To think about the cube method geometrically, I came up with the formula (x+1)^3 + x(2x+1).
(x+1)^3 represents the new top layer of the growing cube (assuming you are building up).
2x+1 is the new "thickness" around each of the existing layers. It is the same concept as you used for the growth of the area in the square example. There would be x layers where this new "thickness" is needed.
If you simplify (x+1)^3 +x(2x+1) it is equal to 3x^2+3x+1 which is the result when working with algebra or calculus.
I am interested if this is the way that other people thought of it, or if you set up a formula for geometric growth in a different way.
I really enjoyed the geometric, algebraic, and calculus connections in this post, but I agree that the geometric one is definitely the most intuitive, at least up to three dimensions.