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

Solids are three-dimensional (3D) figures or shapes, which means that they extend in the three dimensions of the space. It is useful to bear in mind that a line is one-dimensional (1D): it extends only in one direction. Two-dimensional (2D) shapes extend in two directions. And since solids are 3D shapes, thus, unlike 2D shapes, they cannot be drawn on paper as solids extend in the direction perpendicular to the paper as well.

In this lesson, we will be dealing with rectangular prisms.

### Definition: Rectangular Prism

A rectangular prism, also called a cuboid, is a box-shaped solid object. It has six faces that are rectangles. It is described by its length, width, and height.

Before learning how to calculate the volume of a rectangular prism, let us look at the length, area, and volume units. We start with a unit length: it can be represented by a segment of length 1 (in whatever unit we are working with). The unit area is represented by a square of side 1 and the unit volume is represented by a cube of side 1.

Remember that a number squared is the area of a square whose side length is this number. So, the area of a square of side length 3 is indeed 9: as we see in the diagram, there are unit squares in the square of side length 3.

This can be generalized to the area of a rectangle: it is given by the product of its length and width. For instance, the area of a rectangle of length 5 and width 3 is area units.

The same idea is applied to volumes of solid objects. A number cubed is the volume of a cube whose side length is this number. The volume of a cube of side length 3 is indeed volume units.

This can be generalized to any rectangular prism. For instance, the volume of a rectangular prism of sides 2, 3, and 4 is volume units.

This is a general result for any rectangular prism.

### How To: Calculating the Volume of a Rectangular Prism

The volume of a rectangular prism is the product of its length , width , and height :

Let us look at some questions to see how this formula for the volume of a rectangular prism is applied.

### Example 1: Visualizing and Finding the Volume of a Rectangular Prism

Find the volume of the cuboid.

### Answer

The volume of a rectangular prism is given by the product of its length, width, and height:

Here, we have a length of 3 cm, a width of 2 cm, and a height of 5 cm. After having checked that all dimensions are expressed in the same unit, we can substitute them in:

We see that it is the number of unit cubes drawn in the diagram. Each horizontal layer is made of six unit cubes, and there are five such layers, giving 30 unit cubes.

### Example 2: Understanding the Concept of Volume of a Rectangular Prism

Which of the following describes how the volume of a rectangular prism is affected after doubling all three dimensions?

### Answer

We know that the volume of a rectangular prism is given by the product of its length, width, and height:

If each dimension is doubled, the volume is then

Since only multiplication is involved here, we can remove the brackets and multiply the twos together. We find that

And since , we finally find that

When the three dimensions of a rectangular prism are doubled, its volume is multiplied by 8.

### Example 3: Comparing the Capacities of Boxes

A man needs to store 16 170 cm^{3} of rice in a container. He has one box which is a cuboid with dimensions of 35 cm,
22 cm,
and 21 cm and another box which is a cube with length 22 cm. Which box should he use?

### Answer

A box is a cuboid. The space inside a box is called its capacity, that is, the volume of empty space inside the box that can contain something, here rice. A box has thin walls, so we can consider that its volume is the same as its capacity. We need to compare the volumes of the two boxes in order to decide which one is big enough to contain 16 170 cm^{3} of rice.

The first box is a cuboid of dimensions 35 cm, 22 cm, and 21 cm. We know that the volume of a cuboid is the product of its three dimensions (length, width, and height):

The second box is a cube with length 22 cm, so its volume is

The volume of the cubic box is smaller than the volume of rice, while the volume of the other box is exactly the volume needed for the rice. Therefore, the man should use the cuboid.

Before we look at other questions, let us observe something interesting about the volume of a rectangular prism. We know it is given by the product of its three dimensions, but we also know that the product of two of its dimensions gives the area of one of its faces.

Therefore, we can conclude that the volume of a rectangular prism is also given by the area of one of its faces multiplied by the dimension perpendicular to its face, which is then called the height because we imagine that the rectangular prism sits on this face.

The volume of a rectangular prism is then

where is the area of the base and is its height, which should be understood here as the dimension perpendicular to the base.Let us test our understanding with a question.

### Example 4: Finding the Volume of a Rectangular Prism given the Area of Its Base and Its Height

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

### Answer

We want to compare the volumes of both cuboids. We have the three dimensions of cuboid A; therefore, we can work out its volume with

Substituting in the dimensions given in the question, we find that

For cuboid B, we do not have its three dimensions, but we have the area of its base and its height. Thus, we know that its volume is where is the area of the base and is the height. Substituting in the values given in the question, we find that

We find that is greater than , which means that cuboid B is greater in volume than cuboid A.

Now that we have learned how to work out the volume of a rectangular prism when either its three dimensions or the area of its base and its height are known, we are going to look at a question where the volume of the rectangular prism is known but not one of its dimensions.

### Example 5: Finding an Unknown Dimension of a Rectangular Prism

Given that 405 cm^{3} of water is poured into a cuboid-shaped vessel with a square base whose side length is 9 cm, find the height of the water in the vessel.

### Answer

Here, we are given the dimensions of the base of a cuboid-shaped vessel and the volume of the water that had been poured in the vessel. The volume of water when it is in the vessel is then given by the area of the base multiplied by the height of water. Since the base is a square of side length 9 cm, its area is . Therefore, we can write

Dividing both sides by 81 gives

Since we are told that the volume of water is 405 cm^{3},
we find that

The height of the water in the vessel is 5 cm.

### Key Points

- A rectangular prism is a box-shaped solid object. It has six faces that are rectangles. It is described by its length, width, and height.
- The volume of a prism is the product of its length , width , and height :
- Since the product of two dimensions of a rectangular prism is the area of the corresponding face, the volume of a rectangular prism is also where is the area of the base and is its height, which is to be understood here as the dimension perpendicular to the base.
- Remember to check that all dimensions are given in the same unit before working out the volume or the area of a shape.