Key Concepts: Cavalieri's Principle and Applications ⭐

Cavalieri's Principle

If two solids have the same height and every cross section at the same height has the same area, then the two solids have the same volume.

Applications

• Proves that oblique prisms and right prisms with the same base and height have equal volumes.
• Proves the volume formula for a sphere by comparing cross sections with a cylinder-minus-cone figure.

Deriving the Sphere Volume

At height h from the center, the cross section of a sphere of radius r is a circle with area π(r² − h²). This matches a cylinder of radius r with a cone of radius r removed — hence V(sphere) = V(cylinder) − V(cone) = πr²(2r) − ⅓πr²(2r) = (4/3)πr³.