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  • Level: GCSE
  • Subject: Maths
  • Word count: 2796

Fencing Problem

Extracts from this document...

Introduction

Kanwerbir Singh Khalsa        11x2        Mr. Bradford

Fencing Problem

Aim: My investigation is to find the maximum area of a regular polygon with the perimeter of 1000m.

Method: I will calculate the area of different shapes which will consists of regular polygons including triangles, rectangles pentagons hexagons, heptagons and I will conclude my research with circles. I will achieve this by applying a formula for each of the shapes; the formula will include Pythagoras theorem and trigonometry. This will lead me to my conclusion that will tell me which regular polygon has the highest area with the perimeter 1000m.

I started my investigation with rectangles because it is a basic shape and simple to begin the investigation with.

I discovered that when the side lengths of the rectangle are the same, I reached maximum area from the shape.

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

The table shows the different lengths and widths of a rectangle. I have chosen to increase the length and decrease the width by 50m. This pattern continues until the values of the length and width are they same. The table shows that when the two lengths are the same you get the highest value of area.

image00.pngimage01.png

image12.png

This graph shows all the possible information needed to figure out the maximum area of a rectangle with a 1000m perimeter. The graph shows a curve which represents the area of a given rectangle.

...read more.

Middle

     2        2image05.pngimage06.png

        Hero’s formula:                  image07.png

         S (s-a) (s-b) (s-c)

275

400

325

44370.59837

50

490

460

9486.832981

310

320

370

47148.70094

333

333.3

333.4

48155.79269

The maximum area the scalene triangle with a perimeter gives is when all three side lengths are close together. As the scalene triangle has side lengths that are all different lengths it will be harder for me to generate 3 numbers that are close to each other and add up to 1000. I then used hero’s formula which is needed to find the area of the scalene triangle. image08.pngimage09.png

image36.pngimage14.pngimage13.pngimage11.pngimage10.png

image15.png

The graph shows the side lengths in relation to the area of the scalene. The graph shows that as the side lengths get closer together, the area increases. I have completed my research on scalene triangles; I will start a new investigation on pentagons. In this investigation I am going to be using trigonometry, to find the area of the whole pentagon, I am going to split it up into 5 smaller triangles. The perimeter of the pentagon will once again be 1000m.

image16.png

        The area of the whole pentagon is 5T.

        The Circle in the middle of the pentagon represents all of

        360º of the angles in the middle of the pentagon.

        To figure out the value of the angles in the middle I have to

        divide 360 by 5 which gives me 72. This means that the

        angle in the middle for one of the triangles is 72º.

Now that I have the angle for the triangle I have to work out the base length of the triangle.

...read more.

Conclusion

Small Angle Theory

Sin O is roughly the same as theta O

Tan O is roughly the same as theta.

Now the radian formula is similar to the formula of the ‘n’ sided polygon.

Area = 500² X 1 tan(П/n)

  n

As you make the ‘n’ larger the angle gets smaller, as it approaches 0 it makes the angle smaller.

n

Π/n

TanΠ   / h

10

0.314159

0.32492

100

0.03142

0.031426

1000

0.003141592

0.003141602

10000

0.000314159

0.000314159

100000

0.000031415926

                                  0.000031415926

As the angle gets smaller, the approximation will be much clearer and much more accurate.

Conclusion

I have now come to the end of my research; I have found out that the maximum area is achieved from a circle. All the research on the number of sides that give the maximum showed that the maximum area the circle with a perimeter 1000m is 79577.4715m². As the numbers of sides on a shape begin to increase, the shapes become more and more circular. So in conclusion the shape that gives you the highest area is the circle and the more sides you put on a shape the more circle like they become.

image38.png

This is a square in a circle and the edges of the square are touching the circle. If I added more sides to the square the area filled by the shape would increase and would look more like a circle.

image39.png

This is an octagon in a circle and all of the edges are touching the circle. As you can see the octagon is more circular looking and the space left between the octagon and the circle has decreased. This means that the more sides a shape has the more space they fill in a circle.

...read more.

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