$\pi$ is just a number, or also the circumference of a sub-unit circle?

Question

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?

Answer 1

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 $0.5$.

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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 $$\frac{c}{d}=\frac{2\pi}{2}=\pi.$$

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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 $$\frac{c}{d}=\frac{\pi}{1}=\pi.$$

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