# How does the circumference of the top + bottom sides of a cylinder effect our calculations when working out the surface area?

I was watching a video tutorial on khan academy, (I've included the link at the bottom), and the question states that there is a 8cm cylinder, with a radius of 4. Part of the video shows a worked example of finding the surface area (the link cuts directly to this part of the video btw).

He calculated the surface area of the top + bottom sides, and then calculated the circumference, however he multiplied it by the entire length of the cylinder.

Aren't the top + bottom sides not 1cm each though? as they are obviously part of cylinder. Each slice is presumably 1cm thick, with a circumference of a 16pi area. Aren't the top and bottom sides of the circle just one slice each?

Thus, would the surface area of the top + bottom not be 2cm, leaving you with the circumference (or wrapping) of the rest of the cylinder, that being the circumference of one slice which is now known, and multiplying that by 6?

If not, then wouldn't that mean that the circumference of a circle in not included as part of the area? Though if not, it seems strange that we would just exclude the perimeter of circle when calculating it's area. If so however, then why wouldn't we include the circumference as part of the top + bottom areas of the cylinder in our calculations? You see, either way, the logic seems to be self contradictory, and this is why am so puzzled.

Thanks in advance for any help!

Surface area = Area of (top+bottom)+Curved surface area

1. Area of (top+bottom)=$2\pi r^2$
2. Curved surface area:-

Area of this rectangle is $=l\cdot b$ , but here, $l$ is $2\pi r$ and $b$ is $h$. So the area becomes ,

$2\pi rh$

Total area $=2\pi r^2+2\pi rh$

• Hmmm, having seen this diagram I can actually see how the calculations work out. It's kind of like the circumference of the cylinders faces is very small, perhaps infinitely so, and is thus it's exclusion has no noticeable effect. – Jim Jam Jan 1 '15 at 10:46
• @user108262 Yes you are right it is actually infinitesimal. – Dheeraj Kumar Jan 1 '15 at 10:49

If you make a cylinder out of cardboard, you can do it with three pieces. Two are circles, and one is a rectangle.

The width of the rectangle is the height of the cylinder, and the length of the rectangle is the circumference of either of the cylinder's bases.

The two circles have the same radius as the cylinder. If the radius is $r$ and the height of the cylinder is $h$, then the circumference of the circles that form the "top and bottom" sides is $2\pi r$, so the area of the rectangle is $2\pi r h$. This gives the entire surface area as

$$2\pi rh + 2\pi r^2$$

For your question, we have $r=4\,\text{cm}$ and $h=8\,\text{cm}$. This means the surface area of your cylinder is

$$2\pi(4)(8)+2\pi(4)^2\,\,\text{cm}^2 = \boxed{96\pi\,\,\text{cm}^2}$$

• Yes, but when calculating the area of one of the top or bottom, isn't the circumference part of the area? – Jim Jam Jan 1 '15 at 10:42