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Preface: The Windows calculator has an "Inverse" button, which transforms $\ln$ into $\exp$, for example. Oddly enough, it also transforms $\pi$ into $2\pi$, which is nonsensical, since that would imply that $\pi = \sqrt{\frac{1}{2}}$, which leads to the following

Question: Is it possible to have a non-euclidean geometry where the ratio between a circle's circumference and its diameter is equal to $\sqrt{\frac{1}{2}}$ and independent of the circumference? I've read that in general, for a Non-Euclidean geometry this value is not well-defined (in the sense that its not independent of the size of the circle). Are there actually non-euclidean geometries where this ratio is well-defined?

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