Volume of Cylinder/Height and Base Area

Theorem

Let $\CC$ be a cylinder such that:

the bases of $\CC$ have area $A$
the height of $\CC$ is $h$.


The volume $\VV$ of $\CC$ is given by the formula:

$\VV = A h$


Proof

Consider a cylinder $C$ whose base is of area $A$ and whose height is $h$.

Consider a cuboid $K$ whose height is $h$ and whose base is also $A$.


Let $C$ be positioned with its base in the same plane as the base of $K$.

By Cavalieri's Principle $C$ and $K$ have the same volume.


From Volume of Cuboid, $K$ has volume given by:

$V_K = A h$

Hence the result.

$\blacksquare$


Sources

  • 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): cylinder
  • 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): cylinder
  • 2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 7$: Geometric Formulas: Cylinder of Cross-sectional Area $A$ and Slant Height $l$
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