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This may seem simple to a lot of you out there, but I was wondering if there is a way to find the constant, a, in the Archimedian spiral equation:

$$ r=a\theta $$

Specifically, I need to be able to fit the parameter by means of measuring the distance between curves on the spiral. I tried setting this measurable distance to the length (in polar coordinates), but when I use the same first and second angles, my distance is 0. Here is an example where D=32: $$ D= 32= \sqrt{(r_{1}^2+r_{2}^2-2r_{1}r_{2}cos(\theta_{2}-\theta_{1})} $$ I should be able to plug in the expression for r above. Since I measure along the x-axis (same angles) I get D=0. Can someone tell me what I am missing and/or a better method for finding a in r=a*theta.

Thanks!

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    You're on the $x$-axis when $\theta=0$ and again when $\theta=2\pi$. The corresponding $r$-values are $0$ and $2a\pi$. So, the distance is $2a\pi$. If you want the distance to be $32$, you solve $32=2a\pi$ for $a$. Right? – 2017-01-19
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    I had a feeling this solution would be way simpler than I was thinking. Thanks so much! – 2017-01-19
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    You're welcome. Let me encourage you to write up and post an answer, now that you see what's up. Then you can "accept" your answer. – 2017-01-19

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Simply was not considering the fact that I could ignore the change in theta when working along the x-axis. Could simply evaluate r(0) and r(2pi) and use the difference.