A unit circle defined in the Cartesian plane has a radius of $1$ and a diameter of $2$. So making a full round is $2 \pi$. Now, $\pi$ is the ratio of the circumference over the diameter, so if I have a circle with diameter $1$ (radius $0\mathord{,}5$), the circumference would be $\pi$ because its $c/d = c/1$ must equal $\pi$. Hence $c= \pi$. So $\pi$ is actually the circumference of a "subunit" circle? or I am missing something here?
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1$\begingroup$ Yes, $\pi$ is the circumference of a circle of diameter $1$. $\endgroup$– André NicolasMay 17, 2013 at 22:47
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I think you're confusing yourself by talking about two different circles, but you are correct that $\pi$ is the circumference of a circle of radius $0.5$.
For a circle of radius $1$, we have $$r\text{ (radius)}=1\qquad d\text{ (diameter)}=2\qquad c\text{ (circumference)}=2\pi$$ The ratio of the circumference to the diameter is $$\frac{c}{d}=\frac{2\pi}{2}=\pi.$$
For a circle of radius $0.5$, we have $$r\text{ (radius)}=0.5\qquad d\text{ (diameter)}=1\qquad c\text{ (circumference)}=\pi$$ The ratio of the circumference to the diameter is $$\frac{c}{d}=\frac{\pi}{1}=\pi.$$
answered May 17, 2013 at 22:43
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$\begingroup$ why? If I have a circle with diameter 1 then the circonference should equal π ? $\endgroup$– themhzMay 17, 2013 at 22:45
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$\begingroup$ So its true, the circumference is equal to π of a circle with 0.5 radius. Weard thing is that π never ends and it is an irational number but if I unwrap the circumference to a line it does end :D. Anyway thanx, I needed some confirmation on this $\endgroup$– themhzMay 17, 2013 at 22:52
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1$\begingroup$ The decimal expansion of $\pi$ does not end, and $\pi$ is an irrational number. Just because a line segment of length $\pi$ "comes to an end" doesn't imply anything about writing the number down as a decimal. $\endgroup$ May 17, 2013 at 22:53
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1$\begingroup$ @themhz: (con't) On the other hand, trying to measure the circle's diameter with a ruler marked at circumference-length units, the diameter's end falls somewhere between "0" and "1". Sub-dividing the new ruler at every 1/10 of a (circumference) unit, the diameter ends between "0.3" and "0.4", and so on and so forth ... A circumference-based ruler (with equally-spaced sub-divisions) can never precisely measure the diameter, just as a diameter-based ruler (with equally-spaced sub-divisions) can never precisely measure the circumference. The lengths are "incommensurate". (Look it up. :) $\endgroup$– BlueMay 17, 2013 at 23:50