# Fencing problem.

Extracts from this document...

Introduction

Number Coursework

Problem: Farmer Pickles has 1000 metres of fencing. Find the field with the largest area encompassed by the 1000 metre fence for Farmer Pickles.

## Rectangles

I started by looking at rectangles and these are the ones I looked at:

50 length – 450 height

100 length – 400 height

150 length – 350 height

200 length – 300 height

250 length – 250 height

Once I had worked out the areas of all of those rectangles I drew a graph to show my results:

I noticed that the area became larger as the shape became more like a regular quadrilateral. With a 1000 metres of fencing you could create a quadrilateral field with a maximum of 62500m².

## Triangles

I decided to look at triangles next. Right angle triangles

Middle

- = 200

5

From here I used trigonometry:

Tan θ = O A

Tan 36 = 100

A

A = 100

0.7265425

A = 137.6382

Then I did:

Area = (½ x 200 x 137.6382) x 5

Total area of regular pentagon = 68819.1m²

By now I saw a pattern emerging. The more sides a shape had the larger the area, even though the circumference remained 1000m. One problem that emerged was that I couldn’t keep working out things long hand so I had to develop a formula so that I could work out the area of a shape with any number of sides.

## Formula

Conclusion

n² Tan 180

n

Area = 250,000

n Tan 180

n

By using the final simplified formula you can work out the area to any 2D shape, and here is a table to prove it:

Sides of regular shape | Area in m² |

3 4 5 10 100 999 | 48,112.33 62,500 68,819.1 76946.753 79567 79574.92 |

As the number of sides increase so does the area, as the table proves, but also the shapes are becoming more circular. Also the formula doesn’t work with circles so I have decided to check them out.

C = 2πr

1000 = 2πr

1000 = r

2π

r = 159.154 m

Conclusion

As the table proves, no 2D shape has more area than a circle while still having a 1000m circumference. For Farmer Pickles a circular field would be the best option because he wanted the field with the maximum area. Also a circular fence would be relatively easier to create than a 100 sided fence.

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

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