A clasic problem - one which I've always known as "the goat problem" (not cow), and it was a goat on the outside of a circular corral, not silo, but who cares.

You break up the grazing area into 2 parts - call part (A) the area that is covered when the rope is wound part way around the outside of the silo (there are two area A's), and part (b) for the semi-circle of area where the cow's leash is not constrained by the silo. Part B is easy - the area of the semicircle is simply . Part (A) is the hard part - and yes, use polar coordinates. Set at the point of attachment of rope to silo. When the cow's rope is stretched around the silo to angle , the amount of rope that is wound against the silo is . For each that the rope is wound around the silo, the area that the cow has to graze is:

So the total area to graze is:

where is the max angle that the rope can strech around the silo:

Can you take it from here?