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 $\backslash SLE\kappa$, $\kappa =16/\gamma 2\in (4,\infty )$, $\eta$ 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 $\backslash SLE$, $\backslash CLE$, 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 curve for $\kappa \ne 4$; the double points and cut points of SLE for $\kappa >4$; and the intersection of two flow lines of a Gaussian free field. We obtain the Hausdorff dimension of the set of $m$-tuple points of space-filling SLE for $\kappa >4$ and $m\ge 3$ by computing the Hausdorff dimension of the so-called $(m-2)$-tuple $\pi /2$-cone times of a correlated planar Brownian motion.