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We prove a formula relating the Hausdorff dimension of a deterministic Borelsubset of $\mathbb R$ and the Hausdorff dimension of its image under aconformal map from the upper half-plane to a complementary connected componentof an SLE$_\kappa$ curve for $\kappa \not =4$. Our proof is based on therelationship between SLE and Liouville quantum gravity together with theone-dimensional KPZ formula of Rhodes-Vargas (2011) and the KPZ formula ofGwynne-Holden-Miller (2015). As an intermediate step we prove a KPZ formulawhich relates the Euclidean dimension of a subset of an SLE$_\kappa$ curve for$\kappa \in (0,4)\cup(4,8)$ and the dimension of the same set with respect tothe $\gamma$-quantum natural parameterization of the curve induced by anindependent Gaussian free field, $\gamma = \sqrt \kappa \wedge(4/\sqrt\kappa)$.

Dimension transformation formula for conformal maps into the complement of an SLE curve