The big idea
If two solids have the same height and their slices have equal areas at every height, the solids have equal volumes. A hemisphere and a cylinder with a cone scooped out pass that test, so the hemisphere holds the cylinder minus the cone: πr³ − ⅓πr³ = ⅔πr³. Two hemispheres make the sphere, ⁴⁄₃πr³.
Memory trick: “Can minus cone gives the dome: one take away a third is two-thirds, and two domes make four-thirds.”
A bit of history: Around 250 B.C. Archimedes proved, in On the Sphere and Cylinder, that a sphere has two-thirds the volume and two-thirds the surface of the cylinder that just holds it. He asked for a sphere inside a cylinder to be placed on his tomb, and in 75 B.C. Cicero found the forgotten tomb in Sicily by spotting that figure on a small column. In China, Zu Geng used the same slice-by-slice idea around A.D. 500. Bonaventura Cavalieri, a member of a religious order and professor at Bologna, published it in Europe in 1635 in his Geometria indivisibilibus, which is why we call it Cavalieri's principle.
To think about: “Who hath measured the waters in the hollow of his hand, and meted out heaven with the span, and comprehended the dust of the earth in a measure, and weighed the mountains in scales, and the hills in a balance?” (Isaiah 40:12, KJV)