UnCommented

 So anyone can understand:

axiom: in one revolution the center travels one circumference. (no slipping, of course) So, calculate the path the center of the circle travels. answer: as many rotations as necessary to equal the path travelled by the center of the disc. in the example r1=1; r2=3: distance center travels: 2π(r1+r2) => 2π(1+3) = 8π circumference of rotating circle: 2π 2π * x = 8π => x = 4 __________________________ OR - see @user-rx4wo7il2g comment below - if we prefer to focus on the path to be travelled instead of what path the center must travel: put on your Calculus glasses and see that even if the circle is rotating around an infinity small path (r~0, path ~0) it must rotate once! So you always need to start with 1; we must add 1 to the path to be travelled! @GPCTM