The line of the circumference is one dimensional. It has zero area.
A little more than a year ago I decided to pretty much re-learn maths from scratch (and when I say from scratch I mean from very basic things such as long division), then stopped and have gotten back to it recently. Good news is that apart from a few things (mostly purely human constructs such as some statistical concepts) I haven't lost what I had learned, but now that I'm reviewing some of the basic stuff there's all sorts of weird questions that pop up in my mind.
For instance, the formula for the surface area of a right cylinder, is simple enough and intuitive: 2 * area of the base + circumference of the base * height. Well, let me correct that: "it should feel intuitive, but it doesn't to me". Why isn't the formula 2 * area of the base + circumference of the base * height - 2 * circumference? I mean, when you're computing the area of the "wrapped rectangle" that lies between the bases (circumference * height), aren't you overcounting the two circumferences that are part of the area of the base? I don't know if I'm expressing myself very clearly, but I think you get my point. I don't know if I'm having a vizualization problem or if it's because I can't wrap my mind around infinitesimal quantities (I should probably note that I've never learned calculus, although I'll be starting pretty soon) or something else.