# Math Coursework - The Fencing Problem

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The Fencing Problem

Introduction

A farmer has exactly 1000 metres of fencing and wants to use it to
fence a plot of level land. The farmer was not interested in any
specific shape of fencing but demanded that the understated two
criteria must be met:

· The perimeter remains fixed at 1000 metres

· It must fence the maximum area of land

Different shapes of fence with the same perimeter can cover different
areas. The difficulty is finding out which shape would cover the
maximum area of land using the fencing with a fixed perimeter.

Aim

The aim of the investigation is to find out which shape or shapes of
fencing will cover the maximum area of land using exactly 1000 metres
of fencing material.

Prediction

I am predicting that the maximum area of land covered will be achieved
by using the fencing shapes with the greatest number of sides.

Method

I made a list of possible different shapes to be investigated and
assigned measurements to the sides of the shapes making sure that they
fit in within the perimeter of 1000 metres of fencing. I then worked
out the areas of each shape using known mathematical formulae and
techniques such as Pythagoras' theorem to calculate the sides of right
angled triangles; using trigonometrical functions (sine, tangent and
cosine) to calculate either angles or sides of triangles constructed.
Sometimes there are no known exact formulae for working out the area
of certain shapes such as octagon and more complex polygons. In such
cases, given shapes are split into shapes that have known formulae for
areas and the worked out the areas are added together. Areas of the
following shapes were investigated: square, rectangle, kite,
parallelogram, equilateral triangle, scalene triangle, isosceles
triangle, right-angled triangle, rhombus, pentagon, hexagon, heptagon
and octagon.

Results

The results of the analysis are shown in Table 1 and Fig 1.

Table 1 showing the areas for the different shapes formed by using the

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Title Length Color Rating
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### Related Searches

fencing of 1000 metres perimeter.

Shape

Area (m2)

Ranking (highest to lowest areas)

Square

62,500

6

Trapezium

51,961.5

11

Trapezium

59,529.40449

7

Trapezium

58,094.75019

8

Parallelogram

51,961.5

11

Parallelogram

34,641.01615

21

Parallelogram

45,466.3337

16

Rhombus

54,126.59

10

Rectangle

40,000

18

Rectangle

22,500

23

Circle

79,577.47

1

Kite

56,291.643

9

Kite

32,475.95264

23

Kite

47,631.39721

13

Right-angled Triangle

37,500

20

Isosceles Triangle

44,721.36

17

Isosceles Triangle

47,434.1649

14

Isosceles Triangle

38,729.83346

19

Equilateral Triangle

48,112.522

12

Scalene Triangle

23,664.3

22

Scalene Triangle

46,475.80015

15

Scalene Triangle

47,434.1649

14

Irregular Polygon

70,312.5

4

Pentagon

68,819.1

5

Hexagon

72,168.78365

3

Octagon

75,440

2

The results showed that the maximum area of land could be fenced by
using a fencing that has the shape of a circle. The area covered was
79,577.47 square metres. This was followed by the octagon shaped fence
with 75,440 square metres, then the hexagon shaped fence, with area of
72,168.78 square metres and the least area covered was with the
scalene shaped fencing which gave an area of 23,664.3 square metres.
The results did suggest that the area of land fenced appeared to
increase with increase in the number of sides of the shapes of the
fencing. Apart from the circle, the octagon with 8 sides covered the
maximum area, followed by the hexagon and then the pentagon and the
lease were the triangles with only 3 sides.

All the four-sided shapes (parallelogram, rhombus, rectangle, square
and kite) had covered similar areas. In general, the four-sided shapes
covered areas between 51,000 and 63,000 square metres (Table 1 and
Figs 1 & 2). The square shaped fence covered the greatest area, 62,500
square metres, compared with the other four-sided shapes.

Conclusion

The maximum area of land covered with the 1000m perimeter fencing was
achieved by using the circular fencing. The maximum area covered was
79,577.47m2. The area of land covered appeared to increase with
increase in the number of sides of the given shape of fencing material
as well as shapes that appeared wider.