In geometry, the napkin-ring problem involves finding the volume of what remains after a circular hole is drilled through a sphere. Specifically, the hole has the shape of a right circular cylinder (with two spherical caps) whose axis goes through the center of the sphere. Removing the "hole" leaves a circular "band". It is a counterintuitive fact that this volume does not depend on the original sphere's radius but only on the resulting band's height.

The problem is so called because the part that remains resembles the shape of a napkin ring.

Statement

Suppose that the axis of a right circular cylinder represents the height (defined as the distance in a direction parallel

equals the diameter of the sphere. In this case the volume of the band is the volume of the whole sphere, which matches the formula given above.

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

Proof

.

By the Pythagorean theorem, the radius of the cylinder is

The cross-section is the region inside the larger circle of radius given by(2) and outside the smaller circle of radius given by

The radius R. The volume of the band is

.

This is an application of Cavalieri's principle

Another Derivation

using previous results. minus the cylinder's volume minus the volume of two spherical caps

as shown in (1)

plus two spherical cap heights

cancel and one gets

See also

• Visual calculus, an intuitive way to solve this type of problem, originally applied to finding the area of an annulus, given only its chord length

• String girdling Earth, another problem where the radius of a sphere or circle is counter-intuitively irrelevant

References

Further reading

• Problem 132 asks for the volume of a sphere with a cylindrical hole drilled through it, but does not note the invariance of the problem under changes of radius.

• . Levi argues that the volume depends only on the height of the hole based on the fact that the ring can be swept out by a half-disk with the height as its diameter.

• . Reprint of 1935 edition. A problem on page 101 describes the shape formed by a sphere with a cylinder removed as a "napkin ring" and asks for a proof that the volume is the same as that of a sphere with diameter equal to the length of the hole.

• . Reprint of 1954 edition.

External links