We have studied critical circle maps of the type \ensuremath{\theta}\ensuremath{\rightarrow}f(\ensuremath{\theta}), with f(\ensuremath{\theta})=\ensuremath{\Omega}+\ensuremath{\theta}\ensuremath{\Vert}2\ensuremath{\theta}${\mathrm{\ensuremath{\Vert}}}^{\mathit{z}\mathrm{\ensuremath{-}}1}$/2, -1/21/2, and f(\ensuremath{\theta}+1)=1+f(\ensuremath{\theta}), where z is the order of the inflection point of the angular variable \ensuremath{\theta}. Such maps are believed to be useful in the study of the two-frequency quasiperiodic route to chaos. Using a variety of numerical approaches, we have calculated the fractal dimension D associated with such maps as a function of z. The different approaches yield consistent values for D and the completeness of the staircase has also been checked at each order. By comparing 1-D with the width of the mode-locked interval \ensuremath{\Delta}\ensuremath{\Omega}(0/1) (which may be analytically determined as a function of z for this class of maps), we have established that the ratio appears to be roughly independent of z with a value of 1/3. This leads to the empirical formula D\ensuremath{\approxeq}1-[(z-1)/3](1/z${)}^{\mathit{z}/(\mathit{z}\mathrm{\ensuremath{-}}1)}$, which predicts for z=3 that D\ensuremath{\approxeq}0.872, in good agreement with previous precise direct numerical determinations. This general result suggests that the average width of mode-locked intervals of cycle length Q declines at the rate of ${\mathit{Q}}^{\mathrm{\ensuremath{-}}2/\mathit{D}}$ with the 0/1 interval setting the scale for arbitrary intervals for all z and thereby governing the fractal dimension of the set complementary to the devil's staircase.