Scaling affects lengths and areas differently. When we make a scaled copy, all original lengths are multiplied by the scale factor. If we make a copy of a rectangle with side lengths 2 units and 4 units using a scale factor of 3, the side lengths of the copy will be 6 units and 12 units, because $2\boldcdot 3 = 6$ and $4\boldcdot 3 = 12$.

The area of the copy, however, changes by a factor of (scale factor)². If each side length of the copy is 3 times longer than the original side length, then the area of the copy will be 9 times the area of the original, because $3\boldcdot 3$, or $3^2$, equals 9.

Two rectangles. The first rectangle has the vertical side labeled 2 and the horizontal side labeled 4. The second rectangle has the vertical side labeled 6 and the horizontal side labeled 12. Two horizontal dashed lines and 2 vertical dashed lines are drawn in the second rectangle dividing it into 9 identical smaller rectangles.

In this example, the area of the original rectangle is 8 units² and the area of the scaled copy is 72 units², because $9\boldcdot 8 = 72$. We can see that the large rectangle is covered by 9 copies of the small rectangle, without gaps or overlaps. We can also verify this by multiplying the side lengths of the large rectangle: $6\boldcdot 12=72$.

Lengths are one-dimensional, so in a scaled copy, they change by the scale factor. Area is two‑dimensional, so it changes by the square of the scale factor. We can see this is true for a rectangle with length $l$ and width $w$. If we scale the rectangle by a scale factor of $s$, we get a rectangle with length $s\boldcdot l$ and width $s\boldcdot w$. The area of the scaled rectangle is $A = (s\boldcdot l) \boldcdot (s\boldcdot w)$, so $A= (s^2) \boldcdot (l \boldcdot w)$. The fact that the area is multiplied by the square of the scale factor is true for scaled copies of other two-dimensional figures too, not just for rectangles.