Lateral & total surface area of cylinders | Grade 8 (TX) | Khan Academy

We're told the dimensions of a 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! Let'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 going to say bottom or base, and the top, right? The total surface area will include the base and the top.

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.

The perimeter of the circle is 2 pi times its radius. So we could say, in the case of a cylinder, the lateral surface area is going to be 2 pi r times the height. Let me do height in a different color, so times the height.

Now, before I even calculate this, let's just make sure that that makes sense. How 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 the cylinder right over here, and then you were to open it up and flatten it out. What would it look like?

Well, 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. Let 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. But now, let's just figure out what these things are. We know that our radius in this situation is four, 4 cm, and we know that our height is 11.

So we can just calculate this. It's going to be equal to 2 * 4 * 11, which is 88. So it's going to be 88 pi, and we are dealing with square cm.

Now 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 2 pi r h, and then it's going to be plus the area of the base and the top.

Well, what's the area of the base? Well, the area of a circle is pi * the radius squared, or pi r², 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: 2 pi r². It looks like a fancy formula, but all it is is what we just did plus the area of the top and the bottom.

Well, we also just figured out that this is 88 pi. So now we just have to figure out what 2 pi * r² is. We know that r is equal to 4, so this is going to be 4 squared. Let me actually, let me not try to skip too many steps.

So the total surface area is going to be 88 pi plus 2 * 4². This is going to be 16, so 2 * 16 pi, which is 32 pi.

And 88 pi + 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 — let me write the units there just for good measure — 120 pi square cm.