We present a coarse interpretation of the Weil-Petersson distance
d
W
P
(
X
,
Y
)
d_{\mathrm {WP}}(X,Y)
between two finite area hyperbolic Riemann surfaces
X
X
and
Y
Y
using a graph of pants decompositions introduced by Hatcher and Thurston. The combinatorics of the pants graph reveal a connection between Riemann surfaces and hyperbolic 3-manifolds conjectured by Thurston: the volume of the convex core of the quasi-Fuchsian manifold
Q
(
X
,
Y
)
Q(X,Y)
with
X
X
and
Y
Y
in its conformal boundary is comparable to the Weil-Petersson distance
d
W
P
(
X
,
Y
)
d_{\mathrm {WP}}(X,Y)
. In applications, we relate the Weil-Petersson distance to the Hausdorff dimension of the limit set and the lowest eigenvalue of the Laplacian for
Q
(
X
,
Y
)
Q(X,Y)
, and give a new finiteness criterion for geometric limits.