Volume of Right Circular Cylinder

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Theorem

The volume $V_C$ of a right circular 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 1

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

Let $V_C$ denote the volume of $C$.

From Volume of Cylinder in terms of Height and Base Area:

$V_C = A h$

where $A$ is the area of the base of $C$.

From Area of Circle, the area of each base is:

$A = \pi r^2$

Hence:

$V_C = \pi r^2 h$

$\blacksquare$


Proof 2

VolumeOfCylinder.png

Consider a right circular 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 $C$ 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.

$\blacksquare$


Sources

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