Taking the formula for the sum of internal angles, ∑(aº) = 180(n-2) with aº being the degree of one internal angle, and n being the number of sides of the polygon, you can then say that since a circle is either an infinity-sided polygon, or a 0 sided polygon, then the sum of the angles is either ∞º or, -360º. Since -360º results in the same rotation as 360º it is therefore not completely ridiculous to say, that a circle's internal angles add up to 360º.

^{That last part isn't true, I'm doing a bit, the formula isn't meant to be used for shapes below 3 sides.}