Finding volume of a sphere

From: allenroy (allenroy@peoplepc.com)
Date: Fri Oct 03 2003 - 16:56:02 EDT

  • Next message: Ted Davis: "Fwd: Life: Cosmic Accident or Cosmic Destiny, Cambridge, MA, October 10, 2003"

    > The other night I watched the NOVA Program about the discovery of old
    > manuscripts of Archimedes, which got me to thinking about how to develop the
    > formula the volume of a sphere without looking it up in my Engineering
    > Handbook or calculus text book.
    >
    > First we start with a square and a circle
    >
    > Square:
    > Let S=Side, P=Perimiter and A=Area
    > And P=4S and A=S^2
    > Ratios:
    > P/S = 4
    > A/S = 4*S/4 = S
    > A/P = S/4
    >
    > Circle:
    > Let r=radius, D=Diameter, C=Circumference, and A=Area
    > And C=pi*D and D = 2r
    > Ratios:
    > C/D = pi
    > A/D = ?
    > A/C = ?
    >
    > Let us set the ratio of A/C for the circle to be the same as A/P for the
    > square.
    > A/C = A/P = D/4 (S = D)
    > Now we can find A/D = A/C * C/D = D/4 * pi
    > Then A = pi*D/4*D = (pi*D^2)/4 which, of course, reduces to pi*r^2
    >
    > Note the similarities in ratios for the 2D objects
    > P/S = 4 C/D = pi

      A/S = 4*S/4 = S A/D = pi*D/4
      A/P = S/4 A/C = D/4
    > Thus it is apparent the A/C is indeed equal to A/P.
    >
    > Lets move on the the 3D objects: Cube and Sphere
    >
    > Cube:
    > Let S=Side, A=Surface Area and V=Volume
    > And A=6S^2 and V=S^3
    > Ratios:
    > A/S = 6S^2/S = 6S
    > V/S = S^3/S = S^2
    > V/A = S^3/6S^2 = S/6
    >
    > Note the similarities between the cube and the square
    > Square Cube

      P/S = 4 A/S = 6S
      A/S = 4*S/4 = S V/S = 6S*S/6 = S^2
      A/P = S/4 V/A = S/6
    > In rows 1 and 2, the extra "S" is there because the Cube is a 3D object.
    >
    > Sphere:
    > Let r=radius, D=Diameter, A=Surface Area and V=Volume
    > And D=2r
    > Ratios:
    > A/D = ?
    > V/D = ?
    > V/A = D/6 (from Cube)
    >
    > To find A/D we consider the following relationships 4:6S as pi:? and 4:pi as
    > 6S:?
    > Square/Circle P/S = 4 C/D = pi

      Cube/Sphere A/S = 6S A/D = ?
    > It seems that A/D should be a combination of "pi" from the circle and "S" (=D)
    > from the Cube, i.e. pi*D
    > Square/Circle P/S = 4 C/D = pi

      Cube/Sphere A/S = 6S A/D = pi*D
    >
    > Now we can find V/D:
    > V/D = V/A * A/D = D/6 * pi*D = (pi*D^2)/6
    >
    > A/S = 6S A/D = pi*D

      V/S = 6S*S/6 = S^2 V/D = pi*D*D/6 = (pi*D^2)/6
      V/A = S/6 V/A = D/6
    >
    > To find volume of sphere: V = D * (pi*D^2)/6 = (pi*D^3)/6 Which, of course,
    > reduces to (pi*(2r)^3)/6 = 8pi*r^3/6 = 4/3 * pi * r^3.
    > Thus we have found the volume of a sphere -- 4/3 * pi * r^3.
    >
    > Allen

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