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Few questions about surface area and volume

by cowah22
Tags: surface, volume
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cowah22
#1
Feb14-09, 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?
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arildno
#2
Feb14-09, 10:01 AM
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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.
cowah22
#3
Feb14-09, 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.

HallsofIvy
#4
Feb14-09, 04:36 PM
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Few questions about surface area and volume

Quote Quote by cowah22 View Post
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?
nuby
#5
Feb14-09, 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:
V = (4 * circumference) / (6 * radius)

cowah22 was my secondary ID.
nuby
#6
Feb14-09, 09:21 PM
P: 364
Does that make sense?
Office_Shredder
#7
Feb15-09, 08:08 AM
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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)
nuby
#8
Feb16-09, 10:04 AM
P: 364
Quote Quote by Office_Shredder View Post
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.


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