How relate the length of a conveyor belt between two interlocked circles? I'm trying to solve this problem: > Two wheels are one next to another and linked by a conveyor belt. How to express the length of the radius of the smaller wheel in terms of the length of the belt? Note that the size of the radius of A is three times the radius of B. ![Sketch of the problem \(not in scale\)](

Let the length of the belt be $b$.

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\begin{align} d&=\sqrt{(4r)^2-(2r)^2} =2\sqrt{3}r ,\\\ \phi&=\arccos\tfrac{2r}{4r}=\tfrac\pi3 ,\\\ b&=2\,((\pi-\phi)\cdot3r+d+\phi\,r) \\\ &=2(2\pi r+2\sqrt{3}r+\tfrac\pi3 r) \\\ &=(\tfrac{14}3 \pi +4\sqrt{3})r ,\\\ r&=\frac{b}{\tfrac{14}3 \pi +4\sqrt{3}} . \end{align}