(1) Are you ready for your brain to hurt a little bit? (2) If you take a sphere and remove the core from it, you’re left with a shape called a napkin ring – because it looks like a napkin ring. (3) Suppose you first remove the core from a sphere the size of a basketball. (4) Next, suppose you core the earth itself (assuming it is a perfect sphere) to get a napkin ring of the same height. (5) Which napkin ring would have a greater volume? (6) Most students would immediately reply, “The napkin ring from the earth of course!” (7) The diameter of the earth is 12,472 km. (8) Surprisingly, they would not be correct. (9) The two napkins rings of vastly different diameters will have identical volumes.

(10) The mathematics behind the napkin ring problem depends on the Pythagorean Theorem and Cavalieri’s Principle. (11) In order to better understand Cavalieri’s Principle, imagine a vertical stack of 5 quarters and shift the stack to the right until the shape looks slanted. (12) Despite the skew, the stack still has the same height, and every cross section is the same circular quarter. (13) Most importantly, it has the same volume of exactly 5 quarters which has not changed. (14) When computing the cross-sectional area of any napkin ring, the outer and inner circle areas are subtracted and the radius term drops out in the process. (15) The cross-sectional area is the same as that of a sphere of radius h. (16) Also, the volume is the same as that of a sphere of radius h; 4/3πh³. (17) The result seems counterintuitive. (18) Whether the napkin ring of height h is carved from the entire earth or a basketball size sphere, the volume will be the same.