This is not a rule. I hesitate to use that word, since there is no such thing in graphic design. As I said in my comment, the golden ratio is not a rule, it's a tool to aid composition, and it doesn't have to be used. In fact sometimes it's better to ignore it completely. I rarely use it consciously. I find it too limiting.
Remember, design is also about creativity and being original. You don't really want to limit yourself to following some blind rule (like the rest of the sheeple), or the results would be insipid/boring. Nothing would ever surprise the viewer. It's like pop-music that follows the same rhythm and chord progressions - eventually we will want something else.
An analogy for the idea of balance in graphic design that I like to use is when you adjust the balance of the speakers on your hi-fi system, depending on where you are sitting in the room, so that they sound best to you. It's just as valid in design to rely upon what looks good to you, using your eyes and brain. Humans perceive things differently from machines, and I don't think some formula or rule can take that organic factor into account.
Adjusting images visually for the appearance of balance can be done by eye, and the use of some guides if necessary.
For example (see below), using shapes similar to those you posted, I might balance these something like this. This is obviously not the only way you could make a pleasing arrangement. There are other possibilities. Also remember that different people can have different tastes, so there will be differences on how people feel about or react to designs.
The thing to be aware of is that you will probably have to compensate optically for the size of the bigger circle, otherwise the group of circles won't look vertically centred, but too high, or too low. Of course, this assumes that being centred vertically is what you want.
You should note here that on the bottom example, I didn't really like the larger circle centred as I feel the design is too top heavy, and I decided to keep the two circles the same size as those in the upper example, to help fill the space. Of course, this is a personal preference, and might not suit your particular application of the design. My preference for this would pertain to graphic design, and possibly it might not be optimal for a product like a footswitch.