when we expand a metal ring by heating, as we learn from our childhood, its perimeter increases. But, why should it not expand to the inner side and its perimeter remain constant?

when we expand a metal ring by heating, as we learn from our childhood, its perimeter increases. But, why should it not expand to the inner side and its perimeter remain constant?

Not quite sure what that has to do with the original question. You probably should have started a new thread for your question. Doesn't have to stop me from replying, though:

 

What happens when you increase the temperature of a metal ring is that the (most favoured) mean distance between the atoms increases. Think of the metal ring as many tiny, one-atom thick atoms: If you increase the distance between the atoms forming this ring, the radius of the ring will increase. This is true for all of the tiny rings making up the metal ring you heat - even the innermost of them will increase in radius. That's an explanation (not to confuse with the reason) why a metal ring will only increase outwards. Note that the thickness of the ring can still increase. In fact, it probably should, given that mean distances between neighbouring atoms are supposed to increase (and therefore so-should the mean distances between the one-atom thick rings).