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.

Various compactness measures are used. However, these measures have the following in common:

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 ${\displaystyle \Psi }$. Another measure in use is ${\displaystyle (\mathrm {surfacearea} )^{1.5}/(\mathrm {volume} )}$, which is proportional to ${\displaystyle \Psi ^{-3/2}}$.

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