I am really confused over the idea.
If I have a circle I can approximate its area by using a regular polygon inside of it, with $n$ sides, and I can just split that polygon into triangles and compute the area. If I want better accuracy then I can use a polygon with more sides. The accuracy becomes better since the area between the polygon and circle (the "error") decreases as $n$ increases.
So it stands to reason that if we could keep adding sides our answer would keep getting better since the error would get closer to $0$.
I don't know what it means to "add infinitely many sides" because to me the idea doesn't make sense. No matter how many sides there are we can always add one more. But it does make sense to ask what's the value we never actually reach but get closer and closer to? For that we use the concept of a limit.
The error becomes "infinitely small" but I don't understand what this really means. Is "infinitely small" the same as $0$? Because by definition it's never quite getting to $0$. But conceptually then how can we say something with nonzero error gives us an exact answer as if it had $0$ error?