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 fencing of 1000 metres perimeter.
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.
