The fencing problem.

Square

Perimeter = 1000m

Therefore Side = 250m

Therefore Area = 62500m2

There is only one type of square with the perimeter of 1000m, thus this has the maximum area of 62500m2.

Rectangles

The graph shows how as the length increases the area increases, and there it shows the maximum area for a rectangle of perimeter 1000m.

The rectangle with sides 200m and 300m produce a perimeter of 1000m and an area of 62500m2. However, this area does not beat that of the square which therefore means the square is the best shape so far.

Triangles

There are three types of triangles:

Iscosolies          

Equilateral

Scalene

There are also three different ways of calculating the area of a triangle:

1. area = ½ base x perpendicular height
2. area = ½ ab sin c
3. area = square root of ( s(s-a)(s-b)(s-c))

Equilateral Triangle

Side = 333.3

Perimeter = 1000m

Area = ½ ab sin c

        = ½ 333.3 x 333.3 sin 60

        = 48059.60347

This is the maximum area of an equilateral triangle and therefore the square has still the maximum area, for a perimeter of 1000m.