A (rewritten!) Prime Product

Calculus Level pending

Suraj is interested in how the dimensions of a rectangle affect its area. He sets up a ratio of the area of a square to the area of a rectangle of equal perimeter (\(\dfrac { { A }_{ square} }{ { A }_{ rectangle } } \)).The sides of the rectangle are modeled by \(s-a\) and \(s+a\), while the side length of the square is \(s\). Suraj's friend tells him that the product of all of his ratios in which \(s\) is a prime number greater than two and \(a=1\) can be determined. He stated that the product would be equal to the area of a particular circle multiplied by \(\pi \). If the radius of the circle is \(r\), then what is the value of \(\left\lceil 1000r \right\rceil \)?

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