Let $ E \subset \mathbb{C} $ be a compact set, $ g:\mathbb{C} \to \mathbb{C} $ be a$K$-quasiconformal map, and let 0 <$t$< 2. Let $ {\mathcal{H}^t} $ denote$t$-dimensional Hausdorff measure. Then $$ {\mathcal{H}^t}(E) = 0\quad \Rightarrow \quad {\mathcal{H}^{t'}}\left( {gE} \right) = 0,\quad t' = \frac{{2Kt}}{{2 + \left( {K - 1} \right)t}}. $$