Volumes of the ball and its parts
Cavalieri’s principle: if two solids cut by planes parallel to one plane always give sections of equal area, the solids have equal volume. A hemisphere of radius R and a cylinder of radius R and height R with a cone removed are equal in this sense: at height x the section areas are π(R² − x²). So the hemisphere’s volume is πR³ − ⅓πR³ = ⅔πR³ and the volume of a ball is V = (4/3)πR³; that is, a ball is ⅔ of the cylinder circumscribed about it (Archimedes). The spherical shell between concentric spheres of radii R and r (r < R) has volume (4/3)π(R³ − r³). A spherical segment of height h has volume V = πh²(3R − h)/3; a spherical sector (segment plus cone) has V = ⅔πR²h; for a zone between two parallel planes, with base radii r₁, r₂ and height h, V = (πh/6)(3r₁² + 3r₂² + h²). The volumes of similar balls are in the ratio of the cubes of their radii.
“Archimedes’ vessel”: with adult supervision fill to the brim a cylindrical vessel that a tennis ball just fits into (height equal to the ball’s diameter), carefully lower the ball in and measure the displaced water. It should be about ⅔ of the vessel’s volume.