Dragons of the Third Age

[Elladan and Elrohir]

In any case, it is completely impossible that Smaug was even close to being born at the destruction of Angband. He himself says in The Hobbit that when he destroyed Dale and took Erebor for himself, he was young, but that he is now old.

Either this means he stayed young throughout the entire Second Age and most of the Third Age and then suddenly became old in the space of over a century (I can't remember exactly how long) or it means he was not one of the dragons that survived Angband, which means he was one of their offspring.