The circle and the sphere are the most compact planar and solid shapes, respectively.
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.
[1] Compactness measures can be defined for three-dimensional shapes as well, typically as functions of volume and surface area.
[2][5] More sophisticated measures of compactness include calculating the shape's moment of inertia[2][3] or boundary curvature.
The goal is to maximize the compactness of electoral districts, subject to other constraints, and thereby to avoid gerrymandering.
[6] Another use is in zoning, to regulate the manner in which land can be subdivided into building lots.
[7] There is evidence that compactness is one of the basic dimensions of shape features extracted by the human visual system.