How can I prove that in a circle $\frac{circumference}{radius}$ is constant ? I know that $\pi$ is defined like that (because it's indeed constant). But how to prove that it's indeed constant ?
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This is overkill, but you could argue that the radius-$R$ circle centred in $\Bbb R^2$ at the origin has first quadrant $y=\sqrt{R^2-x^2},\,x\in[0,\,R]$ so satisfies $$dy=\frac{-xdx}{\sqrt{R^2-x^2}}\implies ds=\sqrt{1+\frac{x^2}{R^2-x^2}}dx=\frac{Rdx}{\sqrt{R^2-x^2}}.$$Thus the circumference is $4\int_0^1\frac{Rdx}{\sqrt{R^2-x^2}}=4R\int_0^1\frac{dt}{\sqrt{1-t^2}}$.
J.G.
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