Follow the proof Archimedes himself was said to be proudest of. Place a hemisphere and a "cylinder with a cone carved out" side by side, slice both at the same height, and remarkably, the cross-sectional areas are always exactly equal.
Cavalieri's principle: slice two solids at the same height, over and over, starting from the base — if the cross-sectional area is always equal at every height, then the two solids' volumes must also be equal. On this page, we compare a hemisphere of radius r against a cylinder of radius r and height r with a cone of radius r carved out.
The person who first thought up this method was the ancient Greek mathematician Archimedes. He considered this discovery his proudest achievement, and is said to have requested that a sphere and a cylinder be carved onto his tombstone.
Use the slider to change the height and slice both shapes horizontally. You can see with your own eyes that the hemisphere's cross-section (a circle) and the "cylinder minus cone" shape's cross-section (a ring) always have exactly the same area.
① Compare the two solids
Change the radius and compare the volumes of the hemisphere and the "cylinder − cone" shape. The two volumes are always exactly equal, no matter the radius.
② Slice at the same height and compare cross-sections
Use the slider to change the slicing height. See that the area of the circle sliced from the hemisphere, and the area of the ring (donut shape) sliced from the "cylinder − cone" shape, are always equal, at any height.