Math Coursework  The Fencing Problem
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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. [IMAGE] The area of a triangle depends on two things: the height and the base. The base is fixed in this example so the triangle that has the biggest height, i.e. the middle triangle, will have the biggest area. The middle triangle turns out to be an icosoles triangle. I am going to focus only on icosoles triangles. I have constructed a formula linking all three sides in and icosoles triangle. [IMAGE] X X X=any number which is greater than 250 and less than 500 ======================================================== 1000  2X Using Pythagoras theorem I can find and equation linking a side to the area. ====================================================================== Â½(1000  2X)Â² + HÂ² = XÂ² HÂ² = XÂ² + (X 500)Â² H = height X 500  X XÂ²  (500X)Â² H Area 251 249 1000 31.6 7874.1 300 200 50000 223.61 44721.0 333.33 Need Writing Help?Get feedback on grammar, clarity, concision and logic instantly. Check your paper »How to Cite this Page
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83330 288.97 48112.5 350 150 100000 316.23 47434.0 400 100 150000 387.30 38729.8 450 50 205000 452.77 22638.5 499 1 249000 499.00 499.00 As you can see from the table the maximum area is when X = 333 1/3. When this number is plugged into the formula we see that this is actually an equilateral triangle. The next simplest shape is the 4sided shape, namely a rectangle. I have constructed another formula linking the sides. 500  X [IMAGE] X X 500  X From the diagram the area must equal XÂ² +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? Make sides same length Make sides same length [IMAGE]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. [IMAGE] Rotate triangle onto side Equilateral triangle Lets take the square example. Obviously the longer the sides the bigger the area. This means the bigger the length and the bigger the height, the bigger the area. In this investigation we have been given a set perimeter. To make the length longer means you have to sacrifice the height. To make the height bigger you have to sacrifice the length. To get the biggest area you need the sides to be as long as possible. When the sides are equal, it means that the sides are at the biggest they could be simultaneously. This means The closer the sides are in the ratio of 1:1, the bigger the area. Shapes with a ratio of sides that is 1:1 are said to be regular. Regular shapes have numerous properties; they can be split up into icosoles triangles. Irregular polygons can be only split up into scalene triangles. I have already proved why icosoles triangles have a larger area than scalene. This means regular shapes will have a larger area than irregular shapes. From now on I am going to find out which shape has the biggest area with a given perimeter. I will investigate only the regular shapes because I have proved that the regular polygon has the biggest area out of all the irregular polygons with the same perimeter. [IMAGE]Triangle =============== 333.3 Text Box: 333.3 333.3 Using trigonometry. Area = Â½ x 333.3 x 333.3 x sin 60 = 48112.5 [IMAGE]Square ============= 250 250 250 250 Area = 250 x 250 = 62500 Pentagon ======== [IMAGE] Each side = 200 Using trigonometry. 200/sin 72 = Y /sin 54 Y=170.1 Area = 5x( Â½ x 170.1 x 170.1 x sin 72) = 68794.7 Hexagon ======= [IMAGE] Each side = 166 2/3 Using trigonometry Â½ x 6 x 166 2/3 x 166 2/3 x sin 60 Area = 72168.8 Septagon ======== Each side = 142.9 142.9/sin 51.4 = Y/sin 64.3 Y = 164.8 Area = 7x( Â½ x 164.8 x 164.8 x sin 51.4) Area = 74288.7 Octagon ======= Each side = 125 125/sin 45 = Y/sin 67.5 Y = 163.3 Area = 8x( Â½ x 163.3 x 163.3 x sin 45) Area = 75425.4 Lets put all these results in a table 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 As you can see these results will keep on increasing and increasing. This means the shape that can have the largest area must have infinite sides. What shape has infinite sides? I will use the regular polygon symmetry theorem. A triangle has three sides and it has 3 lines of symmetry. A square has 4 sides and it has 4 lines of symmetry. A pentagon has 5 sides and 5 lines of symmetry. A hexagon has 6 sidesâ€¦. You get the idea. The shape with infinite sides must have infinite lines of symmetry. The only shape that has infinite lines of symmetry is the circle. Lets find out the area of a circle with circumference 1000. 2pr = 1000 pr = 500 r = 159.2 A=prÂ² A = p159.2Â² 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 Â¥ 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. 
