Definition:Irrational Area

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An irrational area is a geometric figure whose area is an irrational number of units of area.

In the words of Euclid:

And let the square on the assigned straight line be called rational and those areas which are commensurable with it rational, but those which are incommensurable with it irrational, and the straight lines which produce them irrational, that is, in case the areas are squares, the sides themselves, but in case they are any other rectilineal figures, the straight lines on which are described squares equal to them.

(The Elements: Book $\text{X}$: Definition $4$)

Also see