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FYI just wear armor.
If you *are* wearing armor, then unamored defense is not available. You use the armor's base AC, dexterity modifier up to the armor's maximum; and again, +2 if using a shield.
So, for an example of where wearing certain armor would actually be worse for your AC:
* dexterity 12, constitution 14 -> +1 from dex, +2 from con => AC 13
If such a barbarian put on leather armor, that's AC 11 with no limit on dexterity modifier -- but armor means no CON modifier, so he'd be looking at 11 + 1 -> 12.
However, if the same barbarian found a suit of half plate, that would yield AC 16 (base 15, plus dexterity modifier of +1 as the cap is 2).