In this video, we will investigate how a change in height, width, and length will affect the volume of a rectangular prism. In the first prism, the width is three inches, the height is four inches, and the depth, or length, is two inches. On the right, the prism is six inches wide, eight inches tall, and four inches deep. Each dimension is twice that of the original. Do you think the volume is twice as much?

Let's find out. To find the volume of a rectangular prism, we simply multiply the three dimensions to determine the volume. In the case of the prism on the left, the product of three, four, and two is twenty-four cubic inches. The volume of the prism on the right is the product of six, eight, and four, which is one hundred ninety-two cubic inches. So the volume here not only doubled, it didn't even triple or quadruple. It increased eight-fold. If you recall, the reason for this drastic increase is because the size was increased in each dimension. In this case three-dimensional measurements were doubled. This created an increase of volume eight times the original.

Now we will continue discussing volume with a real-world example. Will the following pieces of lumber fit into a pickup truck bed? Suppose the truck bed is fifty-three inches wide, seventy-two inches long and thirty-eight inches tall. It's limited in height because of the cap on top used to keep the lumber in the truck bed dry. Before we even decide if the pieces of lumber will fit, we should probably determine the volume of the space we are working with--fifty-three times seventy-two times thirty-eight is 145,008 cubic inches. Well this seems like an awful lot of space. This converts to about eighty-four cubic feet (and you can verify on your own). The plywood is eight feet long, or ninety-six inches. It is four feet wide, or forty-eight inches, and it's three-fourths of an inch in depth or thickness. Therefore each sheet is 3,456 cubic inches. Four of these sheets, then, would take up 13,824 cubic inches out of the 145,008 cubic inches we found available in the truck bed. Well this is good news, right? Be careful! Our goal is to fit the lumber in the back of the truck bed so no part is sticking out of the truck, potentially getting wet. The length of the truck bed is only seventy-two inches, or six feet. The plywood would hang out of the back of the truck bed a two full feet. Even though this volume is much less than what we have to work with, the actual dimensions prevent the plywood from fitting.

Looking at the oak slab, we should be okay, correct? It's only six feet long, or seventy-two inches, eight inches wide, and three inches thick. Each of these dimensions is less than the limit of the truck bed, so of course the slab would fit. In fact, the volume of this slab is only 1,728 cubic inches. This is half that of each sheet of plywood.

Let's consider one last volume problem. At a popular shoe manufacturer, boxes for a certain pair of shoes measure eight inches by five inches by thirteen inches. A larger case is needed to ship several pairs of these shoes at one time. A manufacturing engineer has designed a box or case to be twenty-four inches by twenty inches by twenty-six inches. Take a few minutes to determine how many shoe boxes will fit in this case. Resume playback in a moment to see my final solution. I decided that the shoe boxes will fit into the case three boxes wide, four boxes tall, and two long exactly inside the case. Thus twenty-four pairs of shoes would be able to fit inside the case exactly. The engineer did a nice job of designing this case. There was no room left over.