If f is a K-quasiconformal homeomorphism of Jordan domains in the plane, then for any given \(c>1\), f is equal to some composition of \(N1\). Then it is easy to show that \(s_ k\) is an L-quasi-isometry, i.e., \[ (1/L)| p-q| \leq | s_ k(p)-s_ k(q)| \leq L| p-q|,\quad \forall p,q\in \bar D^ 2. \] The main result of the paper shows that it requires \(N\geq k/\sqrt{\alpha ^ 2-1}\) (rather than \(\log _{\alpha}k+1)\) factors to write \(s_ k\) into a composition of \(\alpha\)-quasi-isometries. The conformal structure of planar annuli is used to define a notion of ''twist'' for a homeomorphism. The proof analyses how a given amount of twist can be shared among factors.