Let r = the radius of the small semicircle.



For the right triangle:

The red hypotenuse = r+1.
The green right verticle leg = 1.
So, by the Pythagorean theorem, 
the bottom horizontal leg = 

We add the two parts of the left horizontal radius of the
large semicircle, and get   This 
sum must be equal to the right radius of the large 
semicircle, which is 2.  So we have:









Square both sides:





Subtract r2 from both sides:







cm.    <---answer

Edwin