If you can answer the question easily, humor the interviewer and move on -- because lots of people can't. I actually find it kind of fun, because then you get to see how close you get.
Also, as you become promoted, being able to work autonomously with minimal supervision and incomplete information is indeed part of your job description. If I were hiring someone, I'd want someone with a solid growth trajectory.
The point is that someone talks through their thought process. The solution is to say, "Well, you need to know the volume of a ping pong ball, the volume of the trailer, and the packing density of spheres. Ping pong balls are a few centimeters across, 3 maybe?, so their volume is about..."
The adversarial tone was present in the original followup. I don't see what's wrong with continuing it. It's better than "sorry you feel that way, but we expect people to be capable of looking things up; there's the door."
This is a pretty trivial question. It should take somebody who knows Google and grade-school math about 30 seconds to solve. I'll give you ten minutes just to be generous, though.
Your "confounding factors" are irrelevant. If you crush or melt ping pong balls then what you're carrying are no longer "ping pong balls" so the question as asked can be answered by using the standard 38mm diameter of a ping pong ball and working from there.
See how this could coax a poor performance out of an inquisitive, somewhat introverted candidate, rendering a false negative?
Beyond that, there's the more important adversarial tone: As an employee, will the smallest misstep result in termination? What if I have a problem that cannot be easily solved with a search engine? Is this a cooperative environment where I will feel somewhat secure as an employee, or is my status fragile? Is Mike a mercurial person who gives his employees brain-teasers with job performance implications?
My question is, if you need me to hold your hand from start to finish, then why would I pay you money just so I can do all of your work? No, I don't expect the candidate to solve a non-trivial geometry problem in a high-pressure situation. I do expect the candidate to solve a trivial Google problem in a high-pressure situation.
And, what adversarial tone? The original question was reasonable. bobbytherobot's follow-up question was completely unreasonable. Once the candidate has decided to be adversarial, my decision to continue that no longer really says anything about the work environment.
Can you not see the difference between "smallest misstep" and taking a pretty reasonable question and instantly turning it into a whine? Do you not see how it's ridiculous to ask, "or am I a researcher?" in response to that question? What position would you be interviewing for where you are not going to research some things?