How can Gabriel’s Horn have an infinite surface area, but a finite volume?

Question

Seems rather paradoxical in my opinion.

In: 4

Answer 1

…and hence the name *The Paradox of Gabriel’s Horn*.

The paradox comes from the intuition that if the volume is finite, approaching π units, then we could fill the inside of the horn. But if we can fill, in effect paint, the inside of the horn, then wouldn’t we have enough to paint the outside of the horn?