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I know that the ratio of the circumference to the diameter is Pi - what about the ratio of the circumference to the radius? Does it have any practical purpose when we have Pi? Is it called something (other than 2 Pi)?

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    $\begingroup$ What does "does it mean anything" mean? Does 17 mean anything? Please clarify your question. $\endgroup$ Aug 22, 2011 at 2:25
  • $\begingroup$ "I know that the ratio of the circumference to the diameter is Pi" is incorrect, should also specify in Euclidean geometry to that sentence, try finding the ratio of circumference to the diameter on a sphere just for fun. $\endgroup$
    – jimjim
    Aug 22, 2011 at 8:15
  • $\begingroup$ @Gerry Good point. I've removed that part. $\endgroup$
    – Odinulf
    Aug 22, 2011 at 13:42

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The ratio of the circumference to the radius is $2\pi$, which some people call "One turn". I think you would enjoy to read this article: "$\pi$ is wrong!" by Bob Palais. Other people call $2\pi$ by the name of Tau. See this page: http://tauday.com

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    $\begingroup$ Thanks - those websites were very interesting. $\endgroup$
    – Odinulf
    Aug 22, 2011 at 14:03
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Note that, by definition, $$\text{diameter}=2\cdot\text{radius},$$ so that $$\pi=\frac{\text{circumference}}{\text{diameter}}=\frac{\text{circumference}}{2\cdot\text{radius}}=\frac{1}{2}\cdot\left(\frac{\text{circumference}}{\text{radius}}\right),$$ or in other words, $$\frac{\text{circumference}}{\text{radius}}=2\pi.$$

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That $\pi$ and $2 \pi$ have a very simple relationship to each other sharply limits the extent to which one can be more useful or more fundamental than the other.

However, there are probably more formulas that are simpler when expressed using $2\pi$ instead of $\pi$, than the other way around. For example, there is often an algebraic expression involving something proportional to $(2\pi)^n$ and if expressed using powers of $\pi$ this would introduce factors of $2^n$.

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