Area-Perimeter Relation

For curves that enclose a region, the dimension can be obtained by the comparing the perimeter of the curve and the area of the enclosed region,
P = k⋅Ad/2.
We illustrate this relation for simple Euclidean curves.
Next, we show why the same relation cannot hold for fractal curves.
If the dimension, d, of the curve satisfies d > 1, then the perimeter is infinite yet the enclosed area is finite.
Consequently, P = k⋅Ad/2 cannot hold.
Then we reexpress the Euclidean approach to obtain a form that can be applied to fractal curves.
Looking at geometrically similar shapes and measuring the area and perimeter at the same scale, we find
P1/P2 = (A1/A2)d/2
Here is an example of using the relation between perimeters and areas to calculate the dimension.