Volume of a Cylinder — Formula & Examples

To find the volume of a cylinder, multiply the area of the circular base by the height. For a right circular cylinder with radius $r$ and height $h$,

$$V = \pi r^2 h$$

Here, $r$ is the radius of the 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 cylinder volume formula works

The idea is simple: volume equals base area times height. A cylinder is a prism with a circular base, so the base area is $\pi r^2$. That gives

$$V = (\pi r^2)h = \pi r^2 h$$

That also explains the pattern in the variables. The radius is squared because it belongs to the circle area formula, while the height is multiplied only once. If the height doubles, the volume doubles. If the radius doubles, the volume becomes four times as large because the base area depends on $r^2$.

Worked example: a cylinder with radius $4$ cm and height $10$ cm

Start with the formula:

$$V = \pi r^2 h$$

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

$$V = \pi (4)^2(10)$$

Square the radius first, then multiply:

$$V = \pi (16)(10) = 160\pi$$

So the exact volume is $160\pi\ \text{cm}^3$.

If the problem asks for a decimal approximation, use $\pi \approx 3.14159$:

$$V \approx 502.7\ \text{cm}^3$$

In many classes, the exact form $160\pi\ \text{cm}^3$ is preferred unless the instructions ask you to round.

If you are given the diameter instead of the radius

Suppose the same cylinder is described with diameter $8$ cm and height $10$ cm. The radius is half the diameter, so $r = 4$ cm. Then

$$V = \pi (4)^2(10) = 160\pi\ \text{cm}^3$$

This is one of the most common errors on homework and tests. The formula uses the radius, not the diameter.

Common mistakes with cylinder volume

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 $\pi$.

When to use the volume of a cylinder formula

Use the cylinder volume formula whenever an object can be modeled as a cylinder or close to one. Common examples include cans, pipes, tanks, candles, and circular columns.

If the object is hollow, this formula gives the outer volume unless you subtract the empty inner part. If the radius changes along the height, the shape is not a cylinder, so this formula does not apply directly.

Try a similar problem

Try your own version with radius $6$ cm and height $3$ cm. Set it up before calculating:

$$V = \pi (6)^2(3)$$

If you get $108\pi\ \text{cm}^3$, your setup is consistent. If you want one clean next step, compare this formula with the area of a circle so you can see exactly where the $\pi r^2$ part comes from.