The moving circle’s own circumference is C.
Its center travels on a larger circle whose radius is twice as big
(distance from fixed center to moving center = 2r ).
That path therefore has circumference 2C .
Because the moving circle is rolling without slipping, the arc length it travels along its path must equal the arc length it rotates on its own circumference. So it rotates a total of 2C / C = 2 times.
