Math Coursework - The Fencing Problem :: Math Coursework Mathematics

The Fencing Problem A farmer has 1000m of fencing and wants to fence off a plot of level land. She is not concerned about the shape of plot, but it must have a perimeter of 1000m. So it could be: [IMAGE] Or anything else with a perimeter (or circumference) of 1000m. She wishes to fence of the plot of land with the polygon with the biggest area. To find this I will find whether irregular shapes are larger than regular ones or visa versa. To do this I will find the area of irregular triangles and a regular triangle, irregular quadrilaterals and a regular square, this will prove whether irregular polygons are larger that regular polygons. Area of an isosceles irregular triangle: ======================================== (Note: I found there is not a right angle triangle with the perimeter of exactly 1000m, the closest I got to it is on the results table below.) To find the area of an isosceles triangle I will need to use the formula 1/2base*height. But I will first need to find the height. To do this I will use Pythagoras theorem which is a2 + b2 = h2. [IMAGE] [IMAGE] First I will half the triangle so I get a right angle triangle with the base as 100m and the hypotenuse as 400m. Now I will find the height: a2 + b2= h2 a2 + 1002 = 4002 a2 = 4002 - 1002 a2 = 160000 - 10000 a2 = 150000 a = 387.298m Now I will find the area: 100*387.298 = 3872.983m2 My table shows the areas of other irregular triangles, but to prove that regular shapes have a larger area I will show the area of a regular triangle: Area of a regular triangle: Tan30= 166.6666667/x X= 166.666667/Tan30 X= 288.675m 288.675*166.6666667 = 48112.5224m2 This shows clearly that the regular triangle's area is larger than the