Source: All images created by Anthony Varela
Hello and welcome. My name is Anthony Varela. And today, we're going to talk about volume. So first I'll define what volume is. Then we'll look at formulas for the volumes of different shapes. And then we'll use those formulas to solve problems involving volume.
So first, what is volume? Well, Volume is the space enclosed in three dimensional shapes. So let's take a look at the soup can. It's a cylinder.
So when we're talking about volume, we're talking about all the space inside this cylinder. So at least in terms of liquids, we can also think about volume as capacity. How much can it hold? So volume is the space enclosed in three dimensional objects.
Now we measure volume using cubic units. So two examples of cubic units would be cubic centimeters or cubic feet. Those are just two examples.
So now what about formulas that we can use to calculate volumes? Well, the volume of a rectangular prism is the length times width times height. The volume for a cylinder is pi r squared times h, where r is the radius of the circular base, and h is the height of the cylinder.
And the volume of a sphere is 4/3 pi r cubed where r is the radius of the sphere. Now looking at our formulas for the volumes of a cylinder in a sphere, we see pi. Now remember, what is pi? Pi is the ratio between a circle's circumference and diameter. And it's approximately 3.14.
All right, so let's take a look at this soup can. And I know a couple of things about this cylinder. I know that the radius of the circular base is 1.25 inches, and the height is 4 inches. How can I find the volume of this cylinder? Well, let's bring out our formula, pi r squared times h. I know what r is, and I know what h is. So let's plug those in.
So first, I need to square that radius. So 1.25 inches squared. That equates then to 1.5625 squared inches. And then I'm just bringing down everything else in my equation.
Now, what I want to do is multiply the radius squared by the height of 4 inches. So my volume is 6.25 pi cubic inches. Remember that's our unit for volume. Now to make more sense about what 6.25 pi is, I can multiply 6.25 by 3.14. So this volume is about 19.63 cubic inches.
Let's take a look at this box. And this box is a rectangular prism, and it has the following dimensions-- 2 feet by 3.5 feet by 2 feet. So to calculate the volume of this rectangular prism, I'm going to use v equals lwh, and where the length is 3.5 feet; the width is two feet; and the height is 2 feet.
So let's plug those values in. We just need to multiply these numbers, but we also need to multiply the units. So our volume is 14 cubic feet. All right, well now let's take a look at a sphere. And this is a scale model of the sun. And in this model, the radius of this sphere is 7 inches. So what's the volume then of this sphere?
So let's bring out our formula, 4/3 pi r cubed. And I know what the radius is, r. So let's plug that in. Well, now I know I need to cube that radius. So 7 inches cubed would be 343 cubic inches. And then I'm just bringing down 4/3 pi.
So I can multiply 343 by 4/3 pi. So it doesn't matter if you can multiply it by pi first and then 4/3. You can multiply it by 4/3 and then pi-- doesn't matter. So this volume then is approximately 1,436.03 cubic inches.
All right, so now let's use volume in a different way. We're going to go through those same examples with the soup can, the box, and the model of the sun. But we know what the volume is. And we're going to find a different measurement.
So here I know the volume of a soup can is 48 cubic inches. I know the height is 5 inches. How can I find the radius of the circular base? Well, let's bring on our formula. Except this time, I'm going to plug-in 48 cubic inches. And I'll plug-in 5 inches here for my height. I need to find the radius.
So what I'm going to do is divide both sides by 5 inches. So dividing this side by 5 inches, I just have pi times my radius squared. Dividing the volume by 5 inches, I have 48 divided by 5, which is 9.6. Inches cubed divided by inches is inches squared.
All right, so now you need to divide both sides by pi. So I can just get my radius squared on one side. So dividing 9.6 by pi, I get 3.06. And now to get radius by itself, I need to take the square root of both sides of my equation.
So taking the square root of 3.06 inches squared, I get 1.75 inches. And that is my radius. So my radius is measured in just regular old linear inches.
Now let's return to our rectangular prism. I know that the volume of this box is 27 cubic feet. And I know that the height is 3 feet, and the width is 2 feet, but I don't know the length. So let's go ahead and use our formula and plug in what we know.
We know that 27 cubic feet is 3 feet by 2 feet by some number of feet that we don't know. So I'm going to divide by one of these dimensions. So looks like I've divided by 3. So 27 cubic feet divided by 3 feet is 9 squared feet.
And then I'll divide by 2 feet. So 9 squared feet divided by 2 feet is 4.5 feet. And that is my unknown dimension. This length is 4 and 1/2 feet.
Now I'd like to talk about a special type of rectangular prism, and that as a cube. So in a cube, all of the sides are the same. So we have a different size box here. It's a cube. It's volume is 27 cubic feet. And we need to find the dimensions of this cube.
Now cube has all equal sides. So all I need to do is take my 27 cubic feet and take the cubed root of that. So the cubed root of 27 cubic feet equals 3 feet. And that is the side length to all sides of this cube.
All right, now let's return to our example with the sphere. And here, the volume of a sphere is 523 cubic inches. I need to find the radius of this sphere. So let's bring out our formula and plug in what we know.
I know that the volume is 523 cubic inches. Now this is probably the trickiest example of this entire tutorial. What I need to do is solve for r, but there's quite a bit happening around r. So what I want to do is get rid of the 4/3, so I'm going to multiply both sides by 3/4. That'll cancel out the 4 and cancel the 3 here in that fraction.
So on this side of the equation then, I just multiply 523 cubic inches by 3/4. So I have 392.25 cubic inches equals pi r cubed. Well, before I can take any cube root of anything, I need to divide both sides by pi.
So 392.25 divided by pi is 124.92, still dealing in cubic inches. This equals my radius cubed. So now what I need to do is take the square root of both sides in my equation.
The square roots-- or sorry-- the cubed roots of both sides of my equation, the cubed root of r cubed will just bring me my radius r. And so the cubed root of 124.92 cubic inches is 5 inches. So the radius of this sphere is 5 inches.
All right, so what did we talk about today? We talked about volume being the space enclosed in three dimensional objects always measured in cubic units. We talked about formulas for different three dimensional shapes-- rectangular prism, cylinder, and sphere. And remember, pi is the ratio of a circle circumference to its diameter, and it's approximately 3.14. Well, thanks for watching this video on volume. Hope to catch you next time.