Circles and PiCavalieri

The volume of an oblique cylinder turns out to be exactly the same as that of a right cylinder with the same radius and height. This is due to Cavalieri’s Principle, named after the Italian mathematician Bonaventura Cavalieri: if two solids have the same cross-sectional area at every height, then they will have the same volume.

Imagine slicing a cylinder into lots of thin disks. We can then slide these disks horizontal to get an oblique cylinder. The volume of the individual discs does not change as you make it oblique, therefore the total volume also remains constant: