Volume Formulas
Volume is the amount of space contained within a three-dimensional object. Shapes such as spheres, prisms, and cylinders are three-dimensional objects that take up a certain amount of space, which we can call its volume. We measure volume in cubic units. Here are some examples of cubic measurements:
With these measurements, we can easily tell they are cubic measurements because they include an exponent of three. Not all cubic measurements have this visual clue. For example, the liter is abbreviated with the letter L, and does not contain an exponent of three, yet it is a cubic measurement (it measures liquid volume, and is actually equivalent to 1000 cubic centimeters).
There are several formulas for finding the volumes of different shapes. We are going to focus on the volumes of a rectangular prism, cylinder, and sphere. Here are the formulas to know:
Volume of a Rectangular Prism:
Volume of a Cylinder:
Volume of a Sphere:
Notice that some of these formulas include the number, pi. Pi is an important number.
Pi: the ratio of a circle's circumference to its diameter; approximately 3.14
Volume of a Rectangular Prism
Take a look at the figure below:
To find the volume, we simply multiply all of the dimensions together. This includes the units of measurement.
Volume of a Cylinder
Let's use a formula to find the volume of a cylinder:
To find the volume, first we square the radius of the circular base, and then we multiply by pi and the height. Remember that the order of operations dictates exponents before multiplication, so that's why we square the radius before multiplying anything.
Volume of a Sphere
Let's find the volume of the sphere pictured below:
The only piece of information we are given is the radius of the sphere, but this is all we need to calculate the volume. Just like with the volume of a cylinder, first we apply the exponent. After cubing the radius, we multiply it by pi and then by four-thirds.
To leave your answer as an exact value, you can choose to leave pi as π multiplied by a number. If you need to give a decimal approximation, use π = 3.14
Finding Side Lengths or Radii Given the Volume
In our previous examples, we used given measurements for side lengths and radii to calculate the volume of a three-dimensional object. In the following examples, we are going to use a given measurement of volume to calculate a side length or a radius. We will use the same formulas, but in a different way.
We'll start with finding the side length of a rectangular prism:
Dividing both sides of the equation by the length and the height will do two things: first, it will cancel the length and the width from one side of the equation, leaving just an expression for the width.
Second, it can help us cancel the cubic units on the other side if the equation, leaving our answer in linear centimeters. (Linear centimeters just means regular centimeters we are used to measure. We would only say "linear" if we wanted to make it clear that it isn't square centimeters or cubic centimeters.)
Now we can perform the numeric division on the left side of the equation. This is our solution for the width of the prism:
Lastly, let's use a volume formula to solve for an unknown radius. Consider the sphere below:
We are given the volume as 65.45 cubic inches, with an unknown radius. Substituting known information into the formula for the volume of a sphere, we have:
How can we isolate r? Look at the numbers and operations surrounding the variable r. It is being raised to the power of 3, and then multiplied by (4/3)pi. We need to undo this by applying inverse operations, but also in reverse order. This means first we need to undo the multiplication by (4/3)pi, and then undo the power of 3.
Note that (4/3)pi is a number, so it can be divide in one step. One strategy is to evaluate (4/3)pi first and then divide it through both sides of the equation. Another strategy is to divide by (4/3) first (or multiply by the reciprocal), and then divide by pi. You could also first divide by pi, and then divide by (4/3) (or multiply by the reciprocal). These are all valid methods.
The ratio of a circle's circumference to its diameter; approximately 3.14.