No, crab belongs to the family of Brachyura while hermit crab belongs to the family of Paguroidea.
The hermit crab is a tiny sized crustacean, that is found in ocean waters worldwide. Despite its snail-like look the hermit crab is related to crabs, though they are not that closely related as the hermit crab is not a true crab. There are over 500 different species of hermit crab found in marine habitats all around the world.
Though they do venture into deeper waters, hermit crabs are more commonly found in coastal waters where there is more food and places to hide.
The hermit crab has a soft under-body which it defends by carrying a shell on its back. The shell of the hermit crab is not its own, however one that belonged to another animal. As they grow, hermit crabs continue to find larger shells to accommodate their increasing size.
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