10.10 Volume of rectangular prisms
The volume of a three dimensional shape is the amount of space that is contained within that shape.
A quantity of volume is represented in terms of the volume of a unit cube, which is a cube with side length $1$1 unit. By definition, a single unit cube has a volume of $1$1 cubic unit, written as $1$1 unit3.
1 unit cube has a volume of 1 unit3.
The image below shows a rectangular prism with length $5$5 units, breadth $3$3 units, and height $2$2 units. Notice that the length of each edge corresponds to the number of unit cubes that could be lined up side by side along that edge.
How many unit cubes fit within this shape?
We can find the number of unit cubes that could fit inside the rectangular prism by taking the product of the three side lengths. This gives $5\times3\times2=30$5×3×2=30, so there are $30$30 unit cubes in the prism, which means it has a volume of $30$30 unit3.
Volume of a rectangular prism
In the same way that the area of a two dimensional shape is related to the product of two perpendicular lengths, the volume of a three dimensional shape is related to the product of three mutually perpendicular lengths (each of the three lengths is perpendicular to the other two).
The volume of a rectangular prism is given by
$\text{Volume }=\text{length }\times\text{breadth }\times\text{height }$Volume =length ×breadth ×height , or
$V=l\times b\times h$V=l×b×h
A cube can be thought of as a special type of rectangular prism, one that has all sides equal in length. The formula for the volume of a cube is similar to the formula for the area of a square.
The volume of a cube is given by
$\text{Volume }=\text{side }\times\text{side }\times\text{side }$Volume =side ×side ×side , or
$V=s\times s\times s=s^3$V=s×s×s=s3
Worked examples
Example 1
Find the volume of the following rectangular prism.
Think: The side lengths have units of cm, so the volume will be in cm3.
Do: The base of the prism has a breadth of $2$2 cm and a length of $7$7 cm, and the height of the prism is $9$9 cm. We will use these sides in the formula for the volume of a rectangular prism.
| $\text{Volume }$Volume | $=$= | $\text{length }\times\text{breadth }\times\text{height }$length ×breadth ×height | (Formula for the volume of a rectangular prism) |
| $=$= | $7\times2\times9$7×2×9 | (Substitute the values for the length, breadth, and height) | |
| $=$= | $126$126 | (Perform the multiplication to find the volume) |
So this rectangular prism has a volume of $126$126 cm3.
Example 2
The local swimming pool is $25$25 m long. It has eight lanes, each $2$2 m wide, and its depth is $1.5$1.5 m. What is the volume of water in the pool?
Think: The water in the pool is in the shape of a rectangular prism, so to find its volume we need to find the side lengths of this prism. The length and depth of the pool are two side lengths we can use. The final side length is found by multiplying the number of lanes by the width of each lane.
Do: First we calculate the breadth of the pool using the width of each swim lane: $8\times2$8×2 m $=16$=16 m. Next we use the formula for the volume of a rectangular prism.
| $\text{Volume }$Volume | $=$= | $\text{length }\times\text{breadth }\times\text{height }$length ×breadth ×height | (Formula for the volume of a rectangular prism) |
| $=$= | $25\times16\times1.5$25×16×1.5 | (Substitute the values for the length, breadth, and height) | |
| $=$= | $600$600 | (Perform the multiplication to find the volume) |
So the water in the pool has a volume of $600$600 m3.
Reflect: Even though the volume formula uses the terms "length", "breadth", and "height", when referring to everyday objects it may be more appropriate or more common to use alternative words like "width", "depth", or "thickness". In this example, we could just as well have used the formula $\text{Volume }=\text{length }\times\text{width }\times\text{depth }$Volume =length ×width ×depth .
Volume of composite shapes
Once we are familiar with finding the volume of rectangular prisms and cubes, the same idea can be used to determine the volume of more complicated shapes. If we think of a composite shape as being built out of a number of smaller, simpler shapes, then the volume of the composite shape is the sum of the volume of each shape it is built from.
This complicated shape is built from simple rectangular prisms.
Depending on the configuration of the composite shape, it may be useful to think of building it from a large rectangular prism that then has smaller volumes taken away.
Worked example
Find the volume of the following composite shape.
Think: We can break up this composite shape into two smaller rectangular prisms, as shown below. The height of the top prism is found by subtracting the height of the bottom prism from the total height of the composite shape.
Do: The volume of the entire shape is found by adding together the volume of the two smaller rectangular prisms.
| $\text{Volume }$Volume | $=$= | $\text{Volume of top prism }+\text{Volume of bottom prism }$Volume of top prism +Volume of bottom prism |
| $=$= | $11\times6\times3+11\times10\times2$11×6×3+11×10×2 | |
| $=$= | $198+220$198+220 | |
| $=$= | $418$418 |
So the volume of the composite shape is $418$418 m3.
Reflect: There may be many ways to break up a composite shape. As an alternative approach, we could have found the volume of a larger rectangular prism and subtracted the volume of a smaller prism, as shown in the image below.
In this way, the volume would be given by $11\times10\times5-11\times4\times3=418$11×10×5−11×4×3=418 m3, as expected.
Practice questions
Question 1
Find the volume of the rectangular prism shown.
Question 2
Find the volume of the composite solid shown by breaking it up into rectangular prisms.
Question 3
A box of cereal is in the shape of a rectangular prism. It measures $24$24 cm by $12$12 cm by $19$19 cm.
What is the volume of the box of cereal?
The company that makes these cereal boxes also makes a jumbo size box, which is twice as long, twice as wide, and twice as tall as the regular size boxes. What is the volume of the jumbo box of cereal?