A polar area diagram, also called a Nightingale diagram, represents nonnegative data values by circular sectors arranged about a common origin, with equal central angles and areas proportional to the corresponding values (Understanding Uncertainty 2008). The radii vary, while the central angles remain fixed. Nightingale (1858) used this form of statistical diagram (Numberphile 2021).
For 
categories, where 
is a positive integer, each circular
sector has central angle 
. If 
is the value of category 
and 
is a common area scale, the
area 
of its circular sector
is
 |
The corresponding radius is therefore
 |
Thus a zero value has zero radius, and multiplying every data value by the same nonnegative factor multiplies all radii by its square root.
In a pie chart, the radius is fixed and the central angles vary with the data values. A polar display with radius directly proportional to the value instead makes the circular sector areas proportional to the second powers of the values.
