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?
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.
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.
Yes, but then you re-wrote 8*pi*r^{4} as 4*circumference. So it must be circumference = 2*pi*r^{4}. I think you're reading too much into what's essentially a constant created by integration (r^{2} -> r^{3}/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.