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Is there a ratio / relationship between camera FOV and aspect ratio?

For instance, I have a FOV of 45 degrees which looks right on 16:9, but need to set FOV 55 for aspect ratio of 4:3

I got this by trial and error, but is there a formula for this? What about other aspect ratios?

Hmm, this is how it should work if im not misstaken.

FOV sets the area of the 3d world to be projected on the screen by the angle of a cone at the center of the camera. An FOV of 45 sets the angle of the cone to 45 degrees on one axis. In order to not stretch the view, the FOV on the axis that is not manually set is calculated based on the display aspect.

Most games use a horizontal FOV system, setting the FOV to 45 will set the angle of the cone to 45 degrees on the xAxis. The y axis will then be calculated based on the display aspect. (FOV * (height / width)) which makes the vertical FOV in this case 45 * (9 / 16)

The FOV on the x axis will always stay at 45 degrees but the FOV on the y axis will depend on the display aspect. If you want to use a vertical FOV instead, you need to calculate the FOV based on the display aspect

Changing FOV to 55 worked on the Y axis for me and allowed me to letterbox a modified 16:9 screen to 4:3
(I had already changed aspect ratio from 1.777778 (16:9) to 1.333333 (4:3).
The situation was that I changed FOV from 45 (the 16:9 setting) to 55 (the 4:3 setting), which preserved the x-res, but gave me an expected and acceptable letterbox.

My question was what is the ratio of FOV value vs aspect ratio if there is such a thing, or would I have to manually set FOV for every possible aspect ratio?

I thought FOV was calculated as The Zoq2 suggested but from what you are saying it works differently (using vertical view angle). Some quick calculations required...

Something like this should work:
newFOV# = oldFOV# + (1.0-(newASP# / oldASP#)) * oldFOV#

Edit: Where oldASP# = 16.0/9.0 and newASP# = 4.0/3.0 (for example)