  In order to understand which mechanism is responsible for accretion in
protoplanetary discs, a robust knowledge of the observed disc radius using gas
tracers such as $^{12}$CO and other CO isotopologues is pivotal. Indeed, the
two main theories proposed, viscous accretion and wind-driven accretion,
predict different time evolution for the disc radii. In this Letter, we present
an analytical solution for the evolution of the disc radii in viscously
evolving protoplanetary discs using $^{12}$CO as a tracer, under the assumption
that the $^{12}$CO radius is the radius where the surface density of the disc
is equal to the threshold for CO photo-dissociation. We discuss the properties
of the solution and the limits of its applicability as a simple numerical
prescription to evaluate the observed disc radii of populations of discs. Our
results suggest that, in addition to photo-dissociation, also freeze out plays
an important role in setting the disc size. We find an effective reduction of
the CO abundance by about two orders of magnitude at the location of CO
photo-dissociation, which however should not be interpreted as the bulk
abundance of CO in the disc. The use of our analytical solution will allow to
compute disc sizes for large quantities of models without using expensive
computational resources such as radiative transfer calculations.
