DIR Return Create A Forum - Home
       ---------------------------------------------------------
       gworld
  HTML https://gworld.createaforum.com
       ---------------------------------------------------------
       *****************************************************
   DIR Return to: MATH FACTS
       *****************************************************
       #Post#: 106--------------------------------------------------
       SURFACE AREA OF A SPHERE
       By: eba95 Date: July 30, 2010, 6:28 am
       ---------------------------------------------------------
       Surface Area of a Sphere
       The area of a disk enclosed by a
       circle of radius R is Pi*R2.
       The formula for the circumference of
       a circle of radius R is 2*Pi*R.
       A simple calculus check reveals that
       the latter is the derivative of the
       former with respect to R.
       Similarly, the volume of a ball
       enclosed by a sphere of radius R is
       (4/3)*Pi*R3.
       And the formula for the surface area
       of a sphere of radius R is 4*Pi*R2.
       And, you can check that the latter is
       the derivative of the former with
       respect to R.
       Coincidence, or is there a reason?
       Presentation Suggestions:
       Let your students tell you those
       geometry formulas if they
       remember them.
       The Math Behind the Fact:
       Well, no, it is not a coincidence. For
       the ball, a small change in radius
       produces a change in volume of the
       ball which is equal to the volume of
       a spherical shell of radius R and
       thickness (delta R). The spherical
       shell's volume is thus approximately
       (surface area of the sphere)*(delta
       R). But the derivative is
       approximately the change in ball
       volume divided by (delta R), which is
       thus just (surface area of the sphere).
       So, if I tell you the 4-dimensional
       "volume" of the 4-dimensional ball is
       (1/2)*Pi2*R4, what is 3-dimensional
       volume of its boundary?
       *****************************************************