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Circumference of a Circle

The circumference of a circle is pi times the diameter (the diameter is double the radius).

Let r be the radius of the circle and d be the diameter of the circle.
Then we have: $ Circumference(Circle) = 2\pi r = \pi d $
We can evaluate circle length using integral calculus.F0r $ x(t)=r\cos t,\:y(t)=r\sin t $
Circle length=

$$\int_Cds=\int_{0}^{2\pi}\sqrt {x'^2(t)+y'^2(t)}dt=\int_{0}^{2\pi}\sqrt{r^2\sin^2t+r^2\cos ^2t}dt=r\int_{0}^{2\pi}dt=2\pi r$$

In this figure we have BC = d and AB = r.

$ Circumference(Circle) = 2\pi AB = \pi BC $

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