I am trying the find the distance between two different sized circles, both centred on the horizontal plane. I know the diameter of each circle, and the length around both circles if wrapped like a bike chain around the front and rear sprockets.

I've tried searching for the formula's but everything seems to work with a known distance to find the external tangent length's etc.

What are the steps to calculate this?

Thanks,
Matt

Update:
Found a couple of formulas from timing pulley sites that seem to do the job. Would still like to understand how they get built just out of interest.

**Formula #1**

$C = A + \sqrt{ A^2 + B }$

where

$A = \frac L4 - \pi \frac{D + d}{8}$

$B = \frac{\left(D - d\right)^2}{8}$

---

**Formula #2**

$C = \frac{A + \sqrt{A - 32 \left(D - d\right)^2} }{16}$

where

$A = 4L - 2\pi \left(D + d\right)$

Both come out with the same answer. Which isn't really a surprise as they look very similar.