  We provide an in-detail derivation through the $^3P_0$ pair creation model of
the transition matrix for a baryon decaying into a meson-baryon system. The
meson's analysis was conducted in Ref. [1] and we extend the same formalism to
the baryon sector, focusing on the $\Delta(1232)\to \pi N$ strong decay width
because all hadrons involved in the reaction are very well established, the two
hadrons in the final state are stable, avoiding further analysis, all quarks
are light and so equivalent, and the decay width of the process is relatively
well measured. Taking advantage of a Gaussian expansion method for the hadron's
radial wave functions, the expression of the invariant matrix element can be
related with the mean-square radii of hadrons involved in the decay. We use
their experimental measures in such a way that only the strength of the
quark-antiquark pair creation from the vacuum is a free parameter. This is then
taken from our previous study of strong decay widths in the meson sector,
obtaining a quite compatible result with experiment for the calculated
$\Delta(1232)\to \pi N$ decay width.
