I'm curious if there is any math formula for this.

Let's take a close-up of a human face for instance, given the human face is about 20cm/8 inches, how do I choose the right focal length and from what distance a camera with a APS-C sensor should be so that it fits in the top and the bottom of the frame with the whole face?

I just want to know how to mathematically calculate this. It doesn't matter the effects or distortions caused by a telephoto or a wide-angle lens.

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That is just Field of View, see http://www.scantips.com/lights/fieldofview.html

For a calculator, and if you know trig tangent, the diagram at page bottom will explain how.

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