Your mistake is that you are using only one copy of the base area. You need two since you are including the lids! Just use twice the base area and you'll get your answer. However, I think you are making it unnecessarily difficult for yourself by doing it the way you seem to have done it.
Here's a better way. Let the radius of the small can be and that of the big can . Since they have the same capacity, I will measure the smallness by their height. That is the one where the height is equal to the diameter is the small one and the one with twice its diameter is the big one.
So the small one has diameter , so its height is . Its volume is .
The big one has diameter , so its height is and its volume is .
Since the volumes are the same you have
which means after cancellation. This means
Now for you equation. The ratio should be