These 3D printed objects served as an excellent starting point for a classroom conversation on Cavalieri’s Principle.

Each shape above is an extrusion of the same square. In the middle, the square moves straight up; at left, the square travels in a complete circle from bottom to top; and at right, the square moves along a line segment and back.

The objects all have the same height. Since at every level they have the same cross-sectional area, by Cavalieri’s Principle they all have the same volume!

Cavalieri’s Principle is a simple but powerful idea, and one that can be easily demonstrated around the house. Here are some other examples using CDs and CD cases.