Surface area of a cone

0:00- [Instructor] What we're gonna do in this video is first derive the formula for the surface area of a cone, and then we're gonna apply it to this problem right over here, to this particular cone, that has a height of eight centimeters. We're gonna figure out the lateral surface area and the total surface area.

0:16So first, let's assign some variables, so we could come up with a general formula and then I'll put these numbers here. So this eight centimeters, we already say that this is the height of this cone, and it's just rearranged a little bit differently. We normally don't see cones sideways like this.

0:29Sometimes you see it up down, but this is still a cone. And then, the seven centimeters, this is the radius of it's base, so let's call that equal to r. And to help us visualize at least first the lateral surface area, so that'd be the surface area of the sides of the cone, let's imagine cutting this cone right over here.

0:49And let's just imagine there was no base, and some of you might have done this with a piece of paper. If you do that, and if you were to spread it out and flatten it, what would it look like? Well, you would have part of a circle. It would look kind of like Pac-Man.

1:04So, it would look something like this. It would look something like this. I'm trying to draw a part of a circle, trying to make it as circular as possible. Now what do we know about this? Well, if we wanted to find this length that I'm highlighting in purple right over here, I guess you could say this part of its perimeter or this part of the entire circle's circumference, what would that length be?

1:30Well, that would be this radius, the radius of the base. It makes sense. I have no more edge of that piece of paper. It'd be that radius right there. So what is that going to be? Well, we know what the circumference of this base is. It is two pi r, so this length right over here is going to be two pi r as well.

1:51Now, what is going to be the entire circumference, if I were to include, if I were to include this part here? Well, to know that I would have to figure out the radius of this circle. Now, what is the radius of this circle? Well, that radius is going to be this slant height.

2:09Essentially, how long did I have to make this cut? It would also be the same thing as this length right over here. Now, how do I figure out that slant height? Well, I could use the Pythagorean theorem. I know that this entire length right over here is h. I know that this base, I guess I could say, or this other side, is r.

2:26This is a right triangle, and so I could use L for our slant height. We know that L squared is going to be equal to r squared plus h squared by the Pythagorean theorem, or that L is equal to the square root of r squared plus h squared. Now, L, as we just imagined, if we were to cut it out, that would be this radius.

2:50That would be that. So what would be the area of, let me do this in a new color, of just this part, just this part that I'm kind of shading in in the salmon color? Well, the area of the entire circle would be pi times the radius of the entire circle squared, so that would be pi L squared.

3:13But we aren't thinking about the entire circle. We're thinking about the part of it that forms the Pac-Man, and so what would that be? Well, that would be the ratio of this pink circumference, or this pink parameter, I guess I could say, the two pi r. So it would be times two pi r over the entire circumference of the entire thing.

3:36So if you were to go all the way, if we were to go all the way around, that circumference is going to be two pi times L. I wanted to really think about what this is, pi L squared, that would be this entire area. But if we were looking at that, then we wouldn't have a cone.

3:56There wouldn't be, if we didn't have this cutout here, you wouldn't be able to pop the top up. We always know that this radius of the base, to form the cone, it has to have a smaller value than this slant height. And so we know that this purple part of our entire circumference is not gonna be the entire circumference.

4:17And so the area that we're thinking about right now, it's the ratio of the purple part, the purple circumference or part of the circumference, to the entire circumference of this circle. And if we look at that, this is going to get us, this is this L squared. One of those Ls will cancel out right over there.

4:35The two pis cancel out. We have pi times our slant height times our radius. Or another way to think about it, because this is going to be equal to pi, we know our slant height, or I'll write pi times our radius and our slant height. I've just switched the order here.

4:52The slant height is r squared plus h squared. Now in this particular example, this would be pi times seven times the square root of 49, that's seven squared, plus eight squared plus 64, which is equal to seven, let me write it this way. Seven times the square root of 113 pi, and if I wanted to write that, well, I'll just leave it there and let me get the units there.

5:23That would be square centimeters. You could also put that into a calculator, and if they said round to the nearest unit, you could round. But this is a more exact answer of the lateral surface area, just if we were to go around the side of the cone. And then if you wanted the total surface area, you just have to add the area of the base, and we've done that for many, many years.

5:43The area of this base is just pi r squared. So if you wanted the total surface area of a cone, it would be the base, pi r squared, plus the lateral surface area, I could use this formula or that top one, pi r times the square root of r squared plus h squared. In this case, r squared is 49, so it'd be 49 pi plus all this business which we already figured out, plus seven times the square root of 113 pi square centimeters.