Math Coursework - The Fencing Problem

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:

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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:

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(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.

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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