Volume of cylinders, spheres, and cones
Current time:0:00Total duration:8:07
Let's find the volume of a few more solid figures and then if we have time, we might be able to do some surface area problems. So let me draw a cylinder over here. So that is the top of my cylinder. And then this is the height of my cylinder. This is the bottom right over here. If this was transparent, maybe you could see the back side of the cylinder. So you could imagine this kind of looks like a soda can. And let's say that the height of my cylinder, h, is equal to 8. I'll give it some units. 8 centimeters, that is my height. And then let's say that the radius of one of the top of my cylinder of my soda can, let's say that this radius over here is equal to 4 centimeters. So what is the volume here? What is the volume going to be? And the idea here is really the exact same thing that we saw in some of the previous problems. If you can find the surface area of one side and then figure out how deep it goes, you'll be able to figure out the volume. So what we're going to do here is figure out the surface area of the top of this cylinder, or the top of the soda can. And then we're going to multiply it by its height. And that'll give us a volume. This will tell us essentially, how many square centimeters fit in this top. And then if we multiply that by how many centimeters we go down, then that'll give us the number of cubic centimeters in this cylinder or soda can. So how do we figure out this area up here? Well the area at the top, this is just finding the area of a circle. You could imagine drawing it like this. If we were to just look at it straight on, that's a circle with a radius of 4 centimeters. The area of a circle with a radius 4 centimeters, area is equal to pi r squared. So it's going to be pi times the radius squared, times 4 centimeters squared, which is equal to 4 squared, is 16, times pi. And our units now are going to be centimeters squared. Or another way to think of these are square centimeters. So that's the area. The volume is going to be this area times the height. So the volume is going to be equal to 16 pi centimeters squared, times the height, times 8 centimeters. And so, when you do multiplication, the associative property, you can kind of rearrange these things, and the commutative property. It doesn't matter what order you do it if it's all multiplication. So this is the same thing as 16 times 8. Let's see. 8 times 8 is 64. 16 times 8 is twice that. So it's going to be 128 pi. Now you have centimeter squared times centimeters. So that gives us centimeters cubed. Or 128 pi cubic centimeters. Remember, pi is just a number. We write it as pi, because it's kind of a crazy, irrational number, that if you were to write it, you could never completely write pi. 3.14159 keeps going on, never a repeat. So we just leave it as pi. But if you wanted to figure it out, you can get a calculator. And this would be 3.14 roughly, times 128. So it would be close to 400 cubic centimeters. Now, how would we find the surface area? How would we find the surface area of this figure over here? Well, part of the surface area of the two surfaces, the top and the bottom. So that would be part of the surface area. And then the bottom over here would also be part of the surface area. So if we're trying to find the surface area, it's definitely going to have both of these areas here. So it's going to have the 16 pi centimeters squared twice. This is 16 pi. This is 16 pi square centimeters. So it's going to have 2 times 16 pi centimeters squared. I'll keep the units still. So that covers the top and the bottom of our soda can. And now we have to figure out the surface area of this thing that goes around. And the way I imagine it is, imagine if you're trying to wrap this thing with wrapping paper. So let me just draw a little dotted line here. So imagine if you were to cut it just like that. Cut the side of the soda can. And if you were to unwind this thing that goes around it, what would you have. Well, you would have something. You would end up with a sheet of paper where this length right over here is the same thing as this length over here. And then it would be completely unwound. And then these two ends-- let me do it in magenta-- these two ends used to touch each other. And-- I'm going to do it in a color that I haven't used yet, I'll do it in pink-- these two ends used to touch each other when it was all rolled together. And they used to touch each other right over there. So the length of this side and that side is going to be the same thing as the height of my cylinder. So this is going to be 8 centimeters. And then this over here is also going to be 8 centimeters. And so the question we need to ask ourselves is, what is going to be this dimension right over here. And remember, that dimension is essentially, how far did we go around the cylinder. Well, if you think about it, that's going to be the exact same thing as the circumference of either the top or the bottom of the cylinder. So what is the circumference? The circumference of this circle right over here, which is the same thing as the circumference of that circle over there, it is 2 times the radius times pi. Or 2 pi times the radius. 2 pi times 4 centimeters, which is equal to 8 pi centimeters. So this distance right over here is the circumference of either the top or the bottom of the cylinder. It's going to be 8 pi centimeters. So if you want to find the surface area of just the wrapping, just the part that goes around the cylinder, not the top or the bottom, when you unwind it, it's going to look like this rectangle. And so its area, the area of just that part, is going to be equal to 8 centimeters times 8 pi centimeters. So let me do it this way. It's going to be 8 centimeters times 8 pi centimeters. And that's equal to 64 pi. 8 times 8 is 64. You have your pi centimeters squared. So when you want the surface area of the whole thing, you have the top, you have the bottom, we already threw those there. And then you want to find the area of the thing around. We just figured that out. So it's going to be plus 64 pi centimeters squared. And now we just have to calculate it. So this gives us 2 times 16 pi, is going to be equal to 32. That is 32 pi centimeters squared, plus 64 pi. Let me scroll over to the right a little bit. Plus 64 pi centimeters squared. And then 32 plus 64 is 96 pi centimeters squared. So it's equal to 96 pi square centimeters, which is going to be a little bit over 300 square centimeters. And notice, when we did surface area, we got our answer in terms of square centimeters. That makes sense, because surface area, it's a two-dimensional measurement. Think about how many square centimeters can we fit on the surface of the cylinder. When we did the volume, we got centimeters cubed, or cubic centimeters. And that's because we're trying to calculate how many one by one by one centimeter cubes can we fit inside of this structure. And so that's why it's cubic centimeters. Anyway, hopefully that clarifies things up a little bit.