# Compactness measure of a shape

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.

## Properties

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.

## Examples

A common compactness measure is the isoperimetric quotient, the ratio of the area of the shape to the area of a circle (the most compact shape) having the same perimeter.

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 . Another measure in use is ,[1] which is proportional to .

## Applications

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] Another use is in pattern classification projects so that you can classify the circle from other shapes.