Volume of Cylinder

Theorem

The volume $V_C$ of a cylinder whose bases are circles of radius $r$ and whose height is $h$ is given by the formula:

$V_C = \pi r^2 h$

Proof

Consider a cylinder $C$ whose base is a circle of radius $r$ and whose height is $h$.

Consider a cuboid $K$ whose height is $h$ and whose base has the same area as the base of $C$.

Let the area of those bases be $A$.

Let the cylinder $C$ 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.

The bases of $C$ are circles of radius $r$.

From Area of Circle, the area of each base therefore gives:

$A = \pi r^2$

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

$V_K = A h = \pi r^2 h$

Hence the result.

$\blacksquare$