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Volume

Volume

Author: Sophia Tutorial
Description:

Calculate the volume of a rectangular prism and cylinder using formulas for volume. 

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Tutorial
what's covered
  1. Volume Formulas
  2. Finding Side Lengths or Radii Given the Volume

1. 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:

m cubed
cubic meters
f t cubed
cubic feet
k m cubed
cubic kilometers

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:

formula
Volume of a Rectangular Prism
V space equals space l • w • h

File:5493-volume2.png

Take a look at the figure below:

File:5496-volume5.png

To find the volume, we simply multiply all of the dimensions together. This includes the units of measurement.

left parenthesis 3 space f t right parenthesis left parenthesis 2.5 space f t right parenthesis left parenthesis 2 space f t right parenthesis space equals space 15 space f t cubed

formula
Volume of a Cylinder
A space equals space pi • r squared • h

File:5494-volume3.png

Let's use a formula to find the volume of a cylinder:

File:5497-volume6.png

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.

v equals πr squared straight h
Formula for the volume of a cylinder
V equals straight pi open parentheses 2.5 space in close parentheses squared times 4 space in
Substitute radius length and height
V equals 6.25 straight pi space in squared times 4 space in
Square the radius
V equals 25 straight pi space in cubed
Multiply by 4 inches
V equals 78.5 space i n cubed
Our Solution, using straight pi equals 3.14

Notice that some of these formulas include the number, pi. Pi is an important number.

formula
Volume of a Sphere
V equals 4 over 3 pi • r cubed

File:5495-volume4.png

Let's find the volume of the sphere pictured below:

File:5499-volume8.png

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.

V equals 4 over 3 πr cubed
Formula for the volume of a sphere
V equals 4 over 3 straight pi open parentheses 3.2 space cm close parentheses cubed
Substitute radius length
V equals 4 over 3 straight pi times 32.77 space cm cubed
Cube the radius
V equals 43.69 straight pi space cm cubed
Multiply by 4 over 3
V equals 137.19 space c m cubed
Our Solution, using straight pi equals 3.14

hint
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

term to know
Pi
the ratio of a circle's circumference to its diameter; approximately 3.14


2. 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:

File:5501-volume10.png

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.

fraction numerator 117.6 space c m cubed over denominator open parentheses 3.5 space c m close parentheses open parentheses 11.2 c m close parentheses end fraction equals x space c m

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:

fraction numerator 117.6 space c m cubed over denominator open parentheses 3.5 space c m close parentheses open parentheses 11.2 space c m close parentheses end fraction
Divide volume by length and height
fraction numerator 117.6 space c m cubed over denominator 39.2 space c m squared end fraction
Simplify denominator
3 space c m
Our Solution

Lastly, let's use a volume formula to solve for an unknown radius. Consider the sphere below:

File:9576-Screen_Shot_2019-05-31_at_2.44.22_PM.png

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:

64.45 space i n cubed equals 4 over 3 πr cubed

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.

fraction numerator 64.45 space i n cubed over denominator begin display style 4 over 3 straight pi end style end fraction equals r cubed
Divide both sides by 4 over 3 straight pi
15.62 space i n cubed equals r cubed
Units have not changed
cube root of 15.62 space i n cubed end root equals cube root of r cubed end root
Apply the cube root of both sides
2.5 space i n equals r
Our Solution

hint
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.

summary
Volume is the amount of space that's enclosed in a three-dimensional object. Because volume is the amount of space in a three-dimensional object, it uses cubic units, such as feet cubed or centimeters cubed. Volume formulas for three different three-dimensional objects include volume of a rectangular prism (which includes a cube), the volume of a cylinder, and the volume of a sphere. We can also find side lengths or radii given the volume by using the formulas in a different way.

Terms to Know
pi

The ratio of a circle's circumference to its diameter; approximately 3.14.

Formulas to Know
Volume of Cylinder

V subscript c y l i n d e r end subscript equals pi r squared h

Volume of Rectangular Prism

V subscript r e c t. p r i s m end subscript equals l w h

Volume of Sphere

V subscript s p h e r e end subscript equals 4 over 3 pi r cubed