It seems like an odd number especially fir the metric system.
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360 is even, not odd.
Using 360 degrees for a circle has been done for millennia. Sumerian use is starting around 2600BC. A lot of early usage is in astronomy/astrology. They used a number system with a base of 60 and 360 is 6* 60.
A year is 365 days, quite close to 360 that is divisible with 60. So it is very likely that the number is picked to the sun proximally move 1 degree per day.
The moon moves around the earth relative to the sun in 29.53 day, so there is close to 12 full moons every year.
The Sumerians had 360-day solar calendar based on 12 lunar months of 354 days rounded up
So the 360 degrees for a circle it likely a result of the number of days in a year, the length of a lunar cycle. I would not be surprised if the picked based 60 from observation of them too. We are not sure because they did not write down why they what the selected.
Using 360 degrees for a circle was adopted by Greece, Rome and the usage continued to this day.
A degree was divided into 60 minutes and a minute in 60 seconds. The word minute and the second is from Latin but the practice is Sumerian. It is not until 1000 CE that Al-Biruni use it for the time in regard to Jewish months. In 1267 Roger Bacon ut it for times in regards to full moon. That time usage and it adapted into clocks, that take time because clocks especially watches are not that accurate back then. We call the angular usage arcminutes and arcseconds today even if it is thousands of years older than the time usage.
France tried to change time and angular measurement to with the metric system after the revolution. Ther was a French Republican calendar with 12 months made up of three 10 day weeks. The extra 5 or 6 days per day was complementary days after the end of the last month. A day was 10 hours and an hour was dived in 100 minutes and a minute in 100 seconds. So their minute was 86.4 conventional seconds long.
It was not a popular system. One reason is laboured before has one day of every weed and it continues but it changes from 1 out of 7 to 1 out of 10. That is 36 versus 52 days off per year.
The French Republican calendar was abandoned by Napoleon in 1806 so only used for 12 years.
The did replace degree too a https://en.wikipedia.org/wiki/Gradian , gon or a new degree in many languages used 400 gradians on a circle. It never becomes very popular but there is still some usage.
One reason I suspect is for a lot of scientific and mathematical work neither is used. It is radians where you have 2 pi radians in a circle. The origin is a circle with a radius 1 have a circumfluent of 2 pi so the radian uses the arch length. If the radius is greater the arch lech is the angle in radians multiplied with the radius. Lost of mathematical function works a lot better with radians
Euler’s formula e^(ix) = cos x + i sin x is an example with x in radian, you would need pi/180 constant for it to work with degrees or pi/200 for gons. This is just one example where radians is what is natural.
For practical usage, Milliradians (mrad) are common which is 1/1000 of a radian. that is 6283.185 milliradians in one turn
For military usage, if an object takes up an angle of 1mrad and is 1 meter wide you know it is 1000 meter away. There is often curves in scopes with the height of what they are used against like tanks, humans etc so you can determine the distance from the size, radiation in mrad is also common.
To make it more practical there 6283.185 is not used because it is not a target and you do not get easy cardinal directions, So NATO uses 6400 mils in a circle, The Sovest and the former eastern block use 6000 mils, Sweden dud use 6300 but hav change to 6400 in the 2000s.
You can still say a 1-meter object that is 1 mil wide is 1000 meters away. The error is determined by the angle in just a scope that introduces more error than the change in the number of milliradians. In a more accurate system, you do the maths correctly but then it is not a calculation you do in your head.
So France tried with decimal time and angles but it did not stick. For angles, a major reason is that it is radians that is the more natural unit of 2 pi for a circle, not any number that is a multiple of 10 or any other integer
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