You are changing height and volume in your calculation so it is hard draw conclusions. You are comparing two different pots. You should compare two different pots that hold the same volume.

Let's take your 6" diameter/3" tall pot. It will hold 84.8 in^3 (pi x 3"^2 x 3" height). The weight of the pot will be proportional to the surface area (To get the weight of the pot take the surface area and multiply it by the wall thickness and the metal density). The surface area of your pot is 84.8 in^2 (pi x 3"^2 (for the bottom of the pot) + pi x 6" (diameter) x 3" (height) (for the sidewalls)).

The question then is there a pot with a different height and the same volume that has a smaller surface area. Let's try doubling the height to 6". The diameter would drop to 4.24" (sqrt(4*84.8/pi/6")) but the surface area would increase to 94 in^2 (pi x 4.24^2/4 + pi x 4.24" x 6"). Since the surface area increased the pot would weigh more.

We can also try a pot with the same volume and the height equal to the diameter. The would be a pot with a height and diameter of 4.76" (cuberoot(4*84.8/pi)). The area would be 88.9 in^3 (pi x 4.76^2/4 + pi x 4.76 x 4.76) and would therefore also be heavier than the pot where height is equal to radius.