Involute of a circle by a sheep

Calculus Level 5

A sheep is tied in a silo of radius \(r\), by a rope long enough to reach the opposite end of the silo(see figure above). The total area available for the sheep to graze can be written as,

\[\large\frac{p\pi^{3}r^{2}}{q},\]

where \(p\) and \(q\) are co-prime positive integers. What is the value of \(p+q\)?

Note: The involute (of a circle), is the locus of points traced out by a taut string.
I found this problem in a book, and i thought i would share it.

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