A Minimal Circumcircle Measure of District Compactness

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The United States Constitution prohibits states from "denying any person within its jurisdiction the equal protection of the laws". Various decisions of the United States Supreme Court have interpreted this provision to constrain the otherwise existing freedom of the states to draw political districts in ways that disadvantage racial, ethnic or other protected groups. Districts that are not "compact" are subject to heightened scrutiny for evidence of racial or other prohibited biases.


This Demonstration examines one measure of district compactness: the ratio between the area of the district and the area of the "minimal circumcircle", the circle of least area that contains all points within the district. The Demonstration presents a starting set of points defining the district, which is colored purple. You can move these points around, add points, or subtract points. The Demonstration then computes and shows in yellow the "convex hull" of the district. It likewise shows in red the minimal enclosing circumcircle. A grid below the diagram shows two measures of district compactness: the ratio of district area to convex hull area and the ratio of district area to minimal circumcircle area. Both of these measures have been suggested as potential measures of district compactness.


Contributed by: Seth J. Chandler (March 2011)
Open content licensed under CC BY-NC-SA



The formulas provided in this Demonstration will give an erroneous answer for polygons that self-intersect.

Snapshot 1: a very compact district

Snapshot 2: a very noncompact district

Snapshot 3: a moderately compact district

Snapshot 4: a convex quadrilateral district with moderate compactness

C. Chambers and A. Miller, "Measure of Bizarreness".

E. C. Reock, "A Note: Measuring Compactness as a Matter of Legislative Apportionment," Midwest Journal of Political Science, 5(1), 1961 pp. 70–74.

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