Introduction

In this lesson, you will find the volume of an object by counting how many unit cubes fit within the object. This is how you measure three dimensional (3D) objects.

This video illustrates the lesson material below. Watching the video is optional.

• Introduction to Volume (05:27 mins) | Transcript

Review Area of a Figure

Example 1
An area is 3 units by 2 units. One unit square is 1 unit by 1 unit. So, each square in Figure 1 represents 1 square unit. The total area of the space below is 6 units squared or $6u^2$.

Figure 1

Calculating the Volume of a Rectangular Prism

Volume is three dimensional because you are using three dimensions or directions multiplied together to find the volume. The formula for volume of a rectangular prism is:

$$\begin{array}{r}Volume=length\times width\times height\end{array}$$

• Length is how long the object is.
• Width is how wide the object is.
• Height is how tall the object is.
• Length, width, and height each need to be of the same unit of measurement in order to correctly calculate volume.
• Since the volume is the result of three units of measurement multiplied together, it is measured in units cubed or $c{m}^3$.

Now, suppose you take the same area but make it three-dimensional. In other words, give it a depth of one unit. The volume of this rectangular prism is the total space that it takes up. It is now 3 units by 2 units by 1 unit. To calculate the volume, focus on a unit cube. One cube represents one cubic unit or 1 unit cubed, $1u^3$. What is the volume for this prism?
 

Figure 2

One method to calculate the volume of an object is to count how many unit cubes are in the object or rectangular prism for this example. This object has a volume of 6 cubic units or $V=6u^3$.

Example 2
Another layer of boxes is added to Figure 2. Now, the units are 3 units long, 2 units high, and 2 units wide. How many unit cubes fit in this space? See Figure 3.

Figure 3

You can visually count a total of 12 cubes in Figure 3. Instead of counting, you can calculate the volume by using the formula for a cube:

$$\begin{array}{rl}Volume& =length\times width\times height\\ \\ V& =lwh\end{array}$$

Now, use the formula:

$$\begin{array}{rlr}Volume& =length\times width\times height& \text{Volume of a rectangular prism}\\ \\ V& =(3u)(2u)(2u)& \text{Substitute given terms}\\ \\ V& =12u^3& \text{Multiply}\\ \end{array}$$

This is the same answer you obtained when visually counting the cubes.

Example 3
Another layer of boxes is added to Figure 3. Now the figure is 3 units long, 2 units high, and 3 units wide. Calculate the volume in units.

Figure 4

$$\begin{array}{rlr}Volume& =length\times width\times height& \text{Volume of a rectangular prism}\\ \\ V& =(3u)(3u)(2u)& \text{Substitute given terms}\\ \\ V& =18u^3& \text{Multiply}\\ \end{array}$$

Things to Remember

• Just as variables to the power of 2 are squared, variables to the power of 3 are cubed. The number $4^3$ can be read “four to the power of three” or “four to the third power” or “four cubed.”
• To calculate the volume of a rectangular prism, multiply the length by the width by the height.  
  
  • $V=(l)(w)(h)$
• Remember to cube the units for volume.

Practice Problems

1. A wooden block has a length of 4 inches, a width of 4 inches, and a height of 4 inches. Find the volume of the wooden block. (
   Video Solution

   x

   Solution: $64{\text{ in}}^3$
   Details:
   

   
   (Introduction to Volume #1 (01:26 mins) | Transcript)
   | Transcript)
2. A rectangular swimming pool has a length of 20 ft, a width of 12 ft, and a depth of 6 ft. Find the volume of the swimming pool. (
   Solution

   x

   Solution: $1440{\text{ ft}}^3$
   Details:
   The swimming pool is measured in feet.
    
   When you find the volume of a rectangular solid, you need to find the number of unit cubes in the rectangular solid. In this case, you are measuring the volume in feet, so you want to find out how many 1 foot by 1 foot by 1 foot cubes it would take to fill the pool.
    
   To find the volume of the pool, multiply the $length$ times the $width$ times the $height$.
   
   $20\times 12\times 6=1440$
   
   The volume of the swimming pool is 1440 ${\text{ft}}^3$.
   )
3. A brick has a length of 20 cm, a width of 9 cm, and a height of 5 cm. Find the volume of the brick. (
   Solution

   x

   Solution: $900{\text{ cm}}^3$
   )
4. A cardboard moving box measures 16 in long, 10 in wide, and 12 in high. Find the volume of the moving box. (
   Solution

   x

   Solution: $1920{\text{ in}}^3$
   Details:
   The moving box is measured in inches.
    
   When you find the volume of a rectangular solid, you need to find out how many unit cubes would fit into the rectangular solid. Since you are measuring this box in inches, you need to find how many 1 inch by 1 inch by 1 inch cubes would fit into the box.
    
   To find the volume of the box, multiply the $length$ times the $width$ times the $height$.
   
   $16\times 10\times 12=1920$
   
   The volume of the box is 1920 ${\text{in}}^3$.
   )
5. A large rectangular fish tank is 3 m long and 1 m wide and has a height of 2 m. Find the volume of the fish tank. (
   Solution

   x

   Solution: $6{\text{m}}^3$
   )
6. A small rectangular juice box has a length of 60 mm, a width of 40 mm, and a height of 105 mm. Find the volume of the juice box. (
   Video Solution

   x

   Solution: $252,000{\text{ mm}}^3$
   Details:
   

   
   (Introduction to Volume #6 (01:47 mins) | Transcript)
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