Let $E>1$ and consider an annulus in $\mathbb{R}^2$ with outer radius $R=\sqrt{E}$ and inner radius $R=\sqrt{E-1}$.

How many unit cubes do I need to cover the annulus?

The area of a $2$-dimensional annulus does not depend on the outer and inner radius, so one could think that the number of needed cubes depends only on the dimension. However, for growing $E$, the width of the annulus becomes thinner and the length of the inner and outer circumference becomes longer. So it looks to me that the number of cubes grows with $\sqrt{E}$, but I don't really understand how. Can anybody help me in this?

**Edit:** the squares (not cubes because we are in two dimensions) need to have sides parallel to the $x$ and $y$ axes. Maybe this changes things a little...

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