how do you solve for scaling of surface area to volume ratio of a 1x1x1 cube?

Question

I am trying to understand how to use the scaling of surface-area-to-volume ratio of a cube that is 2X2X2 Cube and a 3x3x3 Cube. The ratio in my book says (scale factor)^2/(scale factor)^3=1/scale factor.

Answer 1

The easiest way to understand these ratios is to take a cube of side a. Its volume is a^3 cubic units.

Its surface area (6 square faces) is 6a^2 square units. The SA to V ratio is: 6a^2/a^3=6/a.

If a=2 (2X2X2 cube), the ratio is 6/2=3. This is measured in units^-1.

If a=3, the ratio is 6/3=2 units^-1.

If a=1, the ratio is 6 units^-1.