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A dimension is something that exist within space, actually we according to general theory of relativity and string theory we have infinite dimension, although we cannot see it because we are living in the 3D dimension. We cannot represent 5th, 6th and so on dimension, yet we can use string theory in order to predict it. A volume isn't a dimension but it is the measure of the amount of space being occupied in our 3D world. Anyone can define the 3rd dimension because there are no precise definition we can give in it, for example 3D is something that has x, y , and z axes. 
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