When we're dealing with areas there will be two dimensions to consider. These can be represented by vertical and horizontal measurements. So if for example we have a rectangle of height y and breadth x its area will be xy.
If the ratio of enlargement is r, then for the enlarged image the height becomes ry and the breadth rx, so the area becomes (ry)(rx)=r2xy. Since the original area A=xy, the new area is r2A. This is the formula. Just find the area of the original 2-dimensional figure and multiply by the square of the enlargement factor to get the area of the enlargement.
This formula applies no matter what shape the original is. For example, if the original is a circle of radius a, then A=πa2 and the enlarged circle has a radius of ra, so its area is π(ra)2=πr2a2=Ar2. The formula still applies.
Note that r can be a fraction so that we get a reduction in the size instead of an enlargement. When 0<r<1 we get a reduction, and when r>1 we get enlargement. When r=1 there's no change in size.