
Derivation of the surface area of a cylinder
Try this
Drag the orange dot to the left to "unroll" the cylinder.
The surface area of a cylinder can be found by breaking it down into three parts:
 The two circles that make up the ends of the cylinder.
 The side of the cylinder, which when "unrolled" is a rectangle
In the figure above, drag the orange dot to the left as far as it will go.
You can see that the cylinder is made up of two circular disks and a rectangle that is like the label unrolled off a soup can.

The area of each end disk can be found from the
radius r of the circle.
The area of a circle is πr^{2},
so the combined area of the two disks is twice that, or2πr^{2}.
(See Area of a circle).

The
area of the rectangle is the width times height.
The width is the height h of the cylinder, and the length is the distance around the end circles.
This is the
circumference of the circle
and is 2πr. Thus the rectangle's area is 2πr × h.
Combining these parts we get the final formula:

where:
π is Pi, approximately 3.142
r is the radius of the cylinder
h height of the cylinder


By factoring 2πr from each term we can simplify this to
However, the first is the one shown in most textbooks and more clearly shows how it is derived.
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