  Prudent walks are special self-avoiding walks that never take a step towards
an already occupied site, and \emph{$k$-sided prudent walks} (with $k=1,2,3,4$)
are, in essence, only allowed to grow along $k$ directions. Prudent polygons
are prudent walks that return to a point adjacent to their starting point.
Prudent walks and polygons have been previously enumerated by length and
perimeter (Bousquet-M\'elou, Schwerdtfeger; 2010). We consider the enumeration
of \emph{prudent polygons} by \emph{area}. For the 3-sided variety, we find
that the generating function is expressed in terms of a $q$-hypergeometric
function, with an accumulation of poles towards the dominant singularity. This
expression reveals an unusual asymptotic structure of the number of polygons of
area $n$, where the critical exponent is the transcendental number $\log_23$
and and the amplitude involves tiny oscillations. Based on numerical data, we
also expect similar phenomena to occur for 4-sided polygons. The asymptotic
methodology involves an original combination of Mellin transform techniques and
singularity analysis, which is of potential interest in a number of other
asymptotic enumeration problems.
