No.
Normally bigger means having more volume!
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Yes. A good example would be styrofoam and steel. The same volume of each of these substances would make for vastly different weights.
1 quart is the same volume as 4 cups. That's bigger than 1 cup.
They would have to have the same base area, if that's what you mean.
they share the same volume. In the Metric system, Liters are a measurement of volume, not of weight. Therefore, 10 Liters of water would share the same volume as 10 Liters of Mercury.
There is no reason for the surface area to remain the same even if the volume is the same.