If you double the radius of a balloon, you increase the volume by a factor of eight (since volume is proportional to the radius cubed). But what about the material on the outside of the balloon? Let’s say I want everything to be just right and I increase the thickness of the material by a factor of two for the largest balloon. Since this material only covers the surface of the balloon, its area would increase by a factor of four. If twice the thickness is included, the material in the larger balloon also has eight times the mass of the smaller one.
But at some point, you don’t need to keep making thicker and thicker balloon skins. I can get some material (say rubber) that is very strong with only a millimeter thick. This means that if I increase the radius of a balloon by a factor of 10, the volume will increase by 1000, but maybe the mass of the shell will only increase by 100. Volume is important because that’s where I get my buoyancy force from.
Now let’s go to the other side. Let’s make an ant balloon. If I reduce the radius of a normal party balloon by a factor of 100 (actually it should be even smaller than that), the thickness of the shell would also have to decrease by 100. These balloons are already quite thin. Go too low and you simply won’t have a structure capable of holding the balloon together. Increase the thickness a little and the dough will rise too much to float. Sorry, no parade balloons for ants.
Bigger balloons are more difficult
Hurrah! I have a giant balloon and it floats. What could be more amazing? Oh, sure, I’m going to need a bunch of people to hold it down (along with a couple of vehicles), but it’s still a giant balloon. But wait. Giant balloons still have problems. Making things bigger may make floating easier, but it adds other problems.
The first problem is the wind. Sure, that breeze in your little handheld balloon is annoying. But what happens when you increase the size of the balloon? This force pushing on the balloon is proportional to the cross-sectional area. If you double the radius of your balloon, you increase this area by a factor of four, which gives four times the air force.