Volume of a Cylinder

In this tutorial, we will learn how to find the volume of a cylinder. So let's start with what a cylinder typically looks like.

So you have two circular and parallel surfaces (bases) joined by a uniform circular cross-section - all of the same radius.

Now a cylinder whose axis - the line joining the centers of the bases - is perpendicular to the bases, is known as a right cylinder or a right circular cylinder.

As is generally the case, for the rest of the tutorial, we'll use the word cylinder to refer to a right circular cylinder.

Volume of a Cylinder

As with any three-dimensional solid, the volume of a cylinder is the total amount of space the cylinder occupies.

Formula for Volume of a Cylinder

For a cylinder with a radius $\hspace{0.2em} r \hspace{0.2em}$ and height $\hspace{0.2em} h \hspace{0.2em}$ , the volume is given by the formula -

V = \frac{4}{3} \pi r^3

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Derivation

For any solid with a uniform cross-section, the volume is given by -

V = A_b \cdot h \hspace{0.25cm} \rule[0.1cm]{1cm}{0.1em} \hspace{0.15cm} (1)

where $\hspace{0.2em} A_b \hspace{0.2em}$ is the area of cross-section and $\hspace{0.2em} h \hspace{0.2em}$ is the length of the solid perpendicular to the cross-section.

For the cylinder with a radius $\hspace{0.2em} r \hspace{0.2em}$ , the area of cross-section would be,

A_b = \pi r^2 \hspace{0.25cm} \rule[0.1cm]{1cm}{0.1em} \hspace{0.15cm} (2)

Combining $\hspace{0.2em} (1) \hspace{0.2em}$ and $\hspace{0.2em} (2) \hspace{0.2em}$ , we get the volume

V = \pi r^2 h

How to Find the Volume of a Cylinder | Examples

Now, let's use what we've learned so far to solve a couple of examples.

Example

A cylinder has a radius of $\hspace{0.2em} 3$ inches and a height of $\hspace{0.2em} 10$ inches. Find its volume.

Solution

We need to find the volume here, so we start with the formula.

V = \pi r^2 h

And plugging in the values for $\hspace{0.2em} r \hspace{0.2em}$ and $\hspace{0.2em} h \hspace{0.2em}$ , we get -

\begin{align*} V &= \pi \cdot 3^2 \cdot 10 \\[1em] &= 282.74 \end{align*}

So the volume of the cylinder is $\hspace{0.2em} 282.74 \text{ in}^2$ .

Example

$\hspace{0.2em} 28 \pi \hspace{0.2em}$ cubic centimeters of metal was melted and cast into an $\hspace{0.2em} 8$ cm high cylinder. Calculate the radius of the cylinder.

Solution

Here's the key to solving this question.

When we melt some metal and cast it into a cylinder (or any other shape), the volume of the cylinder would be the same as the volume of the metal we started with. Assuming there are no losses in the process.

So indirectly, the question tells us the volume $\hspace{0.2em} (V) \hspace{0.2em}$ of the cylinder is $\hspace{0.2em} 28 \pi \text{ cm}^3 \hspace{0.2em}$ .

It also tells us the height $\hspace{0.2em} (h) \hspace{0.2em}$ of the cylinder formed – $8$ cm.

So let's substitute the values of $\hspace{0.2em} V \hspace{0.2em}$ and $\hspace{0.2em} h \hspace{0.2em}$ into our formula for volume.

\begin{align*} {\color{Red} V} &= \pi \cdot r^2 \cdot {\color{Teal} h} \\[1em] {\color{Red} 28 \pi} &= \pi \cdot r^2 \cdot {\color{Teal} 7} \\[1em] r^2 &= 4 \end{align*}

Taking the square root on both sides –

r = 2

Note – We don't need to worry about the negative square root since the radius cannot be negative.

And with that, we come to the end of this tutorial on the volume of a cylinder. Until next time.