Volume of a Cylinder — Formula & Examples

A cylinder holds whatever its circular base sweeps out as it rises to full height, so its volume is the base area times the height. For a right circular cylinder with radius $r$ and height $h$,

In this formula $r$ is the radius of the base, $\pi r^2$ is the area of that circular base, and $h$ is the perpendicular height between the two circular faces. If a problem gives the diameter $d$ instead, convert first with $r = \frac{d}{2}$.

Why The Formula Holds

The reasoning is short: a cylinder is a prism with a circular base, and any prism has volume equal to base area times height. The base area of a circle is $\pi r^2$, so

That also explains why the radius is squared while the height is not. The $r^2$ comes straight from the circle's area; the height is multiplied only once as the base is stacked upward. The practical consequence: if the height doubles, the volume doubles, but if the radius doubles, the volume becomes four times as large because the base area depends on $r^2$.

Worked Example: Radius $4$ cm, Height $10$ cm

Start with the formula:

Substitute $r = 4$ and $h = 10$:

Square the radius first, then multiply:

So the exact volume is $160\pi\ \text{cm}^3$. If a decimal is required, use $\pi \approx 3.14159$:

Many classes prefer the exact form $160\pi\ \text{cm}^3$ unless the instructions ask you to round.

Practice It Yourself

First, redo the example when it is described with diameter $8$ cm and height $10$ cm. The radius is half the diameter, so $r = 4$ cm, and

Same cylinder, same answer, because the formula always uses the radius.

Next, try radius $6$ cm and height $3$ cm. Set it up before calculating:

If you get $108\pi\ \text{cm}^3$, your setup is consistent.

Calculation Traps To Watch For

1. Using the diameter directly in $V = \pi r^2 h$. Convert to radius first.
2. Forgetting to square the radius. The formula uses $r^2$, not $2r$.
3. Multiplying by the slanted side of an oblique drawing instead of the perpendicular height. The formula needs the actual height between the bases.
4. Writing square units instead of cubic units. Volume should be in units such as $\text{cm}^3$, $\text{m}^3$, or $\text{in}^3$.
5. Rounding too early when the problem allows an exact answer in terms of $\pi$.

One more condition to keep in mind: this formula gives the outer volume of a hollow cylinder unless you subtract the empty inner part, and if the radius changes along the height, the shape is not a cylinder, so the formula does not apply directly. Cans, pipes, tanks, candles, and circular columns are the usual fits.

A Clean Next Step

If you want to see exactly where the $\pi r^2$ part comes from, compare this formula with the area of a circle. The cylinder formula is just that circle, given height.