This is my way of , how to derive total surface area of a cylinder,

or in instructables contest rules the area of a cylinder.

Answer to Burning Questions round 6.5, also shows you in a short video

the circumference of a circle and the meaning of pi.

## Step 1: The Formula

This is my way of , how to derive total surface area of a cylinder.

If you look at textbooks, or formula sheets on an exam you will come across this formula.

A =2 pi r ( r + h ), where Pi is the ratio of the circumference of a circle to it's radius

## Step 2: Splitting Up the Shapes

Surface area is the area all around the shape, not to be confused with volume.

We start by splitting the shape up, imagine you have to make a cylinder out of paper.

What shapes would you use to make a 3D cylinder?

2 circles and a rectangle!

Still don't know how these shapes fit, get some paper, tape and scissors, try it!

## Step 3: Area of the Circles

Now the area of the 2 circles. The formula is;

pi x r^{2 }

So in simple terms, (special ratio number) multiplied by the radius

of the circle that is squared.

Squared means the number multiplied by itself

*hehe, it is also the anti derivative of perimeter of a circle...co-incidence or is it?

For curious people the next step is a simple lesson on pi (video no reading, yeah!)

Radius, the point from the middle of circle to the outside

## Step 4: Origins of Pi, and Circumference

Yes, I made the video. It is *FLV format same as you tube videos in your cache, use "windows media player classic" or "totem" on Linux platform

## Step 5: Area of the Rectangles

Now the area of the rectangle. The formula is;

Length multiply by width,

In this case the length of the rectangle is the height, and the width is the circumference.

If you do not know how the circumference is the width, refer to your paper model

and watch my video on the previous step.

The rectangle wraps around the outside of the circle

Formula = h x ( 2 x pi x r)

## Step 6: Adding It All Up

Almost there!, now finding the total Area.

So the cylinder is made up of 2 circles and 1 rectangle lets add them up.

Re call: area of 1 circle = pi x r^{2 }

area of 1 rectangle = h x 2 x pi x r

remember there are 2 circles so adding up is:

Area = pi x r^{2} + Pi x r^{2} + h x 2 x pi x r

circle + circle + rectangle

Simplify: 2 x pi x r^{2} + 2 x pi x r

Then take out the 2 x pi x r as a common factor:

Area = 2 x pi x r x (r + h)

## Step 7: Things You Should Know

Now you have your answer just measure and enter your values of radius and height

and you will find your area of the cylinder.

Things to remember:

- If you find this formula difficult for now, you can always find the areas of the

- radius is the distance from the midpoint of a circle to the outside (not to be confused

- This is the Total surface area not Volume
- pi is more commonly written like it was shown in the video
- pi is irrational (it has many many decimal places), 3.14 is a very basic estimate, for

## Step 8: Practical Uses

Practical uses:

The area of my can;

radius = 3.5 cm

height = 10.5 cm

Area = 2 x Pi x r x (r + h)

= 2 x 3.14 x 3.5 x (3.5 + 10.5)

(solve brackets first)

Area = 307.72 cm2

(remember your units)

Other uses:

*Surface area of a piston in an engine to find how much friction there is.

*Your a product designer and you want to know the amount of materials needed

for a cylindrical cup holder.

*Enter in this competition, Vote for me!

## Discussions