Spin a Rectangle

I had a wonderful time spinning the cube both physically and mathematically. I thought about the rectangle but did not give it much attention. Then, Professor Fardin informed me of Erickson’s piece on the “resolving credit” in the MAA book Aha! Solution (p. 64). I decided to explore the problem a little bit from a design perspective. It turned out to be quite interesting—a cone-rich problem!

In short, if we take a rectangle and spin it around one of its diagonals (from 2D to 3D), what 3D shape do we get? How about the volume of the resulting 3D solid?

The 3D models provided here are based on a 30mm x 40mm rectangle. Several pieces are included for students to perform the physical experiment and appreciate the outcome. Algebraically, the volume of the 3D solid is a function of the two dimensions of the rectangle. One can derive the formula using the cone components of the shape (see the figure for the formula, typeset in LaTeX).

There are several ways to look into the 3D shape. (1) Spin the rectangle; (2) look at the whole 3D shape (smooth whole or cage); (3) look inside the solid. A touch of super glue is needed to make a whole out of two halves.

Among the Files

1. Rectangle (30mm x 40 mm)
2. Stand
3. Spinning rectangle cage (12 copies)
4. One-piece half
5. One-piece half with hole in the middle
6. Three-piece half
7. Three-piece half with hole in the middle

Reference

1. Erickson, Martin. (2009). Aha! solutions. Washington, DC: Mathematical Association of America (“Revolving Credit,” p. 63-64).