The reason why 360 degrees are in a circle is likely because 360 days was used in a year by ancient people, and it was convenient because of factors of 60 and ancient finger counting. I believe they were accurate and the number days in a years has somehow changed, but that is not commonly accepted. Anyway they used to use finger counting to count to 60 with both hands as you can see in the video using 5 fingers and 12 joints on the other hand. The factors are 1,2,3,4, and 5 which makes common division by 3 and 4 easier. Finally 60 times 6 equals 360 which matches 360 degrees and 360 days which as you can see is very convenient with base 60 mathematics(also known as sexagesimal).

(Bonus hexadecimal finger counting in the picture)

How to count hexadecimal from 00 to FF with the 8 fingers and 2 thumbs. (Photo credit: Wikipedia)