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Introduction

To discover the largest obtainable area within a fenced with perimeter of 1000m, I was given the project of finding what shape that has the greatest area for a farmer with perimeter of 1000m. In my investigation I am going to work on different shapes. I will try to find the shape with the largest area by drawing and investigating rectangles first, then triangles and finally regular polygons. Here are some of the shapes I am going to investigate: Now I am going to draw out some rectangles and try to find out the rectangle with the biggest area in the four sided shape family. Formula to find a rectangle: Area: Length x Width Perimeter: Length + Width + Length + Width 450m 50m Perimeter: 450m +50m + 450m +50m = 1,000m Area: 450m x 50m = 22,500m� 400m 100m Perimeter: 400m +100m + 400m +100m = 1,000m Area: 400m x 100m 350m =40,000m� 150m Perimeter: 350m +150m + 350m +150m = 1,000m Area: 350m x 150m =52,500m� 300m 200m Perimeter: 300m +200m + 300m +200m = 1,000m Area: 300m x 200m 250m = 60,000m� 250m Perimeter: 250m +250m + 250m +250m = 1,000m Area: 250m x 250m = 62,500m� This shows that out of the rectangle family the Square has the furthermost area. ` Area: =250m x 250m = 62,500m� Base (x) Height (y) Base x height Area 50m 450m 50m x 450m 22500m� 100m 400m 100m x 400m 40000m� 150m 350m 150m x 350m 52500m� 200m 300m 200m x 300m 60000m� 250m 250m 250m x 250m 62500m� 300m 200m 300m x 200m 60000m� 350m 150m 350m x 150m 52500m� 400m 100m 400m x 100m 40000m� 450m 50m 450m ...read more.

Middle

so we double the triangles. 1000 7 =142.8571429m (Not drawn to scale) 360 7 = 51.42857143O 25.71428571O 51.42857143O h 64.28571429o (Not drawn to scale) 142.8571429m 71.42837143m 90o + 25.71428571o = 115.7142857o 180o - 115.7142857o = 64.28571429o tan 64.28571429o = 71.42837143 h h = 71.42837143m x tan 64.28571429o h = 148.322957m Area of the triangle: 142.8571429m x 148.322957m 2 =10594.4693m2 Area of the pentagon: 7 x 10594.4693m = 74161.4785m2 Because we doubled the length of the side (not aright angle) so we halved the triangles. 1000 8 = 125m 360 8 = 45o 45o 22.5o h 67.5 o 125m 62.5m 90o + 22.5o = 112.5o 180o - 112.5o = 67.5o tan 67.5o = 62.5 h h = 62.5 x tan 67.5o h = 150.8883476m Area of the triangle: 62.5m x 150.8883476m 2 = 4715.260863m2 Area of the pentagon: 16 x 4715.360863m = 75444.1738m2 Because we double the length of the side (not aright angle) so we halved the triangles. 1000 10 = 100m (Not drawn to scale) 360 10 36o 36o 18o 72o (Not drawn to scale) 100m 50m 90o + 18o = 108o 180o - 108o = 72o tan 72o = 50 h h = 50 x tan 72o h = 153.8841769m Area of the triangle: 50m x 153.8841769m 2 =3847.104423m2 Area of the pentagon: 20 x 13763.8192m = 76942.08843m2 Because we haled the length of the side (not aright angle) so we double the triangles.72o I am now going to try and find a formula so I can find out 'n' number of sides. The 'n' number of sides stands for 'any' number of sides. ...read more.

Conclusion

Firstly I am going to show you an 18-sided shape (octadecagon). Now I am going to show you a 30-sided shape (triacontagon). The website I have found this information is: http://en.wikipedia.org/wiki/Polygon#Names_and_types Now I am going to be explaining to you how a tringle will find into a octagon and then a circle. If you take the middle od a tringle and you pull each side outwards the corners of the triangle will be pulled inward a bit and the area inside the triangle will increase as you have maked the shape bigger, this will now look more and more like a octagon. If you keep on repeating this ther shape of the polgyon will turn into a bigger polgyon eg: 12 sided shape and the slowly it will start to look more and more like a circle. As you can see in the digram a lot of spcae is wasted from the triangle and the circle. However not much space is wasted from the octagon to the cricle, also when you have a triacontagon(30-sided shape) and you fit it in a circle there is hardly any wasted spce so this is tell us that, as the sided of the polgyen increases there is less wasted spcae. Perimeter = 1000m Perimeter = Circumference Radius C = 2 ?r� 1000 = 2 ?r� 1000 - 2 = ?r� 500 = ?r� 500 - ? = r� r = 159.15499431m Area of a circle = ?r2 = x 159.154994312 = x 25330.29591 = 79577.47155m2 Finally I have discovered that a circle has the maximum area that the farmer will be able to build his fence over. A circle gives the most area with 1000 metres of fence. Maths Fencing Coursework Rahul Malde 11e ...read more.

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