# The fencing problem

Extracts from this document...

Introduction

Guy McALL Page 5/7/2007

Maths coursework

The fencing problem

## Aim

My aim is to find out the largest area that can be put into a field. That has a perimeter of fencing that is 1000meters long. I am going to find out the biggest area I can get using different shapes that only have a perimeter of 1000meters.

## Investigating rectangles

I have started with rectangles because they are the simplest to find the area of (H*B=area)

The perimeter=1000 because 250 + 250 + 250 + 250 the area is 62500 meters² I used the equation h*b.

### The perimeter=1000 because 400 + 400 +100 + 100 the area is 40000 meters² I used the same equation again h*b.

The perimeter=1000 because 300 + 300 + 200 + 200 the area is 60000 meters² I used the equation h*b

Conclusion for rectangles

### I found that the square had the biggest area than all the other rectangles. I found that the smaller the width the smaller the area (the width 100 and length 400 the are equals 40000.) I also found that the smaller the length the smaller the area (length 100 and width 400 the area equals 40000.)

Middle

250

375

44194.17382

300

350

47434.1649

333⅓

333⅓

48112.52243

350

325

47925.72378

450

275

35575.62368

Investigating a Pentagon

Aim

I have to decided to find out the area of shapes with more sides to see if the area will increase compared with the square and triangle. For this irregular pentagon I am first going to divide the square and triangle up and work out the area of them then add them together to get the area.

Irregular pentagon

The perimeter is 1000 meters because 250 + 250 + 100 + 100 + 300=1000

The area=45000 meters²

Because: -

I split the pentagon in to a triangle and a square then I found the area of the two and add them up: -

Square: -

(B*H)=area

100*300=30000

Square + triangle = area

30000 + 15000= 45000

Regular Pentagon

For this sided shape and the other shapes like this I am going to first divide the pentagon in to triangles. Then work out the area of the triangle and then times by the number triangles I have split.

The perimeter of the pentagon is 1000 the area= 68819.09602meters² because

I first divided the pentagon in to 5 triangles then worked out the angles of the triangle. Then I worked out the height (o) Tan54=O/100

Conclusion

## Investigating a 100 sided shape

The perimeter is 1000 because 10*100=1000

The area=79551.28988meters²

Because

Tan88.2=31.82051595

Tan88.2=O/A

31.82051595*5=

159.1025798

Height=159.1025798

## Investigating a circle

First I have to use a equation to find out what the radius is once I found out the radius is I can find the area.

First I am going to take the equation 2πr

Then I take the equation 1000=2πr

From this I work out the radius: -

1000=2πr /2

500=πr /π (3.141592654)

159.1549431=r

Now I can work out the area of the circle using the equation πr²

πr²=π*159.1549431²= 79577.47155

## Conclusion

### The circle I found has the largest area compared with all the other shapes.

## The area is larger because the more side you have to a shape the larger the area will become this is only when the perimeter equals the same on the shapes. I have worked out a formula to find out all the areas of shapes

Now I can draw a graph that will show how the areas increase as you get more sides.

I would suggest after these results that the farmer should make a circle out of the 1000 meters of fencing. This would give him the largest area for his cows to graze.

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

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