Expert: Jack Cheng - 11/12/2006

Question
I was hoping you could get me started on this question:
A hole is cut through the center of a sphere of radius r. The height of the remaining spherical ring is h. Show that the volumn of the ring is V=pieh^3/6, that is, the volume is indepent of r.

Thank you,
Katt

Answer
Hi Kathrine,

I am assuming that this is an integration problem (I can't think of any other way to do it).

First, imagine a sphere on the coordinate system with its center on the origin. Now, the hole cuts through the sphere horizontally, with the center of the hole passing through the x-axis. Now, what remains is our spheric ring, which goes from -h/2 to h/2 (because the height of the ring is h, and the center at the origin).

Now, you have to find a formula for a vertical cross-section of this spherical ring, namely a formula for the area of the bigger circle minus the area of the circle cut by the hole. Once you have this formula, you should integrate it from -h/2 and h/2.

In finding the formula, you have to write R, the radius of the sphere (not to be confused with r, the radius of the hole), in terms of r and h, which you can do with the Pythagorean Theorem at the left/right end of the sphere. When you this and plug the value in, the r's cancel out.

I hope this is enough to get you started. If you need more help, feel free to ask again.

~ Jack

P.S. This proof is very similar to the proof of the volume of the sphere. If you have that proof, look over it. It should help a lot.