  Given two pants decompositions of a compact orientable surface $S$, we give
an upper bound for their distance in the pants graph that depends
logarithmically on their intersection number and polynomially on the Euler
characteristic of $S$. As a consequence, we find an upper bound on the volume
of the convex core of a maximal cusp (which is a hyperbolic structures on $S
\times \mathbb{R}$ where given pants decompositions of the conformal boundary
are pinched to annular cusps). As a further application, we give an upper bound
for the Weil--Petersson distance between two points in the Teichm\"uller space
of $S$ in terms of their corresponding short pants decompositions. Similarly,
given two one-vertex triangulations of $S$, we give an upper bound for the
number of flips and twist maps needed to convert one triangulation into the
other. The proofs rely on using pre-triangulations, train tracks, and an
algorithm of Agol, Hass, and Thurston.
