As I understand, noun which is singular and start with vowels, we put an article “an” in front of it. ( e.g an eye, an ear )
But why it is not “an earth” and why “the earth”?
You seem to be mixing up two rules here.
We use an instead of a when the next word starts with a vowel sound. Indeed, ‘a ear’ would sound clumsy, so we say ‘an ear’.
We do not use an instead of the; the English have no problem at all saying ‘The ice is thick’ or ‘The ears of a cat are fluffy’ (in fact, I would be hard-pressed to find a sentence with ‘an ice’ in it, since ‘ice’ is a mass noun).
The distinction between a / an and the is well explained in the other answers.