Cavalieri’s Principle and the Volume of a Sphere
This applet shows how to compute the volume of a sphere by using Cavalieri's principle.
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.
Given are a sphere with a radius and a cone with a radius of the base and a height equal to the same . The vertex of the cone is on the plane passing through the center of the sphere.
This applet creates the cross sections of the cone and the upper hemisphere with planes parallel to . It also constructs the combined area of the two cross-sections.
- Drag the slider or click the “Animate slicing” button to see the cross sections and the combined area.
- Click on the Trace ON/OFF button to see an individual cross-section or all of them.
- 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.
- Increase the number of planes to visualize the Cavalieri’s principle.