Eulidean diameter of hyperbolic disk can be controled by distance from centre of the disk to the boundary

Question

Let $\Omega \subset C$ be a hyperbolic domain, $a \in \Omega$, $B(a,d)$ is a hyperbolic disk centered in $a$ with radius $d$. To show that: there exists a constant $C(d)$ which depends on $d$ only, such that $$ diam(B(a,d)) \le C dist(a,\partial \Omega)$$ in the sense of Euclidean distance.

May we need Koebe distortion theorem or Schwarz lemma?