To address the question specifically of when a hill becomes a mountain: there is no official rule. Geologists used to say a hill rises to less than 1,000 feet above sea level and a mountain rises to higher, but they have since abandoned that distinction. Now whether something is referred to as a hill or mountain is due mostly to history, during which time it would be based on subjective criteria like whether the summit was rounded or whether it had a gradual slope.
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