A hole 6 inches long is drilled through a sphere of radius R to form a ring. (See the figure below; the ring is 6 inches high and 10 inches across, and R is 5 inches.)

Find an expression for the volume of the ring. Is there anything remarkable about this result?

Hint 1

The ring is a volume of revolution.

Hint 2

You can find the volume by the shell method or by the disk method, but the disk method is easier. Turn the figure on its side, so we are rotating about the x axis.

The ring is formed by rotating the area bounded by the semi-circle x² + y² = R², y >= 0 and the line of width 6 that intersects it, about the x axis. Let's write y_c for the y value of the circle, and y_l for the y value of the straight line. Then the volume of the ring is

Now, what is surprising about this result?

The Rest of the Solution

The volume is independent of R; no matter how large the sphere is, the ring has the same volume. For very large spheres, the ring is very large in diameter but the material is very thin, and these two factors exactly compensate for each other, keeping the volume constant.

Is this a curiosity--a unique situation?

No. There are many figures of revolution, defined by the intersection of two conic sections, that have similar properties; the volume is independent of some of the parameters defining the curves. In our example the conic sections are a circle and a line, and the volume depends only on the height (in our drawing, the x extent) of the intersection and not on where the curves are placed. The paper by Alexanderson and Kosinski cited below gives a table of many combinations with this property.

References

Morris Kline, Calculus: An Intuitive and Physical Approach, Dover reprint, 1998. Exercise 5 on p. 447 states the problem in the form we used here, asking you "Is the answer remarkable in any respect?"
G. L. Alexanderson and L. F. Kosinski, Some Surprising Volumes of Revolution, Two Year College Math Journal, v. 6 no. 3 (1975) pp. 13-15. Reprinted on pp. 321-323 of: Tom M. Apostol et al., editors, A Century of Calculus, Part II, Mathematical Association of America, 1992.