This is also an interesting problem, like with a circle rotating on the exterior of another circle, the answer is not simply R/r, that is the ratio of the larger radius to the smaller radius, but is actually R/r - 1. By rolling along the interior of a circle, one revolution is lost. An intuitive way of imagining why this happens is by first approximating a circle with a regular polygon, say a hexagon. Imagine rolling a circle around the interior of the hexagon, while the circle rolls along each side of the hexagon, it does not get to roll over the complete length of each side, due to the convexity of the interior of the hexagon, namely, a circle will have completed rotation along one side when it is tangent to two sides of the hexagon, it is clear simply by drawing a picture that the circle does not need to be displaced the entire side length of each side of the hexagon for this to happen, and so some rotation is essentially *lost* because to this.

This is also an interesting problem, like with a circle rotating on the exterior of another circle, the answer is not simply R/r, that is the ratio of the larger radius to the smaller radius, but is actually R/r - 1. By rolling along the interior of a circle, one revolution is lost. An intuitive way of imagining why this happens is by first approximating a circle with a regular polygon, say a hexagon. Imagine rolling a circle around the interior of the hexagon, while the circle rolls along each side of the hexagon, it does not get to roll over the complete length of each side, due to the convexity of the interior of the hexagon, namely, a circle will have completed rotation along one side when it is tangent to two sides of the hexagon, it is clear simply by drawing a picture that the circle does not need to be displaced the entire side length of each side of the hexagon for this to happen, and so some rotation is essentially *lost* because to this.