There have been a couple tests made on this question.
In the Mythbusters episode "Coffin Punch," a test was performed based on the thickest belly fat measurements recorded in a human: sixteen inches. They simulated this using cow fat, which was easily penetrated by a round from a pistol.
While the Mythbusters didn't name the person the body fat measurement came from, it was probably record holder Manuel Uribe, who reached a peak weight of 1,235lbs, making him the fattest person in the world.
The Naked Scientists also performed a similar test at Cavendish Lab's Surface, Microstructure and Fracture Group. In their attempt to answer this question, they used an air gun to fire a ball bearing at 1,640 feet per second. This is slower than most bullets are moving after they've traveled 200 yards. The ball was shot into a perspex tube filled with gelatin which had a density equal to fat. After traveling through it left the tube traveling at 591 feet per second, which is at the edge of what would be considered lethal. The bearing passed through 28 inches of gelatin, nearly twice as far as the Mythbusters' test, and ten inches thicker than the blubber layer of the largest whales.
Can you be so fat bullets can't reach your vital organs? No. Even if you were to match the thickness of the Naked Scientists tests, which still resulted in a lethal wound, you would have to weigh well over a ton.
Posted 3859 day ago