I don't think that argument holds water. Imagine rolling it on the inside of the circle – the center of the circle traces a path with only twice its radius, yet it still rolls 4 times around.
I don’t think that’s right. It if the radius of the inner circle is one third the outer, it only rotates twice rolling around the inside, which makes sense as it’s center traces a circle that’s the the difference of the radii.
Imagine the limiting case, as the inner circle approaches the size of the outer circle - the inner circle completes much less than one rotation per lap around the inside edge of the outer circle, and ‘seizes’ (if we’re imagining these as gears), completing zero rotations per lap when the circles are the same size. However, rolling around the outside, a circle of the same size completes two rotations.
In general the problem is like the old Spirograph toy (which I had to break out to convince myself)
If circle A is rolling around the edge of circle B from within, it is actually revolving around a new, smaller circle C which has the Radius Circle B - Circle A.
Is this correct? wouldn't the radius of the circle it's revolving around be significantly smaller if you had it follow the edge of the larger circle but from the inside? I don't know much about math or physics, so I could be wrong, but I think it would be significantly less, closer to two, right?
The reason that this problem is tricky, and has a counterintuitive solution is that Circle A is rolling around Circle B, and so the 'radius; of the circular path it is following is the radius of Circle B + the radius of Circle A.
I'm no expert, but some quick math:
Circle B has a radius of 9, circumference then is 56.55
Circle A (1.3) has a radius of 3, circumference is 18.84
56.55 / 18.84 = 3. This suggests that you could "unwind" circle A (say it was made of pipecleaner), and you would need 3 circle As to fully ensconce Circle B
BUT that wasn't the question. The question states that Circle A is rolling AROUND circle B.
So the circumference of the circle we are now trying to 'ensconce' is Circle A Radius + Circle B Radius = 12, so the new circumference is 75.39, and divided by Circle A Circumference, we end up with 4, which makes sense, and matches the demonstration from the video.
HOWEVER, if we are 'rolling' around the inside of circle B, then I think you're right, the radius of the circular path Circle A will take is 9 -3 = 6, and the circumference of said circle is 38, so we it only will take two rolls. I think this is correct, i do not think it will be 4 rolls.
Think of it this way: when Circle A is rolling around the inside of Circle B, the way you're picturing it is circle A following the outside of Circle B, which is why I think intuitively it feels like the answer will still be 4 revolutions. BUT a better way to think of it is that Circle A is actually revolving around a new Circle, Circle C, which has a circumference of 38. Circle A is not revolving around Circle B, it's revolving around this new, smaller circle within circle b. Does that make sense?
I made a quick diagram to illustrate my point. I did it in a wire-framing tool that has snap to grid, so it's definitely not perfect:
