<P> ("Base" is in units of area; "height" is in units of distance . Area × distance = volume .) </P> <P> Therefore the volume of the upper half - sphere is 2 3 π r 3 (\ displaystyle (\ frac (2) (3)) \ pi r ^ (3)) and that of the whole sphere is 4 3 π r 3 (\ displaystyle (\ frac (4) (3)) \ pi r ^ (3)). </P> <P> The fact that the volume of any pyramid, regardless of the shape of the base, whether circular as in the case of a cone, or square as in the case of the Egyptian pyramids, or any other shape, is (1 / 3) × base × height, can be established by Cavalieri's principle if one knows only that it is true in one case . One may initially establish it in a single case by partitioning the interior of a triangular prism into three pyramidal components of equal volumes . One may show the equality of those three volumes by means of Cavalieri's principle . </P> <P> In fact, Cavalieri's principle or similar infinitesimal argument is necessary to compute the volume of cones and even pyramids, which is essentially the content of Hilbert's third problem--polyhedral pyramids and cones cannot be cut and rearranged into a standard shape, and instead must be compared by infinite (infinitesimal) means . The ancient Greeks used various precursor techniques such as Archimedes's mechanical arguments or method of exhaustion to compute these volumes . </P>

Use cavalieri’s principle to find a formula for the volume of a body of revolution