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Here is a specific example of what I am asking:

A dog is tied to a fixed point on the outside of a circular silo with a radius of 20 feet. The rope is 50 feet long. How much total area does the dog have to roam around in?

I know that the dog has a semi-circle of radius 50 ft to walk around in front of the silo. When he goes toward the back of the silo, the rope winds around the silo and gets shorter and shorter. This winding starts when the rope is tangent to the circular silo. How do you compute the area he has to walk in as the rope is winding around and getting shorter?

I am more interested in the method you can use to solve this problem than in the exact answer.