Few questions about surface area and volume

  1. When calculating the volume of a sphere, what does (4/3) represent? Why is it (4/3) * pi * r^3 .. and not some other number/fraction?

    I'm also curious about the surface area of equilateral triangle. Why is it sqrt(3)/4 * a^2 ... What does sqrt(3)/4 physically represent in the geometry?
  2. jcsd
  3. arildno

    arildno 12,015
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    Well, if you rewrite the volume for the ball as [tex]V=\frac{1}{3}*4\pi{r}^{3}[/tex], recognize that this can be further simplified as:
    [tex]V=\frac{1}{3}*r*S[/tex] where S is the surface area of the sphere.

    Thus, the volume of the ball is equal to the volume of a cone of height "r" and base area S.

    This is the gist result of how Archimedes proved the formula.
  4. Thanks. Here's what I just came up with for a possible physical (?) representation..

    since, pi is the same as (2*pi*r)/(2*r)

    V = ((4) * (2*pi*r) * (r^3)) / ((3) * (2*r))


    V = (8 * pi * r^4) / (6 * r)

    V = (4 * circumference) / (6 * radius)

    Would the numerator represent 4 dimensions? Seems weird.
  5. HallsofIvy

    HallsofIvy 40,241
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    Since you are working in 3 dimensions, I doubt that! And 2 pi r^4 is the circumference of what?
  6. I meant,

    V = (4/3 * pi * r^3) = (8 * pi * r^4) / (6 * r)

    Which could be considered a ratio between whatever (8 * pi * r^4) is .. and (6 * r) which is (3 * Diameter)

    disregard this:

    cowah22 was my secondary ID.
    Last edited: Feb 15, 2009
  7. Does that make sense?
  8. Office_Shredder

    Office_Shredder 4,500
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    Yes, but then you re-wrote 8*pi*r4 as 4*circumference. So it must be circumference = 2*pi*r4.

    I think you're reading too much into what's essentially a constant created by integration (r2 -> r3/3, and the 4 comes from the surface area of a sphere formula)

  9. Probably. Does (8*pi*r^4), or (Volume * (3*Diameter)) even have any geometric meaning/significance? I just thought it was interesting to see a 4th dimension in a sphere volume equation.
    Last edited: Feb 16, 2009
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