Why does the magnification of a microscope not refer to the area of the magnified image?

Why does the magnification of a microscope not refer to the area of the magnified image?

In fact, it's for convenience. For example, when I say to enlarge the area of a square by 2 times, it's actually the side length by √ 2 times, which is irrational. It's not easy to draw. But I say to enlarge the side length of a square by 2 times, which is easy to do,
The third root sign 2 is an ancient Greek title