Volume of Circular Cylinder/Slant Height

Theorem

Let $\CC$ be a circular cylinder such that:

the bases of $\CC$ are circles of radius $r$

the slant height of $\CC$ is $l$

the inclination of the generatrices of $\CC$ to the base of $\CC$ is $\theta$.

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

$\VV = \pi r^2 l \sin \theta$

Proof

Let $h$ denote the height of $\CC$.

From Relation between Slant Height and Height of Cylinder:

$h = l \sin \theta$

From Volume of Circular Cylinder in terms of Height:

$\VV = \pi r^2 h$

The result follows.

$\blacksquare$

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 4$: Geometric Formulas: Circular Cylinder of Radius $r$ and Slant Height $l$: $4.33$
• 2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 7$: Geometric Formulas: Circular Cylinder of Radius $r$ and Slant Height $l$: $7.33.$