Volume & Surface Area

Derive the solid-volume formulas — stacking, tiling, and Archimedes — then fill them.

The idea

Volume measures how much space a solid fills; surface area measures the area of its boundary. The volume formulas for the standard solids need not be memorized as a list: they all follow from one principle.

The starting point is the box, where volume is counting. An $l \times w \times h$ box holds exactly $lwh$ unit cubes, so $V = lwh$. Every other formula reduces a harder solid to a simpler one.

The tool for reducing is Cavalieri's principle.

Theorem (Cavalieri's principle).

If every horizontal plane cuts two solids in cross-sections of equal area, then the two solids have the same volume.

Push a straight stack of cards sideways into a leaning stack: no card changes, so the space occupied does not change, although the shape does. Cavalieri's principle asserts the same for any two solids that agree slice by slice.

The principle trades a hard comparison for an easy one: comparing cross-sections is plane geometry, while comparing volumes directly is not. Granted as the single assumption, it yields the prism, the pyramid's $\tfrac13$, the cone, and the sphere's $\tfrac43$ by elementary steps. Integration, later, proves the principle rather than assuming it.

Ways to work on it

• Walkthrough. Derive the prism, pyramid, cone, and sphere volume formulas.
• Practice. Apply a derived formula to a cone, cylinder, or sphere.
• Hardest. Compare the volumes of related solids, then run a volume formula backward to find a radius.

Not sure where to start? Take the ten-question placement test.