A Question On Dividing a Sphere

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In summary, the conversation discusses how to divide a sphere's volume equally into 3 parts using two parallel planes. The suggested method is to use the spherical cap formula, which involves solving a tricky cubic equation. Another approach is to modify the method for finding the volume of a sphere by integration or as a volume of rotation. Ultimately, the goal is to find an equation for the distance along the diameter where the cuts should be made to achieve the desired volume.
  • #1
bcarlso2
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I would like to know how to divide a sphere's volume equally into 3 parts, by using two "slices" that are parallel planes. A good example would be cutting a round fruit into 3 equal parts by two slices with a knife. I would like to know the distance (fraction of the diameter) along the diameter where these cuts would be made.
 
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  • #2
Then you'll be needing this formula:

http://en.wikipedia.org/wiki/Spherical_cap

Set V to 1/3 the volume of a sphere, and solve for h. Looks like a tricky cubic equation; your answer is going to have cube roots in it.
 
  • #3
Welcome to PF;
The formula is the way to go ... but this sounds like an exercise for a class.
That means you may be expected to use a different approach. What's the context?
 
  • #4
It is not a class exercise. It is more of a personal curiosity. I figured it would be a simple solution and couldn't find any after doing some online searching. Thanks
 
  • #5
OK then - the spherical cap formula is probably fastest.
The other way is to modify the method for finding the volume of a sphere radius R by integration.

You could also do it as a volume of rotation of the area between x=s and x=R (-R<s<R) under the circle and above the x axis.

In each case you'll be finding an equation for s knowing the desired volume.
 

1. What is the formula for dividing a sphere?

The formula for dividing a sphere is (4πr^2)/n, where r is the radius of the sphere and n is the number of divisions.

2. How many pieces can a sphere be divided into?

A sphere can be divided into an infinite number of pieces, as long as the divisions are continuous and do not overlap.

3. What is the purpose of dividing a sphere?

Dividing a sphere can be useful in various fields such as mathematics, physics, and engineering. It allows for easier calculations and analyses of curved surfaces.

4. Is it possible to divide a sphere into equal pieces?

Yes, it is possible to divide a sphere into equal pieces, but only if the number of divisions is a perfect square. For example, a sphere can be divided into 4, 9, or 16 equal pieces.

5. Can a sphere be divided into irregular pieces?

Yes, a sphere can be divided into irregular pieces. However, it may be more challenging to calculate the area or volume of these pieces compared to dividing them into equal pieces.

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