What is the distance travelled by the circle? A coin of radius 2 is touching a coin of radius 9. The point of tangency is marked as A on the smaller coin and B on the larger coin. The smaller coin is then rotated around the larger coin, without slipping, until the two coins come into contact again at points A and B. Find the distance traveled by the center of the smaller circle. I have a few questions- won't A and B be same? Also, won't they meet after one round?
 A: I'm going to have these labels
Circle 1 = smaller circle
Circle 2 = larger circle
The circumference of the circle 1 is 2pi*r = 2pi*2 = 4pi
Point A on circle 1 is touching point B which is on circle 2. This is the initial state. When circle 1 rolls around circle 2, point A will travel 4pi units along circle 2. When point A gets back to touching circle 2, it will be 4pi units away from point B if you measure just along the circle 2's edge.
Let's figure out the central angle of AOB where O is the center of circle 2 and A is at the new location (4pi units away from B along circle 2's edge)
S = (theta/360)2pir
4pi = (theta/360)*2*pi*9
4pi = (theta/360)*18pi
4pi = pi*theta/20
4 = theta/20
theta = 80
see this image: http://imgur.com/E1tAp8y
This means the smaller coin has traveled 80 degrees, basically 80/360 = 2/9 of the entire larger circle. Notice how 360/80 = 4.5 is not a whole number but 720/80 = 9 is a whole number. This tells us that we need to have the smaller coin do 2 complete revolutions around the larger coin
So in reality, the center of the smaller circle has traveled 2*(22pi) = 44pi units (look at randomgirl's post)
A: Draw both circles. Draw a line from one center of a circle to the other center of the other circle. Then realize that the center of the smaller circle will be traveling on another circle around the circle with radius 9. Find the radius of this newer circle. Then find the circumference. I have a drawing but don't know how to attach it so maybe you can draw it yourself from what I'm saying. 

(There's an "image" button above the edit box...)
A: The marked points will meet again once the small coin has gone around the large coin twice.
Imagine that the two coins are instead cogs, and the width of a tooth is $\frac{\pi}{5}$. Then the small cog has $20$ teeth and the large cog has $90$ teeth. So the marked tooth on the small cog touches the large cog at tooth $\#20, 40, \ldots$  From there, it is easy to see that the marked teeth won't match up on the first time round - when the small cog is touching the large cog's mark again for the first time, the mark on the small cog is on the opposite side.
The centre of the small coin is moving along a path that is$11$ units from the centre of the large coin. so each time around the large coin takes it a distance of $2\pi \times 11$ units, and twice around means that the total distance travelled is $44\pi \approx 138.23$ units.
