Compactness measure of a shape

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The compactness measure of a shape, sometimes called the shape factor, is a numerical quantity representing the degree to which a shape is compact. The meaning of "compact" here is not related to the topological notion of compact space. Various compactness measures are used. However, these measures have the following in common:

They are applicable to all geometric shapes.
They are independent of scale and orientation.
They are dimensionless numbers.
They are not overly dependent on one or two extreme points in the shape.
They agree with intuitive notions of what makes a shape compact.

A common compactness measure, called the circularity ratio, is the ratio of the area of the shape to the area of a circle (the most compact shape) having the same perimeter. That ratio is expressed mathematically as $M = 4π(area) / (perimeter) 2$ . For a circle, the ratio is one; for a square, it is $π / 4$ ; for an infinitely long and narrow shape, it is zero.

Compactness measures can be defined for three-dimensional shapes as well, typically as functions of volume and surface area. One example of a compactness measure is sphericity Ψ. Another measure in use is $(surfacearea) 1.5 / (volume)$ .^[1]

A common use of compactness measures is in redistricting. The goal is to maximize the compactness of electoral districts, subject to other constraints, and thereby to avoid gerrymandering.^[2] Another use is in zoning, to regulate the manner in which land can be subdivided into building lots.^[3]