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This essential trains for: AMC-10, AMC-12, AIME, Math Kangaroo 9-10.

Question: Suppose we have a circle of radius r that rolls along the circumference of another circle, of radius R. How can we calculate the angle of rotation of the rolling circle around its center if we know the angle of rotation around the center of the other circle?

This question can be visualized in the figure below:

The circle with radius r is positioned at the start with its center at point O and is tangent at A to the circle with radius R.

We have emphasized the position of the segment OA at the start of the rolling process. Assume that, at the end of rolling the position of the same segment is O'A'.

If O'C is the position of OA' at the start (parallel to OA), then we need the measure of the angle CO'A'.

Let us denote the angle CO'A' with x.

We also assume all the angles are measured in radians so as to use simpler formulas for the length of the arc.

Because O'C and OA are parallel, the angles below are congruent as alternate internal angles. Let us denote their measure with α:

The measure of the angle BO'A' can now be expressed as the difference between x and α:

The arc BA' must have the same length as the arc BA since they both represent the length of the path on which the two circles have been in contact (tangent):

Using the formulas for the length of the arc:

Now we only have to solve for x: