Area of a Cylinder

In this Instructable you will learn how to find the Area of a Cylinder.

Supplies:
Pencil (needed for it's erase-ability)
Paper
Calculator

Optional:
Can
Can opener
Tin Snips

Warning:
Cutting open a tin can leaves VERY SHARP EDGES. I wore leather gloves when flattening out my can, I suggest you do the same. Please do not cut yourself as this will lead to bleeding and 9 out of 10 doctors agree that bleeding outside of the body is bad.

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Area of a Cylinder
(Surface Area of a Cylinder to be more correct)

This is a concept that is sometimes easier to understand if you can visualize what is happening.
To find the surface area of 3D object, you add of the areas of all the 2D pieces.
The 2D representation of a 3D object is called a Net

You can use an empty tin can to make the Net of a Cylinder.
(not all tin cans are created equal, you need one
with a lip on the top and bottom)

*wear leather gloves for this part to avoid cuts*

1) Use the can opener to open the top
2) Leave just a little bit uncut so that the top stays attached
3) Drain and rinse the can
4) Use the can opener to open the bottom
5) Use the tin snips to cut down the side of the can
6) Flatten the can by stretching open the sides
(I stepped on the can to flatten all the way)

Congratulations, you have just made a 2D Net of a 3D cylinder!

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Now that we can visualize it, let's work an example using 2 different methods.

The Example:
You have a Cylinder that has a:
Radius of 4cm on the base and a
Height of 8cm

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The first method we will use is a "piece-wise" method.
Essentially we will find the areas of the separate pieces and
add them together at the end.

We have to find the area of the:
Circles and
Rectangle

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We need to find the area of the Circular Bases

Area of a Circle is:
A = Pi * r ² ( which is read "a equals pi r squared")

and we have 2 circles (top and bottom)
so we have to double our area.

Thus we get

A = 2 (Pi * r ² )

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We plug in the radius from the example and
solve for the Area of the two bases

r = 4

So we get:

A = 2 ( Pi (4) ²)
A = 2 (Pi (16))
A = 2 (16 Pi)
A = 32 Pi
A = 32 * 3.14
A = 100.48 cm²

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Now we need to find the Area of the Rectangle in the middle.

The formula for Area of a Rectangle is:

A = L * W

In this example however Length of the Rectangle
is equal to the Circumference of the Circle
(remember how it wrapped around the circle when it was whole)

The formula for Circumference is:

C = 2 * Pi * r

So we replace the L in A = L * W with the Formula for Circumference and get:

A = 2 * Pi * r * W

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We plug in the radius and height from the example and
solve for the Area of the Rectangle

r = 4
h = 8

So we get:

A = 2 * Pi * r * h
A = 2 * Pi * 4 * 8
A = 2 * Pi * 32
A = 64 * Pi
A = 64 * 3.14
A = 200.96 cm²

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Now we add the pieces together.

Circles:
A = 100.48 cm²

Rectangle:
A = 200.96 cm²

Total Surface Area:
SA = Circles + Rectangle
SA = 100.48 + 200.96
SA = 301.44 cm²

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The second method is to use a single formula to cover everything in one piece

The Formula for Surface Area of a Cylinder is:
SA = 2 * Pi * r (h + r)

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Again, we are using the same example

r = 4
h = 8

Surface Area of a Cylinder:

SA = 2 * Pi * r (h + r)
SA = 2 * Pi * (4) (8 + 4)
SA = 2 * Pi * (4) (12)
SA = 2 * Pi * (48)
SA = 96 * Pi
SA = 96 * 3.14
SA = 301.44 cm² (WooHoo same answer as Method 1)

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