# 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.

## See also

- Surface area to volume ratio
*How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension*

## References

- ↑ U.S. Patent 6,169,817
- ↑ Rick Gillman "Geometry and Gerrymandering", Math Horizons, Vol. 10, #1 (Sep, 2002) 10-13.
- ↑ MacGillis, Alec (2006-11-15). "Proposed Rule Aims to Tame Irregular Housing Lots".
*The Washington Post*. p. B5. Retrieved 2006-11-15.