[almost]

360 degrees in a circle predates Plato by quite a lot (2000 years I think!). It comes from the Summarians more than 4000 years ago. They used a method of counting on fingers that goes up to 12 on one hand and 60 using both hands, so their numbering system was based on 60. 360 is 6 * 60 and also roughly how many days in a year.

Later societies inherited that from them along with 60 minutes in and hour.

[amelius]

But why wasn't the handcounting inherited too?

It sounds useful to be able to count up until 60 on two hands.

[balamatom]

Probably because it was too contrived. I mean, if you can count up to 12 on one, why can't you do up to 144 on both?

[amelius]

Could have something to do with right-handedness.