52500

200

300

60000

250

250

62500

300

200

60000

350

50

52500

400

00

40000

450

50

22500

500

0

0

Largest area

62500

Graph

Analyse

I analyse that the biggest shape happened to be a square. This makes sense because both the equilateral triangle and square has the largest area and both are regular shapes however the square has more area then the triangle. Because of this I thought of a new theory, the more the sides of the shape the larger the area of the shape. So the only way to test this new theory is to test on the shape with 5 sides (pentagon). So it follows a sequence: 3, 4, and 5.

General formula

The general formula in this case will work out the area of any regular shape with the perimeter of 1000m, I will follow the method that I used to work out the area of a pentagon and hexagon because both are the same an in theory it will work for every other shape. The only variable in the formula should be the number of sides and I will call it "n". I will use a pentagon to help me work out the formula, instead of righting the numbers I will substitute its formulas. e.g.: 100 will by substituted with 1000/10 etc...

Further Polygons

Now I have the general formula I can now work out the area of any regular shape without that much labour, and more efficiently with great accuracy. So now I will find the area of many polygons to see if there is a pattern between area and variables.

Table

Triangle

48112.52243

Square

62500

Pentagon

68819.09602

Hexagon

72168.78365

Heptagon

74161.47845

Octagon

75444.17382

nonagon

76,318.82

decagon

76942.08843

dodecagon

77751.05849

Graph

Table

50 sided shape

79,472.72

00 sided shape

79551.28988

50 sided shape

79565.83568

200 sided shape

79570.92645

250 sided shape

79573.28271

Graph

Analyse

I have noticed as the shape gained more sides the area got larger and larger, I also noticed as the shape gains more sides the difference in area compared to the shape before becomes less and less, in effect levelling out, I have also proven my theory "the rounder the shape the larger the area"

Circle

The next shape I decide to find the area of is the circle, roundest of all shape. However the formula can not work on a circle because the answer will become 0, this is because of the Tan (180/n) part of the formula. 180/1 becomes 180 and if use with tan will become 0, this can be proven by the tan graph, but I will not go into how tan works in this topic of work.

Working

The perimeter of the circle is 1000m and we want to know the radius of the circle to work out the area. To work out the area of a circle you use the formula 2?r.

2 ? r = 1000

Now I will rearrange the formula.

r = 1000

2 ?

Now I will simplify it.

r = 500

? = 159.1549431m

Now I use ?r2 to work out the area of the shape

59.1549431 x 159.1549431 x ? = 79,577.47155m2

Analyse

The circle is the biggest shape of all the shapes I have tested and I think it is the absolute largest shape, but it contradicts one of my theories "The more sides the shape has the larger the area." therefore I have proven which of my statements are correct and which on is incorrect.

Conclusion

After my extensive research the farmer should make a circular shape with his fence to make the largest area, however you can not make a perfect circle with fencing so close to circle with do little difference to the area of the shape as shown in my results.

Ahmed Agabani 10N