Video: Volumes of Rectangular Prisms and Cubes

In this video, we will learn how to calculate volumes of rectangular prisms and cubes given their dimensions and solve problems including real-life situations.

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Video Transcript

In this video, we’ll learn how to calculate volumes of rectangular prisms and cubes, given their dimensions, and solve problems, including real-life situations. We’ll consider what we actually mean by the word “volume” and how the properties of rectangular prisms and cubes can help us to derive and use a formula for their volume.

So, our first question is, what is a prism? A prism is a three-dimensional shape with a constant cross section. In other words, the cross section has the same shape and size throughout its length. A triangular prism, for example, has a triangular cross section. I could slice down here or down here, and the size and shape of that triangle would stay the same. Similarly, a cylinder has a constant cross section; it’s a circle. Now, we’re interested in rectangular prisms, like this, and cubes. A cube is simply a rectangular prism whose dimensions are all the same. We notice its faces — that’s the flat surfaces of the shape — are all squares.

Now, in this video, we’re learning how to calculate the volume of these shapes. The volume of a shape is a measure of its total three-dimensional space. And the easiest way to measure the three-dimensional space is to consider how many cubic units a shape contains. A cubic unit is simply a cube that measures one unit by one unit by one unit. Let’s have a look at an example of this.

Find the volume of the cuboid.

The volume of a cuboid or a rectangular prism is a measure of its three-dimensional space. We can work out the volume by considering how many unit cubes the shape contains. In this case, we see that our cuboid is split into cubes which measure one centimeter by one centimeter by one centimeter. That’s one cubic centimeter. And one method we have is to simply count them, beginning with the front of our shape. Here, we have one, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15 cubes. But is there a quicker way to calculate this? Well, yes! We have five rows, each containing three cubes. So, we could have simply multiplied five and three to see that there are 15 cubes on the front of our shape.

We now look at the depth of our shape. It’s two centimeters. This means we’ve essentially got two identical slices, each containing 15 cubes. The volume is, therefore, 15 multiplied by two, which is 30. Now, each cube is one centimeter cubed. So, the total volume of our cuboid is 30 cubic centimeters. But is there a quicker way we could have done this? Well, yes. Rather than counting cubes, we saw that we can multiply five by three, and then we multiply this by two.

In other words, we can multiply the width by the height by the length, and that’s the formula for the volume of a rectangular prism or a cuboid. It’s the product of its length, its width, and its height. In this case, that was five times three times two. But remember, multiplication is commutative. So, we can actually do this in any order and get the same answer of 30 cubic centimeters.

Let’s now have a look at how we can use this formula to solve problems involving the volume of rectangular prisms and cubes.

Which of the following describes how the volume of a rectangular prism is affected after doubling all three dimensions? Is it (A) 𝑉 sub new is equal to six times 𝑉 sub old? (B) 𝑉 new is equal to 𝑉 old squared. (C) 𝑉 new is equal to two times 𝑉 old. (D) 𝑉 new is equal to four times 𝑉 old. Or (E) 𝑉 new is equal to eight times 𝑉 old.

To answer this question, we’ll begin by recalling the formula for the volume of a rectangular prism. The volume 𝑉 of a rectangular prism is the product of its length, its width, and its height. So, let’s call the volume of our original shape 𝑉 old. It’s 𝑤𝑙ℎ. Now, of course, we could do this in any order. So, we could write 𝑙𝑤ℎ or any other combination. We’re now going to take our original rectangular prism and double all of its dimensions.

The height of our new shape is two times ℎ, which is two ℎ. The width is now two times 𝑤. That’s two 𝑤. And the length is two times 𝑙. That’s two 𝑙. And so, we can now calculate the volume of the new shape. It’s still the product of all of its dimensions, but this time that’s two 𝑤 times two 𝑙 times two ℎ. When we multiply algebraic expressions, such as this, we begin by multiplying the numbers. And so, two times two times two is eight. And the new volume is eight 𝑤𝑙ℎ.

We now compare the original volume to the new volume. And since the original volume is 𝑤𝑙ℎ and the new volume is eight times this, this must mean that the new volume is eight times the old volume. The correct answer, therefore, is (E) 𝑉 sub new is equal to eight times 𝑉 sub old.

We’ll now consider how we can use the formula for the volume of a rectangular prism within a real-world context.

A man needs to store 16,170 cubic centimeters of rice in a container. He has one box which is a cuboid with dimensions of 35 centimeters, 22 centimeters, and 21 centimeters and another box which is a cube with length 22 centimeters. Which box should he use?

Remember a measure of capacity or the amount of space a three-dimensional shape holds is volume. And the formula for the volume of a rectangular prism or a cuboid is width times height times length. Essentially, it’s the product of its three dimensions. Now, we can calculate this product in any order. We also have a cube in this question, but a cube is simply a cuboid whose dimensions are all equal. And so, the volume of a cube is simply the length of one of its edges cubed. So, we’ll begin by calculating the volume of each shape.

The dimensions of the cuboid are 35 centimeters, 22 centimeters, and 21 centimeters. So, the volume is 35 times 22 times 21, which is 16,170. Now, we’re multiplying centimeters by centimeters by centimeters. And so, the units here are cubic centimeters or centimeters cubed. Now, in fact, this is the exact same value as the amount of rice that the man needs to store. But let’s double-check with the volume of the cube.

This time, the length of each edge is 22 centimeters. And so, the volume of our cube is 22 times 22 times 22 or 22 cubed. That gives us a volume of 10,648 cubic centimeters. This is indeed going to be too small, so he should use the cuboid to store the rice.

In our next question, we’ll consider how to calculate the volume of a rectangular prism given the area of one of its faces.

Cuboid 𝐴 has dimensions of 56 centimeters, 40 centimeters, and 34 centimeters. Cuboid 𝐵 has a base area of 2,904 square centimeters and a height of 36 centimeters. Which cuboid is greater in volume?

We begin by recalling the formula for the volume of a rectangular prism or a cuboid. It’s the product of its three dimensions. We could write that as width times height times length in any order. And so, we can calculate the volume of cuboid 𝐴 fairly easily. It’s 56 multiplied by 40 multiplied by 34, which is 76,160. Our units are centimeters, so the units for the volume are cubic centimeters or centimeters cubed. But what about the volume of cuboid 𝐵?

Well, we’ll go back to our formula for the volume of a cuboid. And since multiplication is commutative, we know we can do it in any order. So, we can rewrite this as width times length multiplied by height. But, of course, width times length gives us the area of a rectangle. In this case, that’s the area of the base of our cuboid. And so, we can alternatively say that the volume of a cuboid is equal to the area of its base multiplied by its height, where the height is the side that’s perpendicular to the base.

We also sometimes say that the volume of a cuboid is equal to the area of its cross section multiplied by its length or its height. We can, therefore, calculate the volume of cuboid 𝐵 by multiplying 2,904 — that’s the area of its base — by its height; that’s 36. That gives us a value of 104,544 cubic centimeters. We can quite clearly see that 76,160 is less than 104,544, meaning that cuboid 𝐵 is greater in volume.

We’ll now have a look at how we can use information about the volume to solve problems in a real-world context.

Given that 405 cubic centimeters of water is poured into a rectangular-prism-shaped vessel with a square base whose side length is nine centimeters, find the height of water in the vessel.

In his question, we’ve been given some information about the volume of water being poured into a rectangular-prism-shaped vessel. This vessel has a square base with side length of nine centimeters. So let’s sketch this out. Here is this vessel. Now, we don’t know what the height of the water is in the vessel when it’s poured in. So, let’s call that ℎ centimeters. We do know that the amount of space this takes up in three dimensions is 405 cubic centimeters. And we also know that this is the volume. And the volume of a cuboid is equal to the area of its base multiplied by its perpendicular height.

Now, we can calculate the area of the base of our vessel. It’s simply a square. So, its area is nine multiplied by nine, which is 81. We’re working in centimeters. So, the area of the base of our vessel is 81 square centimeters. We, in fact, also know that the volume of our water is 405 cubic centimeters. And we’ve said that its height is equal to ℎ.

We can, therefore, form an equation in ℎ. We can say that 405, remember, that’s the volume, is equal to the area of the base, that’s 81, times ℎ or 405 equals 81ℎ. We want to solve for ℎ. So, we’re going to divide both sides of our equation by 81. That gives us ℎ is equal to five. And we can, therefore, say that the height of water in the vessel is five centimeters.

In our very final example, we’ll look at how we need to be extra careful when dealing with different units in our question.

Find the volume of the cuboid.

We recall that the volume of a cuboid or a rectangular prism is the product of its three dimensions. So, it’s its width multiplied by its length multiplied by its height. Now, we need to be extra careful here. We are given that the width of our cuboid is 0.5 meters and its length is four meters. Its height, however, is 85 centimeters. The units are different to the other dimensions. And so, before we calculate the volume, we’re going to ensure that all our units are the same.

Now, we could do this in one of two ways. We could convert all of our measurements to centimeters and give our volume in cubic centimeters or convert our measurements to meters and give our volume in cubic meters. We’ll have a look at both examples. We know that there are 100 centimeters in a meter. And so, to convert from meters to centimeters, we multiply by 100. 0.5 meters is 0.5 times 100, which is 50 centimeters. Similarly, four meters is four times 100, which is 400 centimeters. The volume, therefore, of our cuboid in cubic centimeters is 50 times 400 times 85, which is 1,700,000 cubic centimeters.

And now, let’s see what happens when we calculate the volume in cubic meters. This time, to change from centimeters into meters, we’re going to divide by 100. So, 85 centimeters is 85 divided by 100, which is 0.85 meters. In cubic meters then, our volume is 0.5 times four times 0.85, which gives us a volume of 1.7 cubic meters. And so, we’ve calculated the volume of our cuboid in both cubic centimeters and cubic meters.

Now, a common mistake is to think that to convert between cubic centimeters and cubic meters, we multiply or divide by 100. We can see quite clearly that that’s not the case here. In fact, to convert from cubic centimeters to cubic meters, we divide by 100 cubed. And to convert the other way, we multiply by 100 cubed.

In this video, we’ve learned that a rectangular prism is a solid shape that looks a little bit like a box. It has six rectangular faces, and we call its dimensions length, width, and height. We learned that the volume of this prism is the product of these dimensions. It’s width times length times height. And we can calculate this in any order.

We also saw that we can alternatively say that the volume of a rectangular prism is equal to the area of its base or its cross section multiplied by its height, where the height is the dimension perpendicular to the base. And remember, we should always check that dimensions are given in the same unit before working out volume of a three-dimensional shape.

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