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Math Help - Area and Circumference of the circle relation

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    Area and Circumference of the circle relation

    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.
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    Re: Area and Circumference of the circle relation

    Quote Originally Posted by Zyzz View Post
    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.
    To put it in a very informal, non-rigorous way, suppose that you have a circle with radius r, area A and circumference C. Now make it a bit bigger, increasing the radius by an "infinitesimal" amount dr. The added amount of area, dA, consists of a thin strip of length C and width dr. So its area is Cdr. Therefore dA = Cdr, or \tfrac{dA}{dr} = C. In other words, the circumference is the derivative of the area.

    That sort of argument would not be allowable in mathematical circles these days, but it would probably have been acceptable to Newton. You may find that it gives an intuitive feel for why that result should be true.
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    Re: Area and Circumference of the circle relation

    Quote Originally Posted by Zyzz View Post
    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.
    A similar way to look at it is...

    The area of a circle is the integral of the circumferences of circles
    with radii from 0 to R.

    A=\int_{0}^R 2 \pi r dr
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