Strange thing i noticed about centroid of a cone vs triangle

[bhh1988]

To calculate the centroid of a cone, it seems that you have to use calculus. It comes out to be h/4, where h is the height from the base of the cone. But intuitively I thought that the centroid would have been h/3 because that's a triangle's centroid, and the cone can be obtained by rotating a triangle.

Now notice that if you slice the cone in half so that you have two halves with semi-circle bases, the height of the centroid of each half should still be h/4. Now if you slice in half again so that you have quarter cones, the height of the centroid of each of the quarter cones will be h/4 again. You can continue slicing this way until you have virtually an infinite number of fraction cones, each with their bases being a tiny slice of a circle. Each of these should have their centroid height be h/4

You can also obtain one of these cone slices by taking a right triangle and rotating it by a tiny angle theta. But the triangle's centroid itself is h/3. So the moment you rotate it a little bit the centroid of the solid generated jumps down to h/4.

I'm not too sure what I'm asking here, but I just find this "jump" to be strange and am looking for some intuition as to why this happens. Can anyone explain?

 

[bhh1988]

I realize now that there shouldn't really be any more of a reason to get stuck on this as there is to get stuck on why the centroid of a pyramid is h/4 from the base while a triangle is h/3. The fractional cone as theta goes to 0 basically becomes a pyramid.

I think I'm satisfied enough with this reason now. It's not any more weird that the centroid of a triangle drops to h/4 the moment I rotate than it is weird that the centroid of a pyramid is h/4. But if anyone else has any other insights, please post.