This is a combinations problem disguised as a geometry problem. We're looking for the number of lines (that is, combinations of two points) that can be drawn inside a regular octagon (a set of 8 points), with the exception of the sides of the octagon. This is a combinations problem with a set of 8 and a desired subset of 2. The combinations formula will tell us how many distinct sets of 2 points exist among the 8 vertices of an octagon. Each of those sets of two points is a distinct line segment as soon as we draw a line between them.

To solve:
8! / (8 - 2)!2!
= 8! / 6!2!
= (8)(7) / 2
= 28
However, 8 of those lines are the 8 sides of the octagon. Thus, our answer is 28 - 8 = 20, choice (A).