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How do you find the surface area of an n-dimensional sphere?

Thanks!

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- Thread starter fizixx
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- #1

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How do you find the surface area of an n-dimensional sphere?

Thanks!

- #2

DaveC426913

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The surface is the derivative of the volume.

In 3D, volume is: [tex]4 \pi r^3[/tex] and surface is: [tex]\frac{4}{3} \pi r^2[/tex]

So,

A 4D sphere's surface area will be : [tex]4 \pi r^3[/tex] while its volume will be [tex]16 \pi r^4[/tex]

I think.

- #3

rachmaninoff

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I'm 99.999% sure the other response is not going in the right direction, but I appreciate the responses!

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mathwonk

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try to recall how to get the volume ofa 3 sphere by integrating area of circular slices, then apply that one dimension up.

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saltydog

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fizixx said:

I'm 99.999% sure the other response is not going in the right direction, but I appreciate the responses!

I think the following web site is a good one:

Area and Volume formulas for n-dim spheres and balls

I'll go over the integrals too!

Edit: Ok I did:

Hey guys, I found the volume formulas for n-balls interesting: simply integrate a cross-section of the object over appropriate bounds. For the ball, it was:

[tex] V_2=2\int_0^{\pi/2} (line)dheight[/tex]

[tex]V_3=2\int_0^{\pi/2} (circle)dheight [/tex]

[tex]V_4=2\int_0^{\pi/2} (sphere)dheight [/tex]

and so on if you check the reference I gave above. I wonder if this is the case in general. That is, if I calculate the volume of my coffee cup then would the volume of a 4-D coffee cup be:

[tex]V_4=\int_a^b \text{(3-D coffee cup)} dp[/tex]

for appropriate values of a, b, and p.

(Hope I don't make it confussing for you Fizixx).

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- #7

saltydog

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Can we say in general to calculate the volume of ANY object in n-dimensional space, integrate the n-1 dimensional object over appropriate bounds? What then is the volume of the n-D torus? Same dif? I would say so. Maybe we need to calculate it.

Also, I found it interesting in the MathWorld reference that the volume of the sphere reaches a maximum value at around 5-D and then drops getting closer and closer to zero as the dimension is increased. Is that the case with all n-D volumes? Does this mean that my coffee cup holds smaller and smaller amounts of n-D coffee?

I can see it now: Salty's n-D Coffee Cafe' . . . where the coffee never runs out.

- #8

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Great in concept, but it does not answer my question, unfortunately. I think I asked my question on the wrong board.

Providing an aswer that tritely says to just integrate your way up the dimension-ladder despite isn't sufficient, but I do apreciate the responses a great deal, and thanks to everyone for sending a response, but I'm closing this thread and going elsewhere for further inquiry.

Good day all, and hapy thoughts to everyone!

- #9

rachmaninoff

Did you read my link? ( http://mathworld.wolfram.com/Ball.html )fizixx said:Great in concept, but it does not answer my question, unfortunately. I think I asked my question on the wrong board.

Providing an aswer that tritely says to just integrate your way up the dimension-ladder despite isn't sufficient,

It gives an

[tex]V_n(r) = \frac{\pi ^{n/2} r^n}{\Gamma \left( 1 + \frac{1}{2}n \right)}[/tex]

From which the 'area' of a hypersphere's surface naturally follows, in the usual way:

[tex]S_n(r) = \frac{ dV_n }{ dr } = \frac{n \pi ^{n/2} r^{n-1}}{\Gamma \left( 1 + \frac{1}{2}n \right)}= \frac{2 \pi ^{n/2} r^{n-1}}{\Gamma \left( \frac{1}{2}n \right)}[/tex]

If you don't like this, there's a table at the bottom of the page which works out the first ten dimensions; and a more complete listing is Sloane's A072478 and A072479 .

edit: More of the same is here (also from Mathworld).

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- #10

rachmaninoff

http://mathworld.wolfram.com/Hypersphere.html

(item 10)

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