actually a very good exercise to understand why this argument doesn’t work. you’ll learn something about the nature of continuity and what it really means to repeat a process infinitely many times
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you actually don’t need a background in real analysis or topology even, although it would help you go deeper; the basic conceptual mistake can be explained more simply
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That at some point you'll run out of corners when you run up against the circle?
Or am I just not learned enough in math/science to follow along with the point?
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the simplest way to explain the issue with the argument is that you never actually “reach” the circle; even if you repeat the process infinitely many times you only end up creating these more and more jagged shapes, none of which are circles
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if you took a calculus or real analysis class you might ask, okay, but what about limits? and then the issue with this is that there are multiple ways you might define the limit of a sequence of shapes, and the argument requires a definition with two contradictory properties
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the reason it ends “problem, archimedes?” is that archimedes pioneered a very similar technique for computing areas and volumes called the “method of exhaustion”; the trollishness of the argument comes from the observation that the method of exhaustion fails to apply to lengths