Few questions about surface area and volume

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Discussion Overview

The discussion revolves around the mathematical representation of the volume of a sphere and the surface area of an equilateral triangle. Participants explore the significance of specific constants and fractions in these formulas, as well as their physical interpretations and geometric meanings.

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Conceptual clarification

Main Points Raised

  • One participant questions the meaning of the (4/3) in the volume formula for a sphere, asking why it is not another number or fraction.
  • Another participant suggests that the volume can be related to the surface area of the sphere and compares it to the volume of a cone, referencing Archimedes' proof.
  • A participant proposes a physical representation of the volume formula, manipulating the equation to explore the significance of the numerator and its relation to dimensions.
  • There is a suggestion that the numerator in a derived equation might represent four dimensions, although another participant expresses skepticism about this interpretation.
  • Participants discuss the geometric meaning of expressions like (8 * pi * r^4) and (3 * Diameter), questioning their significance in the context of the volume of a sphere.

Areas of Agreement / Disagreement

Participants express varying interpretations of the mathematical constants and their physical meanings, with no consensus reached on the significance of certain terms or the dimensional implications of the equations discussed.

Contextual Notes

Some participants reference integration and constants derived from geometric formulas, but the discussion remains open-ended regarding the implications of these mathematical manipulations.

cowah22
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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?
 
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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.
 
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))

or

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


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


Would the numerator represent 4 dimensions? Seems weird.
 
cowah22 said:
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))

or

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


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


Would the numerator represent 4 dimensions? Seems weird.
Since you are working in 3 dimensions, I doubt that! And 2 pi r^4 is the circumference of what?
 
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:
V = (4 * circumference) / (6 * radius)
cowah22 was my secondary ID.
 
Last edited:
Does that make sense?
 
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)
 
Office_Shredder said:
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)
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:

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