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

Note that, by definition, $$ ext{diameter}=2cdot ext{radius},$$so that$$pi=frac{ ext{circumference}}{ ext{diameter}}=frac{ ext{circumference}}{2cdot ext{radius}}=frac{1}{2}cdotleft(frac{ ext{circumference}}{ ext{radius}}

ight),$$or in other words,$$frac{ ext{circumference}}{ ext{radius}}=2pi.$$

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

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If $dfrac{

adechworld.comrm {circumference}}{

adechworld.comrm {diameter}}$ is the same for all circles, does the surface have to be flat?

Why do we say pi is the ratio of the circumference to the diameter, and not diameter to the circumference?

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