Well, had time on my hands and thought I'd do some vaguely useful until the time came that I didn't have as much free time. All the awesomeness in Shounen seems to be emanating from One Piece, only. Naruto is good sometimes but lacks consistency. Bleach is Bleh.
How 'bout you come back or something? You weren't half bad.
I think it has to do with the queerness of infinity (which is usually the source of most things that seem counter-intuitive in mathematics)
How many possible chords (a mathematical 1-D line with 0 thickness) can you draw in a circle? Infinity.
How many of those chords are entirely inside the area between the 2 circles and do not intersect the smaller circle at all (hence shorter than 1)? Also infinity.
By using the area approach, we obtain that the first number is supposed to be four times as many as the second one because the area of full circle is of course greater than only part of it. But in fact, despite the area being smaller, the number of possible chords we can draw in the area between the 2 circles is still infinity. Can we say one infinity is 4 times greater than another infinity? We can't. But the area method does seem intuitive and logical on the surface, which is why we use it, I guess. As for whether the answer it gives is correct, my opinion is there is no one absolutely correct answer to this question. It all depends on which method you prefer.
You know what a locus is right? (not the genetic locus, mathematical one. )
The locus of any chords shorter than the radius 1 is actually the area between the r=1 circle and a smaller concentric circle inside it. You just need to find the radius of that smaller circle and you will be able to find that area.