Because bullets are designed with very specific performance parameters in mind. In the case of the 5.56 NATO they were:
* .22 Caliber
* Bullet exceeding supersonic speed at 500 yards
* Rifle weight of 6 lb
* Magazine capacity of 20 rounds
* Select fire for both semi-automatic and fully automatic use
* Penetration of US steel helmet through one side at 500 yards
* Penetration of .135-inch steel plate at 500 yards
* Accuracy and ballistics equal to M2 ball ammunition (.30-06 Springfield) out to 500 yards
* Wounding ability equal to M1 Carbine
On top of all that, they wanted a round with less recoil than the M2.
After development and testing of many different new ammunitions, the brand new .223 remington was the round that had the best results for the specified parameters. It was later officially adopted by the US military and renamed “Cartridge, 5.56mm Ball, M193.” This round was further developed and became the 5.56×45mm NATO with the subsequent military designation of SS109 in NATO and M855 in the U.S.
While 5.56 may sound arbitrary, it was the result of a 13 year project that spanned multiple countries and involved every major gun and ammunition company of the time, to create a small caliber, high velocity firearm for military use.
To add to what others have said, the bullets weren’t designed around nice round numbers because a) they weren’t necessarily designed with modern units and b) they were almost all designed for military contracts, to be as cheap as possible while still being effective.
So while making them slightly bigger to get a nice round number might seem attractive, a government that’s buying ten million bullets isn’t going to want to pay for that extra material. Add this to the various ways scaling up or down can affect projectile performance and you can see why that sort of thing was never taken into consideration.
In Engineering and Science, precision is important. The more decimal digits are specified, the tighter the tolerance must be when manufacturing., plus or minus half the lowest digit. So whether 5.56mm or 6.00mm, but never 6mm.
A 6mm would be 5.5mm to 6.5mm
A 6.0mm would be 5.95mm to 6.05mm
A 6.00mm would be 5.995 to 6.005mm
And that sort of precision is required to keep the bullet the right size for the barrel.
Some caliber in metric come from the convertion from imperial dimension. But for most of them, the main reason is that the caliber of the ammunition is like thier brand or model. It need to be unique so that people don’t confuse them with another similar but different ammunition. If all ammunition would be round up, you would end up with hundred of ammunition with the same number, that’s bad marketing. It just make it harder to the consumer to find the specific type of ammunition his weapon use.
The 5.56mm NATO cartridge was in fact designed using the metric system. The bullet ended up as exactly 5.70mm in diameter, or in imperial it would be .224″. The designation of 5.56mm is from the diameter of the barrel which is smaller then the bullet in order to create an airtight seal around the bullet and allow the rifling to cut into the bullet.
7.62mm is exactly .30″ which were the caliber for the earlier rifles that these cartridges are based on. And even though most .30″ rifles are bigger, some are even over 8mm they were all called 30 caliber which were converted to 7.62mm. It should also be noted that some countries did call these 8mm instead of 30 caliber, for example the 8mm Label and the 8mm Mauser from the French and German manufacturing respectively. However these did not stick around. On the other hand the 9mm Parabellum round have kept its name until today.
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