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"Ice area only? Wouldn't volume also be a important measurement?"
I think area and volume are directly proportional to each other. They may not be in perfect alignment, but they are certainly related.
Nope. Like a puddle with ice on the surface, is not the same as the entire puddle frozen solid.
The surface freezes, then continued freezing conditions build more ice, increasing volume. A one time freeze will not replace the ice already gone.
And you don't have to believe in Global Warming any more than you have to believe that the plane you're on is crashing.
There are no known laws of physics that require specific chemical reactions in a human brain to function.