Background size for 4K UHD Shooting I would like to create a background/backdrop for a video, but I am having quite a hard time deciding on the best dimensions for the backdrop. I will be shooting at 3840 x 2160. Is there a way to determine the dimensions based on the resolution? The idea would be to have a backdrop that completeoly 'fills' the frame of the camera, if that makes sense.

I tried playing with different options in Photoshop and Illustrator. i.e. I created a document with the dimensions 3840 x 2160 at 300DPI (since it's for print), determined it would come out as roughly 32.51 x 18.29cm, and then multiplied until I got a reasonable set of dimensions (i.e. multiplying 8 times = 260.8 x 146.32cm). But somehow, it seems wrong because I would need to place the camera at a certain distance from the background to capture it all.

I'm quite sure there's no rule or anything like that for this, but I would really appreciate any help/pointers on this one!
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.