Hello 9-05 and 9-06! So Mr. B assigned 9-05 (because we're going to be the class) to make a post explaining to 9-06 why we use 360 all the time. I hope this short and concise post will be helpful to you!
First of all, 360 is a very round and friendly number to use. It's divisible by a lot of numbers (24 to be exact). Secondly, the Babylonians were devoted to geometric figures, and they based their number system on the number 60 rather than 10 like we usually do. This number system is called the Sexogonal System. Their calendar even had 360 days and their complete cycle was divided up into 360 units. We have adapted from the Babylonians because our clocks have 60 minutes and each minute is 60 seconds.
Here is an example of the Babylonian's number system.
The Babylonians thought that there were approximately 360 days in a calendar, since the Earth rotated about 1 degree around the Sun per day. So, when the Earth makes a full rotation around the Sun that would be a full circle. That's why the Babylonians used 360 to represent a full circle. Our place value is ones (1) tens (10) hundreds (100) thousands (1000) etc. because our number system is based on 10. As mentioned previously, Babylonians used the Sexagismal system. In the chart above, you can see all the symbols for each number character. I think that the 59 symbol and the 1 symbol would be the representative for the number 60, or perhaps the 50 symbol and the 10 symbol would be the representative for 60. I am not sure, because of the sites I've encountered, they were unclear about this. I think that the next place value would be something like 120 and it could be a combination of two 50 symbols and two 10 symbols or a 20 symbol, which is actually two 10 symbols. I think that an easier way to write these numerals using our number system would be 50 and 10 for 60. 50 and 50 for 100, etc.
So that is basically why we use 360̊̊ to describe a full circle. If I've missed anything, feel free to comment!