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# Fencing problem

Extracts from this document...

Introduction

Page  of

Maths Coursework

## Fencing problem

I have been asked to investigate the maximum area enclosed by 1000 metres of fencing. To do this I need to use various shapes, to start with a square.

Secondly I shall look at rectangles.

 Rectangle Length (M) Height (M) Area (M) A 450 50 22500 B 400 100 40000 C 350 150 52500 D 300 200 60000 E 250 250 62500 F 200 300 60000

The biggest quadrilateral in the table was E which is a square, meaning the square has a  larger area than rectangles. I only investigated six rectangles as after E the rectangles have the same numbers but on different sides. This means the area would be the same and so it was pointless to continue at this rate of increase so I looked

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Middle

 Triangle Equal sides  (m) Base Total area (m  ) A 300 400 44721.35955 B 350 300 47434.1649 C 400 200 38729.83346 D 450 100 4330.127019

The biggest area I found was with triangle B although as I increased by 50m each time I think that I could investigate this triangle in more depth with smaller intervals to find a larger area.

I continued with the same formula as before but just applied it to another group of triangles. These are my results.

 Triangle Equal sides (m) Base (m) Total area (m  ) A 305 390 45731.553658 B 310 380 46540.3051128808 C 315 370 47165.9305007334 D 320 360 47623.5235991628 E 325 350 47924.7237817015 F 330 340 48083.2611206854 G 335 330 48105.353132 H 340 320 48000 I 345 310 4774.208523

Although I could go further into depth with Triangle G I am

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Conclusion

Now that I know the perpendicular height I can use the same formula as before to get the area.

Base X height ÷ 2

125 X 150.8883476 ÷ 2 = 9430.521725

Area = 9430.521725m

This gives me the area of one of eight equal triangles in an octagon so now I must multiply this number by eight.

9430.521725 X 8 = 75444.1738

As expected the area has again increased in size. I think that for regular shapes the more sides the shape has the bigger the area is. A circle has infinite sides in theory so I think this will have the biggest area.

Radha Campbell 10L

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