What is the moment of inertia of a cone?!?

I will warn you in advance that you might appreciate this post more if you have taken physics.  But I will write it anyway, with the hopes that it might amuse some people (and hoping not to scare anyone away from taking physics--I think all of you should!)

We were going to have a test today about rotational mechanics.  At lunch, some people from my physics class were joking about the strange, complicated problems that sometimes appear on our tests.

"I've determined what the first problem on the test will be," someone said, grinning.  "There's a vertical rod with a cone on top, and it's in equilibrium, so it stays in that position.  But then a torus that someone threw lands on the cone, and the system starts to rotate."  The cone was because we have a continuing joke that our teacher tells us that we only have to know the moment of inertia for a disk and a hoop, and then he accidentally puts some other weird shape on the test. 

This was enough to make people groan, but we continued, various people adding to the problem.

"Since the rotation is vertical, you have to consider gravity."

"And the rod rotates about an axle, but there's friction in the axle."

"And the axle shifts around an elliptical path, like one of those baby toys where you push the thing through the path.  And the question is, how far along the path does the axle go!"

"No, the axle is connected to three gears, but the gears don't mesh perfectly, and sometimes they slip--"

"And each one has a different coefficient of friction."

"And one of them is a triangle!"  

By the end of the musings, we had created quite a nightmarish problem.  Someone sitting nearby said to me and another boy, "Can you fail the test so that Mr. H. decides that it was impossible and puts it on a curve?"

I was thinking to myself, that of course I wouldn't do such a thing, that I tend to be very good at physics, I did all of my homework relatively easily, and I would probably get an A.  

But the sad thing is that the actual test didn't seem much better than our musings.  Maybe my brain wasn't running at full speed, or maybe the test was harder than usual.  And I really do think that there were some moments of inertia on there that were not hoops or disks.  Yes, I really do think so.  

-Philip