I am told that fat men have apparently shorter penises because when men get really obese, fatty tissue obscures (buries) part of the penis, making it appear shorter than it actually is.

Another reason to keep the bulge away, I suppose.

A side note: there is very little fat in the penis shaft itself, so the penis does not get larger as the body gets fatter.

Short answer: no.

Most fat is subcutaneous, however there is basically none under the skin of the penis. Those of us who are male should be able to find this out ourselves, by palping the penis when it is erect. (Yes, there is a difference between palping and wanking.)

On mine, at least, :) I can plainly feel every ripple of the engorged corpus cavernosum beneath a thin layer of skin. There is no subcutaneous fat that I can detect.

By the way, one popular method of 'penis enlargement' some years ago involved harvesting fat from the abdomen and injecting it under the skin of the penis. In the words of the author of The Guide To Getting It On!, "you pay $$$ to give your penis cellulite." The injected fat could also get hardened and clumpy. Yummy.

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