Today we're going to talk about volume. Volume is the amount of space that's enclosed in a three-dimensional object. So you might want to find the volume of a milk carton, so how much milk could that carton hold, or the volume of a swimming pool, so how much water would that pool hold?
And because we're measuring volume as space within a three-dimensional object, we use units that are cubed units. So for example, instead of measuring in feet for length or feet squared for area, we're measuring in feet cubed, which sometimes could be written as cubic feet, or "cu" to abbreviate cubic. Similarly, you could have inches cubed, which would be abbreviated as cubic inches.
So we're going to look at some formulas for finding the volume of a rectangular prism, a sphere, and a cylinder. And we'll do some examples using each of those formulas.
So let's do some examples with the volume of a rectangular prism. Our formula tells us that the volume is equal to the length times the width times the height of that rectangular prism. So for my first example, I have a length of 5 inches, a width of 10 inches, and a height of 15 inches. So I can substitute those values into my formula to find the volume.
So I have the volume is equal to 5 inches times 10 inches times 15 inches. Multiplying these values together, I find that my volume is equal to 750. And my units are going to be inches cubed. I'm multiplying three inch units together, so my units will be inches cubed.
So for my second example, I've got a rectangular prism with a length of 7 centimeters and a height of 4 centimeters and a volume of 140 centimeters cubed. So I can use my formula to find the width of this rectangular prism.
So substituting my values into my formula, I have my volume is 140 centimeters cubed. My length is 7 centimeters. My width I don't know. And my height is 4 centimeters.
So I'm going to start by simplifying by multiplying 4 centimeters and 7 centimeters. That's going to give me 28 centimeters squared. Bringing down my other values.
Then I'm going to divide both sides by 28 centimeters squared to isolate my variable from my width. This will cancel. And here this is going to simplify to 5 centimeters which is going to be the value of my width of this rectangular prism.
So for my third example, I've got a cube, which is a special type of rectangular prism where all of the side lengths are the same. So my length, width, and height are all the same value. And I know that my volume is 1,000 feet cubed.
Since I know that my side lengths are all the same, I'm going to introduce another variable, s, to show that the length, width, and height of this cube are all the same. And so we can use our formula to find the length of our cube, the side lengths of our cube.
So I'm going to substitute 1,000 feed cubed in for my volume. And I know that this is going to be equal to s to the third power. I'm multiplying my three side lengths together, which would give me s to the third.
To isolate my s variable, I'm going to use a cubed root to cancel out the 3 exponent. And I'm going to do that on both sides. So now I'll just have my s variable. And on the other side, I will have 10 feet, so I found that the side length of my cube is 10 feet.
So let's look at some problems involving the volume of cylinders. The formula tells us that the volume is equal to pi, which we'll approximate with 3.14, times our radius squared times the height. So here I see my height is 8 inches and my radius is 4 inches. So I can use my formula to find the volume. Substituting in those values, I've got pi times 4 inches squared times 8 inches.
I'm going to start by squaring my radius. So this will give me pi times 16 inches squared times 8 inches. And now multiplying my values together, 3.14 times 16 times 8 will give me a volume of approximately 401.92. And my units, inches squared times inches, will become inches cubed.
So for this example, I see the volume of my cylinder is 150 centimeters cubed and the height is 10 centimeters. So substituting those values into my formula, I'm going to leave my radius as r, because I don't know what that is, and 10 centimeters for my height. So to solve for my radius, I'm going to divide both sides by 10 centimeters.
Here it will cancel. And I'll be left with 15 centimeters squared. Dividing both sides by my approximation for pi, 3.14, I'll have 4.78 centimeters squared is approximately equal to r squared. And then taking the square root of both sides, when I take the square root I only need to consider my positive answer since I'm talking about a radius, which is a distance. This will give me 2.19 centimeters is approximately equal to my radius.
All right. Let's look at a couple of examples using the volume of a sphere. This formula tells us that the volume is equal to 4/3 times pi, which we will approximate with 3.14, times my radius cubed. So if I know for this example that my radius is 4 feet, I can use that to figure out my volume.
So my volume of this sphere is going to be equal to 4/3 times pi times my radius, 4 feet, to the third power. I'm going to start with my exponent by cubing the 4 feet. And that is going to give me 64 feet cubed times my pi times 4/3 is equal to my volume.
Now I can simply multiply 4/3 times my approximation for pi, 3.14, times 64 feet. And that is going to give me a volume of approximately 267.95. And my units are feet cubed.
All right. Let's do one more example using the volume of a sphere. So here I know the volume of this sphere is 450 inches cubed. So I can use this formula to solve for my radius.
So I'm going to substitute my value in for my volume, 450 inches cubed. That's equal to 4/3 times pi times r to the third. Now to cancel out my 4/3 fraction, I can multiply both sides by the inverse or the reciprocal of that fraction, 3 over 4. So now these two things will cancel.
3/4 times 450 is going to give me 337.5 inches cubed. And that's going to be equal to pi times my radius cubed. Now I'm going to divide both sides by 3.14, my approximation for pi. And that's going to give me 107.48 inches cubed is approximately my radius cubed.
Now I just need to cancel out my 3 exponent. I'll do that by taking the cubed root of both sides. So this is going to simplify to be approximately 4.75. My inches cubed will turn into just inches. And that will be approximately equal to my radius.
So let's go over our key points from today. Make sure you get these into your notes if you don't have them already so you can refer to them later.
So we talked about the fact that volume is the amount of space that's enclosed in a three-dimensional object. And because it's the amount of space in a three-dimensional object, it uses cubic units, such as feet cubed or centimeters cubed. And then we looked at the formulas for three different three-dimensional objects, volume of a rectangular prism, which includes a cube; the volume for a cylinder; and the volume for a sphere. So again, make sure you get these formulas into your notes if you don't have them already.
I hope that these key points and the examples that we did today helped you understand a little bit about calculating volume. Keep using your notes. And keep on practicing. And soon you'll be a pro. Thanks for watching.
