Want a hint? Think about a circle bound by an n-sided polygon. What happens to the space between the bounding polygon and the circle as n increases? And when n is infinite?

So of three possible regular tilings, which will be most and least efficient?

(Btw, strictly speaking, I shouldn’t have said tri/hex before, as it’s really just hex tiling.)

You could also use some fancy trig to calculate the efficiency %, but that’s way too much work for me. :)