Volume and surface area
Let's think a little bit about the volume of a cone. So a cone would have a circular base, or I guess depends on how you want to draw it. If you think of like a conical hat of some kind, it would have a circle as a base. And it would come to some point. So it looks something like that. You could consider this to be a cone, just like that. Or you could make it upside down if you're thinking of an ice cream cone. So it might look something like that. That's the top of it. And then it comes down like this. This also is those disposable cups of water you might see at some water coolers. And the important things that we need to think about when we want to know what the volume of a cone is we definitely want to know the radius of the base. So that's the radius of the base. Or here is the radius of the top part. You definitely want to know that radius. And you want to know the height of the cone. So let's call that h. I'll write over here. You could call this distance right over here h. And the formula for the volume of a cone-- and it's interesting, because it's close to the formula for the volume of a cylinder in a very clean way, which is somewhat surprising. And that's what's neat about a lot of this three-dimensional geometry is that it's not as messy as you would think it would be. It is the area of the base. Well, what's the area of the base? The area of the base is going to be pi r squared. It's going to be pi r squared times the height. And if you just multiplied the height times pi r squared, that would give you the volume of an entire cylinder that looks something like that. So this would give you this entire volume of the figure that looks like this, where its center of the top is the tip right over here. So if I just left it as pi r squared h or h times pi r squared, it's the volume of this entire can, this entire cylinder. But if you just want the cone, it's 1/3 of that. It is 1/3 of that. And that's what I mean when I say it's surprisingly clean that this cone right over here is 1/3 the volume of this cylinder that is essentially-- you could think of this cylinder as bounding around it. Or if you wanted to rewrite this, you could write this as 1/3 times pi or pi/3 times hr squared. However you want to view it. The easy way I remember it? For me, the volume of a cylinder is very intuitive. You take the area of the base. And then you multiply that times the height. And so the volume of a cone is just 1/3 of that. It's just 1/3 the volume of the bounding cylinder is one way to think about it. But let's just apply these numbers, just to make sure that it makes sense to us. So let's say that this is some type of a conical glass, the types that you might see at the water cooler. And let's say that we're told that it holds 131 cubic centimeters of water. And let's say that we're told that its height right over here-- I want to do that in a different color. We're told that the height of this cone is 5 centimeters. And so given that, what is roughly the radius of the top of this glass? Let's just say to the nearest 10th of a centimeter. Well, we just once again have to apply the formula. The volume, which is 131 cubic centimeters, is going to be equal to 1/3 times pi times the height, which is 5 centimeters, times the radius squared. If we wanted to solve for the radius squared, we could just divide both sides by all of this business. And we would get radius squared is equal to 131 centimeters to the third-- or 131 cubic centimeters, I should say. You divide by 1/3. That's the same thing as multiplying by 3. And then, of course, you're going to divide by pi. And you're going to divide by 5 centimeters. Now let's see if we can clean this up. Centimeters will cancel out with one of these centimeters. So you'll just be left with square centimeters only in the numerator. And then to solve for r, we could take the square root of both sides. So we could say that r is going to be equal to the square root of-- 3 times 131 is 393 over 5 pi. So that's this part right over here. Once again, remember we can treat units just like algebraic quantities. The square root of centimeters squared-- well, that's just going to be centimeters, which is nice, because we want our units in centimeters. So let's get our calculator out to actually calculate this messy expression. Turn it on. Let's see. Square root of 393 divided by 5 times pi is equal to 5-- well, it's pretty close. So to the nearest, it's pretty much 5 centimeters. So our radius is approximately equal to 5 centimeters, at least in this example.