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No, Trump is not the fattest president
He probably is lying about his height, though.

“Girthers” suspect that President Donald Trump’s vital statistics are misrepresented, and that he has a serious obesity problem.
Even considering the uncertainty over Trump’s height and weight, he is definitely not the fattest president in history. He might be pushing second place, but even that isn’t likely.
The first thing to understand is that Trump does seem to have padded his vital statistics. His 2012 driver’s license and 1961 draft card both list a height of 6′2″. In other words, there is an official chain of records stretching from 1961 to 2012 showing him with a height of 6′2″. Trump currently claims to be 6′3″. It is most likely that his actual current height is about 6′1″.

Height reporting errors are common
First, it’s worth understanding that it’s very common for people, especially men, to add 1–2″ to their heights. Men’s self-reported heights are generally around 1″ taller than those measured by medical professionals; OKCupid published an estimate that men add an average of 2″ to their height on their online dating profiles.

It is common for a man who reports his height as 6′3″ to actually be 6′1″ or 6′2″. It’s very uncommon for any height measurement, whether a self-report or an official measurement, to be an underreport, so the older reports of 6′2″ are unlikely to be underestimates.
Additionally, people do tend to lose a little bit of height in their old age. From age 30 to age 70, the average man loses about 3 centimeters (1.2″) of height. Usually people don’t re-measure their height regularly after reaching adult age, so this goes unnoticed. Someone who is 6′2″ as a young man may inaccurately measure themselves as 6′3″; but, by the time they reach Trump’s age, will have shrunk to 6′1″.