soft question - Simple definition of the Hausdorff measure using squared paper

I am giving a "non-technical" seminar in which I would like to give an elementary introduction to the Hausdorff dimension and measure.

For simplicity, I was hoping to give a more intuitive introduction using 'squared paper': suppose you draw your figure on squared paper with squares of size $2^{-n}$, and let $M(n)$ the number of squares that your figure touches; take the limit of $M(n)(2^{-n})^d$ for $ntoinfty$.

This seems to be ok for defining the dimension $d$ (the only issue is that you only allow coverings with very specific sets, the 'elementary squares'), but not for the measure. In fact, if you just consider line segments, then this definition converges to their 1-norm length ($|x_1-x_0|+|y_1-y_0|$) instead of the usual Euclidean length.

So, my questions are:

1- is there any hope to save this approach, or do I really have to introduce the measure using balls (this would be unpleasant as I would also have to deal with the proper constants for the volume of a ball in fractional dimension, ouch)

2- is there a simpler introduction that I am overlooking here?