relationship between area and volume expansivity

Answer 1

I am not sure I know what you mean.
Perhaps I can guess by taking a simple example, a sphere.
A sphere of radius r has a surface area of A = (4) pi r^2
how much does the volume change if I increase the radius a little?
well, I add a volume on the outside of (4) pi r^2 times the the change in radius.
If I keep doing that from r = 0 to r = R I will get (4) pi R^3 / 3
or in other words (4/3) pi R^3
( If you have had calculus you know that the integral of r^2 dr from 0 to R is R^3 / 3 )
in other words the area is proportional to the length dimension squared
and the volume is proportional to the length dimension cubed

Answer 2

and the increase in volume is proportional to the area times a length increase

Answer 3

for similar figures
If you double a length dimension
you multiply all areas by four
and volumes by eight