In the diagram above, we have a hemisphere with radius [math]r[/math], side by side with a cylinder with radius [math]r[/math] and height [math]r[/math]. The cylinder has had an inverted cone with the same radius and height removed from its interior.[br][br]The diagram at the right shows the cross-sections of both figures. To change the height of the plane, use the slider at the bottom of the diagram. [br][br]Both cross-sections have the same area. Which can be shown using two triangles in the diagram.[br]Since the cross-sections have the same area at every height of the plane, the two figures have the same volume by Cavalieri's Principle.[br][br]Thus,[br][br]The volume of the hemisphere is:[br][math]V=\pi r^2r-\frac{1}{3}\pi r^2r=\pi r^3-\frac{1}{3}\pi r^3=\frac{2}{3}\pi r^3[/math][br][br]Making the volume of a sphere:[br][math]V=\frac{4}{3}\pi r^3[/math]