Surface Area of a 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.

A cylinder

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.

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.

Surface Area of a 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.

Surfaces of a cylinder

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

Formula for Surface Area of a Cylinder

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

1. Total surface area

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

2. Lateral surface area or curved surface area

LSA=2πrh\text{LSA} = 2 \pi rh

3. Area of each base

Abase=πr2A_{base} = \pi r^2
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Derivation

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

Deriving the surface area of a cylinder

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.

TSA=LSA+FSA=2πrh+2πr2=2πr(h+r)\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*}

How to Find the Surface Area of a Cylinder | Examples

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

Solution

The formula for the lateral surface area is -

LSA=2πrh\text{LSA} = 2 \pi r h

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

LSA=2π⋅8⋅5≈251.33\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

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

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

TSA=2π⋅8⋅(8+5)≈653.45\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 251.33 in2\hspace{0.2em} 251.33 \text{ in}^2 \hspace{0.2em} and its total surface area is 653.45 in2\hspace{0.2em} 653.45 \text{ in}^2 \hspace{0.2em}.

Example

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

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

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

60π=2π⋅3⋅(h+3)10=h+3h=7\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.