Question

Cavalieri’s principle states that if two solids of equal height have equal cross-sectional areas at every level parallel to the respective bases, then the two solids have equal volume.

The two shaded solids both have a height of 2r units. At every level, the areas of the cross sections of both solids equal π(r2 – b2).

A cylinder and a sphere are shown. 2 cones are cut out of the cylinder. The cones have a radius of 4. The cylinder has a height of 2 r. The sphere has a radius of 4. The sphere has a height of 2 r. Cross-sectional areas are shown on each figure.

Therefore, the formula for the volume of the sphere can be derived by writing an expression that represents the volume of

one cone within the cylinder.
the two cones within the cylinder.
the solid between the two cones and the cylinder.
the cylinder.

Answer 1

Write the volumes in terms of r:

- One cone (base radius r, height r): V1 = (1/3)π r^2·r = (1/3)π r^3.
- Two cones: V2 = 2·(1/3)π r^3 = (2/3)π r^3.
- Cylinder (radius r, height 2r): Vcyl = π r^2·2r = 2π r^3.
- Solid between the two cones and the cylinder: V = Vcyl − V2 = 2π r^3 − (2/3)π r^3 = (4/3)π r^3.

Thus the sphere’s volume equals the solid between the two cones and the cylinder, Vsphere = (4/3)π r^3.

(If r = 4, Vsphere = (4/3)π·4^3 = (256/3)π.)