Why does the calcualtion include the perimeter squared (P^2) and not the perimeter?

This ratio is designed to produce a dimensionless measure of the shape. A dimensionless shape appears when taking the ratio removes the dimensions of measurement.

Example: 6 meters/ 2 meters = 3

Or, in our case: (6 meters)^2/3 meters^2 = 36 meters^2/3 meters^2 = 12.

This then presents the image in terms of its shape and not its measurements of size.