TITLE: Does the staircase paradox apply to areas or volume?
QUESTION [6 upvotes]: So there is the “staircase paradox” that is sometimes used to “show” that $\pi = 4$ (in the case of approximating a circle), or that $\sqrt{2} = 2$ (in the case of approximating the hypotenuse of a triangle).
But when we define things like the integral, don’t we also speak of approximating something, in a similar fashion to approaching the hypotenuse of a triangle by staircases from above and below? For example, for the Riemann integral, we speak of approaching the area by looking at the supremum of the lower sums and the infimum of the upper sums; if these two numbers agree, then we say it is integrable.
So why do we seem to not “run into such problems” when considering area/volume (or at least, the problems we run into are not quite the same), whereas we have problems when we (naively) consider lengths? Is it a matter of defining the area/volume as the limit (but this doesn’t seem satisfactory, because after all isn't a big purpose to model phenomena and solve problems, and avoid paradoxes that conflict with basic geometry)? Does it perhaps have to do with measure or dimension, e.g. approaching a line segment by line segments is sort of like approaching a 1-dimensional object by another 1-dimensional object, allowing some “space for things to go wrong”? (And, in general, approaching an $n$-dimensional object by $n$-dimensional ones?)
My basic question is that, a priori, there might not appear much of a reason (not that I’m saying we need one) to confidently/fully believe that our definitions and axioms of area/volume do not run into such paradoxes or do not conflict with some basic geometric properties that we seek to have. So how do we work around this and ensure that our definitions and considerations are consistent and good and all that (whatever that means)?

REPLY [2 votes]: It's because the staircase paradox refers to a 1-dimensional object in 2-dimensional space. If you are computing the area of a 2-dimensional object in 2-dimensional space you do not encounter the paradox (e.g. a naive calculation of the Riemann integral is an 2-dimensional area in 2-dimensional space).
You can easily encounter a staircase type paradox for the area of an $n$-dimensional surface in $(n+k)$-dimensional space, but things get a bit more complicated. The relevant 'more advanced' formula is the 'Area formula'
A basic example is the $n$-dimensional area of the portion of the graph of the function $f : \mathbb{R}^n \to \mathbb{R}$ that lies in the cylinder $(B_1^n(0) \times \mathbb{R})$ is given by $$ \int_{B_1^n(0)} \sqrt{1 + |\nabla f|^2}.$$ And since it is possible for another function $g$ to be uniformly close to $f$ without the derivatives being pointwise close, you can achieve the same "paradox".