



Tools Math 'Creating Measures' Compactness Task, Example #3 (solution)
Squareness, Example #1 (solution)  Steepness, Example #2 (solution) Compactness, Example #3 (solution)  Crowdedness, Example #4 (solution) Awkwardness, Example #5 (solution)  Sharpness, Example #6 (solution)
Malcolm Swan
Jim Ridgway
Over recent years, a number of geographers have tried to find ways of defining the shape of an area. In particular, they have tried to devise a measure of 'compactness'. You probably have some intuitive idea of what "compact" means already. Below are two islands. Island B is more compact than island A. "Compactness" has nothing to do with the size of the island. You can have small, compact islands and large compact islands.
Sketch a large 'lesscompact' island and a small 'lesscompact' island. One person has suggested the following way of measuring "compactness."
This question is to make sure that students have followed the introduction. They should all understand that more 'rounded' shapes tend to be compact, while 'long spindly' shapes tend to be less compact.
The results for area ÷ perimeter for the six shapes are shown below:
A better measure of compactness might therefore be area ÷ (perimeter)^{2}. This is dimensionless and does give measures in accord with intuition:
Thus, in order of "compactness", we have that
We could further improve the measure, so that it always lies between 0 and 1 by multiplying by 4. This would make a circle a perfectly compact shape. The ratio 4A/p^{2} is quoted in Selkirk (1982) as the "circularity ratio." One criticism of its use by geographers is that it is difficult to define and calculate the perimeter of a very large irregular boundary such as a country or a river basin. (Fractal geometry might suggest that such perimeters may even be infinite!) Instead, there are several other possibilities:
These are all quoted in Selkirk (1982) as methods which geographers have used. For interest, Selkirk offers the following calculations for four countries, using these ratios:
Selkirk's results place the countries in a similar order, apart from the circularity ratio, which suggests that Czecholslovakia is less compact than Austria.
Compactness, Example #3 (solution)  Crowdedness, Example #4 (solution) Awkwardness, Example #5 (solution)  Sharpness, Example #6 (solution)

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