How to Divide a Sphere's Volume Equally into 3 Parts Using Parallel Planes?

[bcarlso2]

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.

 

[Ben Niehoff]

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.

 

[Simon Bridge]

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?

 

[bcarlso2]

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

 

[Simon Bridge]

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.