Calculating the scale factor to resize a polygon to a specific size

When you scale a polygon (or any subset of the plane whose area you can define) by $\alpha$, its area scales by $\alpha^2$. Proving this rigorously takes some work and requires you to have a rigorous definition of "area", but you can see that this makes sense by considering a rectangle. If you scale a rectangle by $\alpha$, you scale each of its sides by $\alpha$, so since the area is the product of the two side lengths, you scale the area by $\alpha^2$.

So in your case, you want $\alpha^2A=A'$, or $\alpha=\sqrt{A'/A}$.

If the ratio of areas is $A:B$, then the ratio of corresponding lengths must be $$\sqrt{A}:\sqrt{B}$$