Napkin Ring Problem

Napkin Ring Problem

In geometry, the volume of a band of specified height around a sphere—the part that remains after a hole in the shape of a circular cylinder is drilled through the sphere—does not depend on the sphere's radius.

Specifically, suppose the axis of a right circular cylinder passes through the center of the sphere and the height (defined as distance in a direction parallel to the axis) of the part of the boundary of the cylinder that is inside the sphere is h, and the radius of the sphere is R. The "band" is the part of the sphere that is outside the cylinder.

The result is that the volume of the band depends on h but not on R.

As the radius R of the sphere shrinks, the diameter of the cylinder must also shrink in order that h can remain the same. The band gets thicker, and that would increase its volume. But it also gets shorter in circumference, and that would decrease its volume. The two effects exactly cancel each other out. The most extreme case, involving the smallest possible sphere, is that in which the diameter of the sphere is the same as the height h. In that case the volume of the band is the volume of the whole sphere:

An early study of this problem was written by 17th-century Japanese mathematician Seki Kōwa. According to Smith & Mikami (1914), Seki called this solid an arc-ring, or in Japanese kokan or kokwan.

The "Napkin Ring Problem" is called so because after removing a cylinder from the sphere, the remaining band resembles the shape of a napkin ring.