Can a black hole have a finite perimeter but an infinite radius/diameter? Recently, I attended a talk at UNC Chapel Hill by Kip Thorne and I recall that he mentioned that black holes can have a finite perimeter but an infinite radius/diameter (since spacetime is curved with the instance of a black hole, general euclidian geometry isn't obeyed) Is this true, or am i misinterpreting/misremembering what he said?
 A: In special relativity, lengths aren't well-defined unless you specify what frame of reference you're measuring them in, due to the phenomenon of length contraction. In general relativity, we have the additional issue that there is no such thing as a global frame of reference, only local ones.
So subject to these two limitations, if we want to define something like the diameter or circumference of a black hole, we need to effectively specify a chain of local observers, each in some state of motion.
At any point outside a black hole, there is a pretty natural frame of reference to prefer, which is the frame of an observer who is at rest with respect to the black hole, at that point. After all, if I was going to measure some ordinary object like a tree, I would naturally prefer to measure it in the frame where it was at rest. This means that we can define the circumference $C$ of any circle centered on the black hole, as long as the circle lies outside the black hole. This also works in the limit as shrink the circle down as far as the event horizon.
For these circles, it's also conventional to define a radius $r=C/2\pi$. However, this $r$ is really just an arbitrary coordinate, not a distance that can be measured from the center. The reason is that once we go inside the black hole, it's no longer possible to have observers at rest relative to the black hole. So this "radius" isn't really a radius in the Euclidean sense. It's known as the Schwarzschild $r$ coordinate, and its value when the circle is shrunk down to the horizon is called the Schwarzschild radius $r_s$.
Because there is no preferred frame of reference at points inside the black hole, there is no good way to define notions such as the geometrical radius or diameter of a black hole, or its volume.
