[From Girl’s Angle: A Math Club for Girls, via Albany Area Math Circle.]
Do you know why this proof works? How can we be sure the red and yellow areas don’t change as they slide around?
For my math club students, I call it the “stack of cards” rule: If you have a stack of playing cards (or any other rigid shapes), the volume of the stack stays the same when you slant the stack different ways.
Here is an interactive demonstration of Cavalieri’s Principle (with very thick, 2-dimensional “playing cards”) from Jim Loy’s Mathematics Page.
Wikipedia has several examples of area and volume problems solved by Cavalieri’s Principle.