This applet shows how to compute the volume of a sphere by using Cavalieri's principle.[br][br]Cavalieri’s principle says that if two solids are included between two parallel planes and if every plane parallel to these planes intersects the solids in cross-sections of equal area, then the solids have equal volumes.[br][br]Given are a sphere with a radius [math] r[/math] and a cone with a radius of the base and a height equal to the same [math]r[/math]. The vertex of the cone is on the plane [math]\alpha[/math] passing through the center of the sphere. [br]This applet creates the cross sections of the cone and the upper hemisphere with planes parallel to [math]\alpha[/math]. It also constructs the combined area of the two cross-sections.[br][list][br][*]Drag the slider or click the “Animate slicing” button to see the cross sections and the combined area.[br][*]Click on the Trace ON/OFF button to see an individual cross-section or all of them.[br][*]Use Cavaliei’s principle and the formulas for the volume of a cone and the volume of a cylinder to find the formula for the volume of the sphere.[br][*]Increase the number of planes to visualize the Cavalieri’s principle.[br][/list]