A line contains an infinite amount of circles? And is a rectangle??What? I don't know what my math professor is talking about anymore. His statements don't make intuitive sense as they used to. He says a line contains an infinite amount of circles and is also a rectangle with an infinite length. Last time I checked a line contained no circles, its just made up of an infinite number of points. And a line is not a rectangle. A rectangle can contain line segments, but a line can not be an rectangle with infinite length because a line has no width. A rectangle must have a width. The formula for area is length * width. If there is no width there is no area, therefore no rectangle. 
Please explain to me, if I am approaching this the right way. Or if my professor is right? If my professor is right please explain how he is right?
BTW, This is for a calculus class. He said learning these geometric definitions may help us later in the year for optimization.
 A: He probably is speaking of degenerate figures, and his statements do make some sense.
One definition of a circle is the locus of points at a given distance from a given point. If that given distance is zero, the locus is indeed just a point. We may consider a point to be a degenerate circle, but still a circle.
One definition of a rectangle is a rotated figure of a set of ordered pairs where the first coordinates are taken from an interval and the second coordinate also from an interval. The real number line may be considered as an interval $(-\infty,+\infty)$. A set containing just one real number may be considered as an interval $[a,a]$ for some real number $a$. Those intervals may be degenerate, but they are still intervals. The set $\{(x,y) \mid x\in(-\infty,+\infty),\ y\in[a,a]\}$ is a rectangle under that definition, as are all rotations of that set. The areas of those sets may be considered to be zero, but that just means they are degenerate, not that they are not rectangles. Those are, of course, all the straight lines.
I teach in my calculus class that degenerate figures should also be considered when finding the figure that maximizes or minimizes some attribute. The rectangle with minimum area just may be degenerate. A straight line could in some sense be considered be a rectangle with maximum perimeter. And so on. If you like, if you find that the optimal "rectangle" is a straight line you could say that there is no optimal rectangle. That approach is sometimes easier than not considering the degenerate figures at all. This is partly a matter of taste, but it can reduce the amount of work required to solve the problem. And who knows, in many cases the degenerate figure may be an acceptable answer.
