Develop a formula for finding the surface area of the outside of a cylinder

Answer 1

Let's begin by thinking about the shape of a cylinder. To "generate" a cylinder, you start with a circle. Then, you continue to stack as many circles on top as you want to reach a desired height. So far so good?

We begin with a circle, with area equal to π*r². As we stack on top, we also stack the total areas, which would end up giving us volume. For just the surface area, we only really care about the perimeter of each circle. Hence, we only care about the circumference.

So the next question becomes, how many circles do we stack? Well it depends on our height, h!

So, for a height h, we stack "h" circumferences. So, for the surface area of a cylinder, we take the perimeter/circumference of a circle (2*π*r) and multiply it by how many we want to stack, h.

Now your formula for the surface area of a cylinder becomes 2*π*r*h, and you've got the area of the entire tube. Finally, we add the area of the top and bottom faces -- which, we know have an individual area of π*r².

So, in summary, you sum all the circumferences of every circle that makes up the cylinder (the number of circles is h) by multiplying circumference by h. Then, you add the area of the top and bottom face.

Your total area becomes 2π*r*h + 2*π*r² :)