ORCCAOpen Resources for Community College Algebra

SectionA.7Volume

Volume is the quantity of three-dimensional space enclosed by a closed surface. We measure area by unit squares; similarly, we measure volume by unit cubes. Let's look at the following figure:

The 3D shape in the figure is called a rectangular prism. It is composed of \(1\text{ in}\times1\text{ in}\times1\text{ in}\) unit cubes, with each cube's volume being \(1\) cubic inch (or \(\text{in}^3\)). The shape's volume is the number of such unit cubes. The bottom layer has \(4\cdot5=20\) unit cubes. Since there are \(3\) layers, the shape has a total of \(3\cdot20=60\) unit cubes. In other words, the shape's volume is \(60\text{ in}^3\text{.}\)

Let's look at the same prism without cutting it into unit cubes:

Earlier, we found the number of unit cubes in the bottom layer by doing \(4\cdot5=20\text{,}\) and then multiply \(20\) by \(3\) because there are three layers. Note that the area of the prism's base can also be calculated by \(4\cdot5=20\text{,}\) so a prism's volume formula is

where \(V\) stands for volume, \(B\) for base area, and \(h\) for height.

Now we can use the formula to calculate the prism's volume in Figure A.7.2:

A cylinder is not a prism, but its volume can also be calculated by the same formula. Let's look at an example.

ExampleA.7.3

Find the volume of a cylinder which has a radius of \(3\) meters and a height of \(10\) meters. First, write your result in terms of \(\pi\text{,}\) and then round the result to two decimal places.

Solution

We will use the formula \(V=Bh\text{.}\) The base area can be calculated by \(A=\pi r^2\text{,}\) so we have:

\begin{align*} V\amp=Bh\\ \amp=\pi r^2 h\\ \amp=\pi (3\text{ m})^2(10\text{ m})\\ \amp=90 \pi\text{ m}^3 \end{align*}

The cylinder's volume is \(90 \pi\) cubic meters.

Next, we will use a calculator to calculate the decimal value:

\begin{align*} V\amp=90\pi\text{ m}^3\\ \amp=90(3.1415926\ldots)\text{ m}^3\\ \amp\approx282.74\text{ m}^3 \end{align*}

The cylinder's volume is approximately \(282.74\) cubic meters.

When we solve application problems, it's good practice to quickly check whether our solution is reasonable. If not, we must have made a mistake somewhere. Let's look at an example.