Why is a circle 360 degrees? Here are some answers from the 
Dr. Math archives:

   Why is a circle 360 degrees?
   What is the origin/basis of the degree measurement? Why is a circle divided 
   into 360 degrees rather than some other number? 

   Why Is a Circle 360 Degrees?
   Why is a circle defined as 360 degrees?

   Degrees in a Circle
   Why does a circle measure 360 degrees - is there a special reason for this 
   calculation?

   Origin of the 360-Degree Circle
   Did the Babylonians discover the 360-degree circle?