Surface area of a cylinder

To derive the formula of the surface area of a cylinder, we will start by showing you how you can make a cylinder. Start with a rectangle and two circles.

Template of a cylinder

Then, fold the rectangle until you make an open cylinder with it. An open cylinder is a cylinder that has no bases. A good real life example of an open cylinder is a pipe used to flow water if you have seen one before.

Folded rectangle

Next, using the two circles as bases for the cylinder, put one on top of the cylinder and put one beneath it.

Two circle and a folded rectangle will make a cylinder

Of course, the two circles will have the exact same size or the same diameter as the circles obtained by folding the rectangle.

Finally, you end up with your cylinder!

Cylinder made with 2 circlea and a folded rectangle

Now, what did we go through so much trouble? Well if you can make the cylinder with the rectangle and the two circles, you can use them to derive the surface area of the cylinder. Does that make sense?

The area of the two circles is straightforward. The area of one circle is pi × r2, so for two circles, you get 2 × pi × r2

To find the area of the rectangle is a little bit tricky and subtle!

Let us take a closer look at our rectangle again.

Cylinder template
Thus, the longest side or folded side of the rectangle must be equal to 2 × pi × r, which is the circumference of the circle.

To get the area of the rectangle, multiply h by 2 × pi × r and that is equal to 2 × pi × r × h

Therefore, the total surface area of the cylinder, call it SA is:

SA = 2 × pi × r2  +   2 × pi × r × h

A couple of examples showing how to find the surface area of a cylinder.

Example #1:

Find the surface area of a cylinder with a radius of 2 cm, and a height of 1 cm

SA = 2 × pi × r2  +   2 × pi × r × h

SA = 2 × 3.14 × 22  +   2 × 3.14 × 2 × 1

SA = 6.28 × 4  +   6.28 × 2

SA = 25.12 + 12.56

Surface area = 37.68 cm2

Example #2:

Find the surface area of a cylinder with a radius of 4 cm, and a height of 3 cm

SA = 2 × pi × r2  +   2 × pi × r × h

SA = 2 × 3.14 × 42  +   2 × 3.14 × 4 × 3

SA = 6.28 × 16  +   6.28 × 12

SA = 100.48 + 75.36

Surface area = 175.84 cm2

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