Few questions about surface area and volume

[cowah22]

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?

 

[arildno]

Well, if you rewrite the volume for the ball as $$V=\frac{1}{3}*4\pi{r}^{3}$$, recognize that this can be further simplified as:
$$V=\frac{1}{3}*r*S$$ 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]

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]

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]

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]

Does that make sense?

 

[Office_Shredder]

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]

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.