Surface Area of Cylinder

What is a Cylinder?

The term cylindrical means to have a straight parallel sides and circular cross sections.

The flat circular ends of a cylinder are parallel to each other and they are separated or joined together by a curved surface. See the figure below.

Fig. 1. Parts of a right cylinder.

Some examples of cylindrical shapes we see every day are canned food and canned soup. The individual parts of a cylinder are shown below. The ends are circles, and if you roll out the curved surface of a cylinder you get a Rectangle!

Fig. 2. The individual part of a cylinder.

There are different types of cylinders, including:

• Right circular cylinders, like in the picture above,

• Half cylinders;

• Oblique cylinders (a cylinder where the top is not directly above the base); and

• Elliptic cylinders (where the ends are ellipses rather then circles).

In particular you will be looking at right circular cylinders here, so from now on they will just be called cylinders.

Total Surface Area of a Cylinder

Let's look at the definition of the total surface area of a cylinder.

Usually people leave off the word "total", calling it just the surface area of a cylinder. As you can see from the picture in the previous section, there are two parts to the area of a cylinder:

• The surface area occupied by just the Rectangle of the cylinder is called lateral surface area.

• The surface area of the ends is the area of two circles.

Let's take a look at each part.

Lateral Surface Area of a Cylinder

To make life easier, let's use some variables. Here:

• \(h\) is the height of the cylinder; and

• \(r\) is the radius of the circle.

Generally the area of a rectangle is just the length of the two sides multiplied together. One of those sides you are calling \(h\), but what about the other side? The remaining side of the rectangle is the one that wraps around the circle that makes up the end of the cylinder, so it needs to have a length that is the same as the circumference of the circle! That means the two sides of the rectangle are:

• \(h\); and

• \(2 \pi r\).

That gives you a lateral surface area formula of

\[ \text{Lateral surface area } = 2\pi r h.\]

Let's take a look at an example.

Now on to the total surface area!

Formula for the Surface Area of a Cylinder

A cylinder has different parts which means it has different surfaces; the ends have their surfaces and the rectangle has its surface. If you want to calculate the surface area of a cylinder, you need to find the area occupied by both the rectangle and the ends.

You already have a formula for the lateral surface area:

\[ \text{Lateral surface area } = 2\pi r h.\]

The ends of the cylinder are circles, and the formula for the area of a circle is

\[ \text{Area of a circle } = \pi r^2.\]

But there are two ends to the cylinder, so the total area of the ends is given by the formula

\[ \text{Area of cylinder ends } = 2\pi r^2.\]

The surface area occupied by both the rectangle part and the ends is called the total surface area. Putting together the formulas above gives you the total surface area of a cylinder formula

\[\text{Total surface area of cylinder } = 2 \pi r h + 2\pi r^2.\]

Sometimes you will see this written as

\[\text{Total surface area of cylinder } = 2 \pi r (h +r) .\]

Calculations for the Surface Area of Cylinders

Let's take a look at a quick example that uses the formula you found in the previous section.

Let's take a look at another example.

What happens if you cut a cylinder in half?

Surface Area of a Half Cylinder

You have learned about the surface area of a cylinder, but let's see what happens when the cylinder is cut in half lengthwise.

The figure below shows what a half-cylinder looks like.

Fig. 4. A Half Cylinder.

When you hear the word 'half' in mathematics, you think about something divided by two. So, finding the surface area and the total surface area of a half cylinder involves dividing the formulas for a right cylinder (a complete cylinder) by two. That gives you

\[\text{Surface area of half cylinder } = \pi r (h +r) .\]

Let's take a look at an example.

Surface Area of a Capped Half Cylinder

With the surface area of a capped half cylinder, it is more than just dividing by two. There is something else you have to consider. Remember the cylinder you are dealing with is not complete, in other words it certainly wouldn't hold water! You can cap it by adding a rectangular section over the cut part. Let's take a look at a picture.

Fig. 6. Showing the rectangle surface of a half cylinder.

You just need the area of that rectangle surface you capped the cylinder with. You can see it has the same height as the actual cylinder, so you just need the other side. It turns out that is the diameter of the circle, which is the same as twice the radius! So

\[ \begin{align} \text{Surface area of capped half cylinder } &= \text{Surface area of half cylinder } \\ &\quad + \text{Area of rectangle cap} \\ &= \pi r (h +r) + 2rh.\end{align}\]

Let's take a look at an example.