Surface area of cylinders formula | Grade 8 (TX) | Khan Academy

We're told the dimensions of a cylinder are shown in the diagram. Fair enough, so the radius of the base, which is going to be the same as the radius of the top, is 3 m. The height is 5 m. What is the lateral surface area of the cylinder? Pause this video and try to figure this out before we do this together, and it's okay if you use a calculator for this one.

All right, now let's do this together. So the first thing is, what do they even mean by lateral surface area? Well, when you're referring to the lateral of anything, you're referring to its side or it's related to the sides. So we're talking about the surface area if we're going around this cylinder, so all the way in the backside. But essentially, the total surface area without the base and without the top right over there.

So if you imagine, well, just imagine it without a top or a bottom. That's the surface area we care about. Now, you might have seen some different formulas here where lateral surface area is equal to the perimeter of the base times the height. In this situation, what is the perimeter of the base? When the base is a circle, the perimeter of a circle is pi times its diameter or Pi times 2 times its radius or 2 pi r.

So we're talking about a cylinder like this. This perimeter of the base is going to be the same thing as two pi r, and then we're going to multiply that times the height. Now, why does that make sense that we could take the perimeter of the base and multiply it by the height in order to get the lateral surface area? Well, imagine if we were to cut this cylinder right over here, and if we were to open it up, then it would look something like this.

It would look something like this, where the height here is still 5 m, and the perimeter of that base would now be stretched out like this. And so, then it makes sense, we're just taking the area of a rectangle in this situation: the perimeter of the base times the height. Perimeter of the base times the height.

So let's just do that right over here. So the perimeter of the base is going to be 2 pi. What's our radius? It is three. So, 2 times 3, and then what is our height? We've already talked about that; it is 5 m. So, times 5. So this is going to be equal to 2 times 3 times 5 times pi.

So it's 6 times 5, so that's 30 pi, which is the answer. But it looks like they want us to have a number without writing something times pi. So I could get a calculator out, and my calculator has a pi button on it, but they want us to use 3.14 for pi.

So let me just do 30 times 3.14, which is equal to, and they want us to round to the nearest tenth. I didn't even have to round here. Here, 94.2. So this is approximately 94.2, and we're now dealing with square meters, and we are done.

And if someone said, "What is the total?" well then you would add the area of the top and the bottom. Sometimes you might see a formula like this: the total surface area, especially if you're dealing with a cylinder. Well, then it's going to be 2 pi RH, or sometimes you might see that in different orders.

It might be, well, I'll just do this right over here, plus two times the area of the base because that'll take into account the base and the top. So the area of each of them is pi r squared. This might look complicated, but once again, this is the one we just dealt with, and then this is just adding the areas of the top and the bottom to it.

I like to always just reason through it without having to memorize formulas because you're probably going to forget these formulas. But it's really important to know where the formulas come from.