'Creating Measures' Compact-ness 
Task - Example #3 (solutions)


Malcolm Swan
Mathematics Education
University of Nottingham
Malcolm.Swan@nottingham.ac.uk

Jim Ridgway
School of Education
University of Durham
Jim.Ridgway@durham.ac.uk


This problem gives you the chance to:
· criticise a given measure for the concept of "compact-ness"
· invent your own way of measuring this concept
· refine your scale so that it measures from 0 to 1.

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.  "Compact-ness" has nothing to do with the size of the island. You can have small, 
compact islands and large compact islands. 

____________________________________________________

 

Warm-up

Sketch a large 'compact' island and a small 'compact' island.
Sketch a large 'less-compact' island and a small 'less-compact' island.

One person has suggested the following way of measuring "compactness."

   

Comment:
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.

1. Calculate the "compactness" of each of the following 'islands' using the above definition.
 
  
 
Solution:
The results for area ¸ perimeter for the six shapes are shown below:

Shape
A
B
C
D
E
F

Area
12
28.26
3
16
16
4

Perimeter (cm)
16
18.84
8
16
32
8

Area ¸ Perimeter
0.75
1.5
0.375
1
0.5
0.5


 
 
 
 
2. Use your results to explain why Area ¸ Perimeter is not a suitable definition for 
"compactness."
 
Solution:
· While at first the measure area ¸ perimeter might seem reasonable, it is not dimensionless, 
so it will give different results for enlagements of shapes.
· Thus in the examples D and F are both squares, but they give different measurements of 
compactness.  Similarly with A and C.
 
 
 
 
 
3. Invent your own measure of "compactness".
 Put the shapes A to F in order of "compact-ness" using your measure.  Discuss whether or not 
your measure is better than 'Area ¸ Perimeter.'
 
Solution:
A better measure of compactness might therefore be area ¸ (perimeter)2.

This is dimensionless and does give measures in accord with intuition:
 
Staircase
A
B
C
D
E
F

Area ¸ Perimeter2
0.047
0.080
0.047
0.063
0.016
0.063


 
Thus, in order of "compactness", we have that
B (most compact), D and F (tie), A and C (tie), E (least compact).
 
 
 
 
 
4. Adapt your measure so that it ranges from 0 to 1.
 A perfectly compact shape should have a measure of '1,' while a long, thin, shape should have 
should have a measure near to 0.
 
Solution:
We could further improve the measure, so that it always lies between 0 and 1 by multiplying by 
4p. This would make a circle a perfectly compact shape. 

The ratio 4pA/p2 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:
 
 Form Ratio:  
 A = area
 L = length of line joining the area's two most distant points
 
 Compactness Ratio:  
 A = area
 R = radius of the smallest circle which will surround the shape
 
 Radius Ratio:  
 r = radius of the largest circle which will fit inside the shape
 R = radius of the smallest circle which will surround the shape
 
 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: 
 

Czechoslovakia
Austria
Bulgaria
Romania

Circularity ratio
0.33
0.30
0.43
0.50

Form ratio
0.28
0.30
0.52
0.56

Compactness ratio
0.28
0.30
0.47
0.56

Radius ratio
0.28
0.40
0.46
0.60

 
 Selkirk's results place the countries in a similar order, apart from the circularity ratio, which
 suggests that Czecholslovakia is less compact than Austria.