
#1
Feb1409, 09:48 AM

P: 15

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
Feb1409, 10:01 AM

Sci Advisor
HW Helper
PF Gold
P: 12,016

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. 



#3
Feb1409, 10:41 AM

P: 15

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. 



#4
Feb1409, 04:36 PM

Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 38,879

few questions about surface area and volume 



#5
Feb1409, 05:19 PM

P: 364

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. 



#6
Feb1409, 09:21 PM

P: 364

Does that make sense?




#7
Feb1509, 08:08 AM

Mentor
P: 4,499

Yes, but then you rewrote 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) 



#8
Feb1609, 10:04 AM

P: 364

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. 


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