  The peanosphere construction of Duplantier, Miller, and Sheffield provides a
means of representing a $\gamma$-Liouville quantum gravity (LQG) surface,
$\gamma \in (0,2)$, decorated with a space-filling form of Schramm's
SLE$_\kappa$, $\kappa = 16/\gamma^2 \in (4,\infty)$, $\eta$ as a gluing of a
pair of trees which are encoded by a correlated two-dimensional Brownian motion
$Z$. We prove a KPZ-type formula which relates the Hausdorff dimension of any
Borel subset $A$ of the range of $\eta$ which can be defined as a function of
$\eta$ (modulo time parameterization) to the Hausdorff dimension of the
corresponding time set $\eta^{-1}(A)$. This result serves to reduce the problem
of computing the Hausdorff dimension of any set associated with an SLE, CLE, or
related processes in the interior of a domain to the problem of computing the
Hausdorff dimension of a certain set associated with a Brownian motion. For
many natural examples, the associated Brownian motion set is well-known. As
corollaries, we obtain new proofs of the Hausdorff dimensions of the
SLE$_\kappa$ curve for $\kappa \not=4$; the double points and cut points of
SLE$_\kappa$ for $\kappa >4$; and the intersection of two flow lines of a
Gaussian free field. We also obtain the Hausdorff dimension of the set of
$m$-tuple points of space-filling SLE$_\kappa$ for $\kappa>4$ and $m \geq 3$ by
computing the Hausdorff dimension of the so-called $(m-2)$-tuple $\pi/2$-cone
times of a correlated planar Brownian motion.
