  Let $ \text{Mod}(S_g)$ denote the mapping class group of the closed
orientable surface $S_g$ of genus $g\geq 2$, and let $f\in \text{Mod}(S_g)$ be
of finite order. We give an inductive procedure to construct an explicit
hyperbolic structure on $S_g$ that realizes $f$ as an isometry. In other words,
this procedure yields an explicit solution to the Nielsen realization problem
for cyclic subgroups of $ \text{Mod}(S_g)$. Furthermore, we give a purely
combinatorial perspective by showing how certain finite order mapping classes
can be viewed as fat graph automorphisms. As an application of our
realizations, we determine the sizes of maximal reduction systems for certain
finite order mapping classes. Moreover, we describe a method to compute the
image of finite order mapping classes and the roots of Dehn twists, under the
symplectic representation $\Psi: \text{Mod}(S_g) \to \text{Sp}(2g;
\mathbb{Z})$.
