Lateral & total surface area of cylinders

0:00- We're told the dimensions of the cylinder are shown in the diagram. Fair enough. What is the lateral surface area of the cylinder, and what is the total surface area of the cylinder? Pause this video and have a go at this before we do this together. All right, now let's do this together.

0:16Let's first focus on the lateral surface area. Lateral refers to the sides of something. So we're really thinking about the sides of the cylinder. We're going to think about the area without the base, I was gonna say bottom, or base, and the top right over here.

0:30The total surface area will include the base and the top, and so how will we do that? Well you might have seen formulas like lateral surface area is equal to perimeter of the base times the height, and what is the perimeter of the base in this situation? Well the perimeter of the base in this situation is the perimeter of this circle.

0:51A perimeter of the circle is two pi times its radius. So we can say in the case of a cylinder, the lateral surface area is going to be two pie R times the height, and let me do the height in a different color. So times the height. Now before I even calculate this, let's just make sure that that makes sense.

1:13How does this formula actually work? I'm a big fan of not just memorizing formulas, but really thinking about why it's intuitive. Well, imagine if you were to cut this roll or this cylinder right over here and then you were to open it up and flatten it out, what would it look like?

1:29Well, the perimeter of the base would then be stretched out like this, and then you would have its height like this, and you essentially now have a rectangle. And how would you find the area of a rectangle? Well, you would multiply the base times the height. This right over here, the base that is the perimeter...

1:51Let me do this in the same color. This is the perimeter of the base and then the height is the same height right over here. So we're just finding the area of this rectangle when we take the perimeter of the base times the height, hopefully something you're familiar with.

2:06But now let's just figure out what these things are. We know that our radius in this situation is four centimeters, and we know that our height is 11 and so we can just calculate this is going to be equal to two times four times 11 is 88, so it's going to be 88 pi and we are dealing with square centimeters.

2:30Now let's think about the total surface area of the cylinder. So once again, you might see a formula like this, the total surface area is going to be equal to the lateral surface area, which we've already talked about is two pi RH and then it's going to be plus the area of the base and the top here.

2:47Well what's the area of the base? Well, the area of a circle is pi times the radius squared. Pi R squared, and we have not just one base, but we have that area again right over here, so that's why we have two of them. Two pi R squared. It looks like a fancy formula, but all it is is what we just did, plus the area of the top and the bottom.

3:11Well we also just figured out that this is 88 pi, so now we just have to figure out what two pi times R squared is. We know that R is equal to four. So this is going to be four squared. Actually let me try not to skip too many steps, so the total surface area's going to be 88 pi plus two times four squared, which is going to be 16 times pi.

3:40This is 32 pi and 88 pi plus 32 pi... Let's see, we're going to use 12 of those pi to get to 100 pi, so this is 120 pi. 120 pi and we're done. If they actually want us to calculate this we might use a calculator of sorts, but this is the... Let me write the units there just for good measure.

4:02120 pi square centimeters.