Let \(\mathrm{Mod}(S_g)\) represent the mapping class group of a closed orientable surface \(S_g\) with genus \(g \geq 2\). We present a method to estimate the Weil-Petersson distance between the unique hyperbolic structures realizing certain irreducible cyclic subgroups of \(\mathrm{Mod}(S_g)\) in the moduli space \(\mathcal{M}(S_g)\). Our approach begins with deriving an explicit description of a pants decomposition of \(S_g\) in which the lengths of the curves are bounded above by the Bers' constant. We then establish an upper bound for the distance between a pair of such decompositions in the Pants graph \(\mathcal{P}(S_g)\). Finally, we utilize Brock's quasi-isometry between the Teichmuller space and the Pants graph \(\mathcal{P}(S_g)\) to obtain our distance estimate in \(\mathcal{M}(S_g)\).
