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
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
Next, using the two circles as bases for the cylinder, put one on top of the cylinder and put one beneath it.
Of course, the two circles will have the exact same size or the same diameter as the circles obtained by folding the rectangle
You
Finally, you end up with your cylinder!
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 × r
^{2}, so for two circles, you get 2 × pi × r
^{2}
To find the area of the rectangle is a little bit tricky and subtle!
Let us take a closer look at our rectangle again.
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 × r
^{2} + 2 × pi × r × h
Example #1:
Find the surface area of a cylinder with a radius of 2 cm, and a height of 1 cm
SA = 2 × pi × r
^{2} + 2 × pi × r × h
SA = 2 × 3.14 × 2
^{2} + 2 × 3.14 × 2 × 1
SA = 6.28 × 4 + 6.28 × 2
SA = 25.12 + 12.56
Surface area = 37.68 cm
^{2}
Example #2:
Find the surface area of a cylinder with a radius of 4 cm, and a height of 3 cm
SA = 2 × pi × r
^{2} + 2 × pi × r × h
SA = 2 × 3.14 × 4
^{2} + 2 × 3.14 × 4 × 3
SA = 6.28 × 16 + 6.28 × 12
SA = 100.48 + 75.36
Surface area = 175.84 cm
^{2}

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