Background size for 4K UHD Shooting
There is a rule for something like this, however, it has nothing to do with image resolution.  To calculate the area of a scene a camera will cover, you need to know a few things:  A) Distance to subject.  This one's easy.  B) the focal length of the lens, and C) the sensor size, also sometimes called "film back" or "gate".  If your image width to height ratio is the same as your sensor's you probably won't need to know D) aspect ratio, but for some cameras you will, so I'll include it for good measure.

From there, you should be able to use any number of online FOV calculators to tell you the area of your image.  Here is a link to one I found with a Google search.  The website also explains most of this better than I can and includes the mathematical formulae.

If you've ever used a magnifying glass to focus light onto a point, you're probably aware that the shape of the light it creates is circular.  The same phenomenon happens inside a camera.  Your camera's sensor sits inside of this circle.  Now, if you imagine your sensor divided into many tiny pixels, you should get an appreciation for why resolution does not affect this equation.  It doesn't matter how finely you chop up the rectangle -- whether it's 3840, 1920, 960, or eight pixels wide -- the sensor doesn't grow or shrink.  It always fits within the image circle.