Math Coursework - The Fencing Problem

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