# Fencing - maths coursework

Extracts from this document...

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.

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

## Found what you're looking for?

- Start learning 29% faster today
- 150,000+ documents available
- Just £6.99 a month