It seems like an odd number especially fir the metric system.
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Actually has it’s roots in ancient times
The Babylonian usage of 60 as their number base, they decided that each of the angles of an equilateral triangle would be 60 degrees.. 6×60= 360
60 was just a neat number to work with be it for commerce and trade even in the Mesopotamian days.
The Egyptians also rolled with it for the same reasons and kept going due to their obsession of perfect triangles and 6 of those equal ones fit into a circle.
Since most of our golden age of early science came from this rough middle east direction from istanbul to egypt.. and a lot of influence on occasion coming over from asia.. that’s where we are now.
Its just defined as that, there is no mathematical trick to it. And there is other units like radian that do not work on the 360 base.
And 360 is a nice number to divide, its divisible by 2, 3, 4 ,5 ,6 and much more. So like the clock thats in some multiple of base 6 too, we addopted this system from the mayans i think.
It’s arbitrary, it’s just how we defined degrees. We could have 100-degree circles, but that would be kind of annoying because it wouldn’t evenly divide into thirds or sixths like 360 does. 360 divides evenly into 2, 3, 4, 5, 6, 8, 9, 10, 12, making it very convenient for a lot of calculations.
This is, incidentally, the same reason why we have 12 hour clocks instead of 10 hour clocks (and both have a related historical origin.)
We use degrees in the same way America uses Imperial units – we inherited it, some people find it easier not to change, etc., but it’s far from the only angle measurement in town. Radians are arguably “better” in that they’re inherent in geometry thanks to Pi, but at the end of the day it’s a scaling factor for the same thing (like metres and yards are)
The number itself goes back to ancient times, like Babylon etc.
The rational is relatively simple though.
Imagine you have 10 coins and want to split them up equally among a group of people.
If you have 2, 5 or 10 people that works easily, but if you have 3, 4, 6, 7, 8, 9 or more than 10 people it won’t work.
Now imagine you have 12 coins instead. You can split them up equally among 2, 3, 4, 6 and 12 people but get problem with 7, 8, 9, 10, 11 and 13 and more people.
This is why old units to measure things were often split up into 12 sub-units.
Nowadays we just use fractions and decimals and split every unit up into sub-units that are powers of 10 and even the British have given up on their weird non-decimal currency a while ago, but for people in older times were math was not yet as much of a thing and the common people weren’t well educated choosing some units that were easily divisible was an important thing.
Having something big that was divisible into 12, 24, 30, 60 or 360 units was beneficial.
it also helped that the universe itself seemed to be organized along similar line with a year having close to 360 days and a month having almost 30 days, it didn’t fir quite, but it was close enough.
So for time keeping and geometry which were closely linked through astronomy something with 12 as the basis made the most sense.
A circle might needed to be divided into quite fine sub-units.
360 was a good fit for that.
You could do half, a thirds, quarters, fifths, sixths, eights, ninths, tenths, twelfths, fifteenths, twenty-fourths, thirtieth, forty-fifths, sixtieths, ninetieths and one-hundred-eightieths.
100 you can only divide by: 1, 2, 4, 5, 10, 20, 25, 50
If you have an equilateral triangle and aren’t able to divided a circle into three equal parts that would be a problem.
In modern times we have subdivided a degree of a circle into minutes and seconds the same as we did hours. 60 minutes to a degree and 60 seconds to a minute.
60 is another of those numbers that are easily divisible by a number of small numbers.
We have also come up with other ways to divide a circle like gradians which have 100 to a right angle and radians which have half pi to right angle, but they are not used as much in every day life.
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
There is a metric system of angles. Gradians. 100 gradians in a quarter circle (so 400 in a circle). They never caught on although there are niche uses.
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