Thursday, October 21, 2010

Volume of a bead vs. volume of a sphere

Imagine you drill a hole through the center of a sphere. The remaining object (a bead, if we think of a little pearl with a hole) has an interesting property as pointed out by Pat Ballew on his blog at the end of this post. Whatever the radius of the initial sphere, the volume of the bead depends only on its height h (see following figure taken from Pat's blog). More precisely, its volume is the same as the volume of a sphere with diameter h. A proof of this property has been given by Pat in a later post.

As I was thinking about it, I thought about another way to prove it, inspired by Mamikon's visual calculus method.

If we think of a plane parallel to the axis of the bead and tangent to its inner cylindrical surface, the intersection of the bead and the plane is a disk of diameter h. If we now rotate the plane around the axis for a whole turn such that it remains tangent to the inner surface, the disk will also rotate a whole turn, as if it sweeps the volume of a sphere of diameter h.

Post scriptum. Afterwards I found a drawing of such a bead in Mamikon's original notes (see second page of his drawings).


  1. Beautiful... why couldn't I see it.. Thanks for the inspiration, and then a brilliant solution

  2. Thanks for the mention on your blog. My first explanation in the comment on your first post was a bit confused. I came up with this tangent disk later. There are also nice related illustrations on Mamikon's site, for example

  3. Hi Arjen,
    This idea of proof based on Mamikon's theorem is really cool!
    Best regards,