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**Zyzz** Circumference and Area of circle

Why is it that the circumference of the circle is given by 2rPi which is the derivative of r^2Pi by r, which gives the area of it. Same if we transfer it to cartesian coordinates, y= Sqrt(r^2-x^2) the arc length is given by r(arcsin(x/r)) which is the derivative of r(arcsin(x/r) + r/2(Sqrt(r^2-x^2))) which on the other hand gives the area of the quarter circle with radius r. The same applies to the sphere. Can this be extended to the n-dimensional circle? (sphere: V=4/3Pir^3 SA= 4Pir^2)

its just something interesting i found today and was wondering abt the explanation behind this. this.