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.

In this tutorial, we will learn how to find the surface area of a cylinder. And we'll 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.

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

There are three surfaces that make up the total surface of a cylinder - two circular bases and the curved surface joining them. The surface area of the cylinder is the total area covered by these surfaces.

The area of the curved surface is known as the curved surface area or lateral surface area of the cylinder.

Here are the formulas for a cylinder with a radius $\hspace{0.2em} r \hspace{0.2em}$ and height $\hspace{0.2em} h \hspace{0.2em}$.

1. Total surface area

$\text{TSA} = 2 \pi r (h + r)$

2. Lateral surface area or curved surface area

$\text{LSA} = 2 \pi rh$

3. Area of each base

$A_{base} = \pi r^2$

Imagine cutting open a cylinder's surface. You will have two circles and a rectangle.

Now the height (or width) of the rectangle will be equal to the height of the cylinder. And its length will be equal to the circumference of the cylinder.

So the curved or lateral surface area (LSA) of the cylinder would be -

For the flat surface area (FSA), we just need to double the area of each circle -

Adding the area of the base and the top, we get the total surface area (TSA) of the cylinder.

$\begin{align*} \text{TSA} &= \text{LSA} + \text{FSA} \\[1em] &= 2 \pi r h + 2 \pi r^2 \\[1em] &= 2 \pi r (h + r) \end{align*}$

Alright. Let's now apply the concepts and formulas we have learned so far to solve a couple of example problems.

Example

A cylinder has a radius of $\hspace{0.2em} 8$ inches and a height of $\hspace{0.2em} 5$ inches. Find its lateral and total surface areas.

Solution

The formula for the lateral surface area is -

$\text{LSA} = 2 \pi r h$

Plugging in the values of $\hspace{0.2em} r \hspace{0.2em}$ and $\hspace{0.2em} h \hspace{0.2em}$, we have

$\begin{align*} \text{LSA} &= 2 \pi \cdot 8 \cdot 5 \\[1em] &\approx 251.33 \end{align*}$

Now, for the total surface area, the formula is

$\text{TSA} = 2 \pi r (h + r)$

Let's substitute the values of $\hspace{0.2em} r \hspace{0.2em}$ and $\hspace{0.2em} h \hspace{0.2em}$ into the formula.

$\begin{align*} \text{TSA} &= 2 \pi \cdot 8 \cdot (8 + 5) \\[1em] &\approx 653.45 \end{align*}$

And that’s it. The lateral surface area of the cylinder is $\hspace{0.2em} 251.33 \text{ in}^2 \hspace{0.2em}$ and its total surface area is $\hspace{0.2em} 653.45 \text{ in}^2 \hspace{0.2em}$.

Example

The total surface area of a cylinder is $\hspace{0.2em} 60 \pi \hspace{0.2em}$. Find its height if it has a radius of $\hspace{0.2em} 3 \hspace{0.2em}$.

Solution

It's clear that we need to work with the total surface area of the cylinder, so let's start with its formula.

$\text{TSA} = 2 \pi r (h + r)$

Next, we substitute the values of TSA and $\hspace{0.2em} r \hspace{0.2em}$ in the formula. And solve for $\hspace{0.2em} h \hspace{0.2em}$. So,

$\begin{align*} 60 \pi &= 2 \pi \cdot 3 \cdot (h + 3) \\[1em] 10 &= h + 3 \\[1em] h &= 7 \end{align*}$

Done.

And that brings us to the end of this tutorial on the surface area of a cylinder. Until next time.

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