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Why Cubed?

Date: 11/27/2001 at 00:30:59
From: Bob Henry
Subject: Why do we use cubed

Why do we use cubed in the volume of a figure?

Example: Find the volume of a rectangular prism of length 4ft, height 
10ft, and width 5ft. I know that you would just mutiply all the 
numbers to get the answer 200ft. But why do they put 200ft cubed? Also 
why do they use a 3 to symbolize cubed?

Date: 11/27/2001 at 08:44:25
From: Doctor Peterson
Subject: Re: Why do we use cubed

Hi, Bob.
What is written as  ft  or ft^3 in our e-mail, and read as "cubic 
feet," is the unit of volume equal to a 1x1x1 foot cube. You could fit 
200 of these "cubic feet" into your rectangular prism.

The volume of a cube is the cube (third power) of the length of a 
side. A 2-foot cube has volume 2^3 ft^3, or 8 cubic feet.

When you multiply the dimensions, you are not just multiplying

    4 x 10 x 5 ft

but rather

    4 ft x 10 ft x 5 ft = 4 x 10 x 5  ft x ft x ft

If it makes it clearer, you can think of "4 ft" as meaning 4 times one 
foot. So this is actually

    4 x ft x 10 x ft x 5 x ft = 4 x 10 x 5 x ft x ft x ft

Just as the product of three 2's would be written as 2^3, or two 
cubed, the product of three "feet" is written as the cube of a foot, 
or ft^3. So the volume is 200 ft^3, not 200 ft.

As this page points out, the older, more traditional abbreviation for 
a cubic foot would be cu. ft.; the form you are asking about is used 
by scientists, usually using the metric system.

   Writing 'Square Foot'   

Does that help?

- Doctor Peterson, The Math Forum   

Date: 11/27/2001 at 23:49:48
From: George Henry
Subject: Why the word cubed

Why do we use the word cubed in the answer to the volume of a cube or
a rectangular prism? Why do we use the number 3 to abbreviate cubed?

Date: 11/28/2001 at 08:39:21
From: Doctor Rick
Subject: Re: why the word cubed

Hi, George.

The reason we use the word "cubed" in the answer to the volume of a
cube is exactly because it is the answer to the volume of a cube. In
the same way we use the word "squared" in the answer to the area of a

If a square has sides of length 5 inches, then its area is the product
of its length and width, which are both 5 inches. Therefore its area

  (5 inches)*(5 inches) = 25 inches*inches

(I'm using "*" for the multiplication sign.) Likewise the volume of a
cube with length, width, and height equal to 5 inches is the product
of these three numbers:

  (5 inches)*(5 inches)*(5 inches) = 125 inches*inches*inches

We don't want to write "inches*inches" for the units of an area, and
even more, we don't want to write "inches*inches*inches" for the units
of a volume. We'd like a shorter way to write these units. And there
is a way: what we call "exponents."

An exponent is the number of times that you multiply by the same
number. Suppose you start with 1, and multiply it by 5 four times:

  5       =   5 = 5  (5 to the first power)

  5*5     =  25 = 5  (5 to the second power)

  5*5*5   = 125 = 5  (5 to the third power)

  5*5*5*5 = 625 = 5  (5 to the fourth power)

The exponent is the number above the line after the 5. When typing in 
e-mail, we type "5^4" for 5 to the 4th power, so we don't need to take
up extra lines. When you write it, or see it in books, the exponent is
usually not just above the line, but a little smaller than the base
(the 5 in my examples).

It's easier to write 5^4 than 5*5*5*5. 

There is more that we can do with exponents, but this is enough for
now. It shows us an easier way to write the units of areas or volumes:

  inches*inches = inch^2

  inches*inches*inches = inch^3

That explains why we use the little 2 for areas and the little 3 for 
volumes. The last thing to explain is the easiest part: why we use the
words "squared" and "cubed" for these units. We could say "inches to
the second power" or "inches to the third power," but again we want
something shorter and easier to say. Because inches to the second
power are the units of the area of a square, we call them inches
squared (or square inches). Because inches to the third power are the
units of the volume of a cube, we call them inches cubed (or cubic

Does this answer your question? I'd be glad to help you with anything
else you still wonder about; we at Ask Dr. Math like to talk with kids
who wonder!

- Doctor Rick, The Math Forum   
Associated Topics:
High School Definitions
High School Euclidean/Plane Geometry
High School Exponents
High School Geometry
High School Higher-Dimensional Geometry
Middle School Definitions
Middle School Exponents
Middle School Geometry
Middle School Higher-Dimensional Geometry
Middle School Two-Dimensional Geometry

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