2021 Fall AMC 10A Problems/Problem 19

$19$ . A disk of radius $1$ rolls all the way around in the inside of a square of side length $s>4$ and sweeps out a region of area $A$ . A second disk of radius $1$ rolls all the way around the outside of the same square and sweeps out a region of area $2A$ . The value of $s$ can be written as $a + \dfrac{b\pi}{c}$ , where $a,b,$ and $c$ are positive integers and $b$ and $c$ are relatively prime. What is $a+b+c?$

$\textbf{(A) }10\qquad\textbf{(B) }11\qquad\textbf{(C) }12\qquad\textbf{(D) }13\qquad\textbf{(E) }14$

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2021 Fall AMC 10A Problems/Problem 19