It may involve no calculations by you, but that is because they have already been done for you by the person who worked out where to put the DoF scale marks on the lens.
Just the same as using a DoF table. You look up the value in a table for which someone else has done the calculations.
Just the same as using a DoF app. You read the result after the computer does the calculations for you.
All DoF calculators assume a value for Circle of Confusion (CoC). The equation is DoF is approximately (2 u^2 N CoC), where u is distance in number of focal lengths or actual distance divided by focal length, N is f-number and of course the CoC. The CoC is a subjective value having to do with human visual perception and it varies depending on how the image is viewed. If one pixel peaks at 100% the CoC will be on the order of just a few pixels. If the image is viewed on a display at a normal viewing size and distance, it will be different (and larger than for the pixel peaker). If one prints it will be different still.
Personally, I rely on 50 years of experience with multiple camera formats from APSC to 4x5. I might add I also consider the final display of the image is the fine print. If one is wants to pixel peak, the DoF will be very small unless one goes into the zone of diffraction limited where nothing will be really sharp.