直观感受：放大时面积成平方比变换

Since area must be compared to a reference shape, usually a "unit square", the "area" of a shape is simply the ratio between it's area, and the unit square's area. Now, suppose, instead of dilating the shape, we dilate our "unit square" inversely! The ratio change is equivalent to that of dilating the shape, and since we know that squares dilate quadratically, it follows that the shape dilates quadratically as well!

To prove that squares dilate quadratically, we simply let the unit square's length be $x$, and the comparison square's length be $kx$, for some $k$. Then, the ratio of areas is $\dfrac{(kx)^2}{x^2}$, or $k^2$, so area is multiplied by $k^2$ when side length is multiplied by $k$, for a square.