Cups and Volume

Date: 12/06/2001 at 15:31:16
From: Kedra
Subject: Volume

How can I calculate the volume of a box, if I know how many cups of
rice (or something like that) fill it? 

I don't understand how 2 cups is a volume measure, since volume =
length x width x height.  For example, if I calculate the box to have
a volume of 24 cubic inches, how is that the same as the 2 cups of
rice it holds?

Date: 12/07/2001 at 09:44:38
From: Doctor Ian
Subject: Re: Volume

Hi Kedra,

The _formula_ for the volume of a rectangular prism is 

  volume(rec. prism) = length * width * height

but _volume_ itself is a measure of the amount of 3-dimensional 
'stuff' that an object can hold. 

Suppose I have a box with dimensions 2 x 3 x 4 inches.  The volume of 
the box is 2 * 3 * 4 = 24 cubic inches, right?  So if I fill the box 
with rice, I have 24 cubic inches of rice.  

Now, suppose I have a cylindrical bowl that is just large enough to 
hold that rice.  That is, if I pour all the rice from the box into the 
bowl, the bowl is completely filled. 

The box and the bowl hold exactly the same amount of 3-dimensional 
stuff, whether that's rice, water, flour, sand, or just air. That's 
what it _means_ for them to have the same volume.  

It turns out that a gallon container can hold 231 cubic inches of 
stuff. This is independent of the shape of the container - whether 
it's a box, or a cylinder, or a sphere, or a truncated cone (like a 
disposable coffee cup, or a measuring cup), or a torus (i.e., a 
doughnut), or just some weird arbitrary shape, like many perfume 
bottles. 

Now, this means that any quart container will hold 1/4 of that, or 
231/4 cubic inches of stuff, and any pint container will hold half of 
that, or 231/8 cubic inches of stuff, and any cup container will hold 
half of that, or 231/16 cubic inches of stuff. 

Different common shapes have different formulas that can be used to 
compute their volumes.  You can find many of those formulas in our Dr. 
Math FAQ:

  http://mathforum.org/dr.math/faq/formulas/   

In the example of the box and the bowl, suppose I know that the bowl 
is 6 inches high.  I can use the formula for volume, 

  volume(cylinder) = pi * radius^2 * height

to find out the radius of the bowl by solving for radius:

  radius = sqrt( volume / (pi * height) )

Or, if I know that the radius of the bowl is 4 inches, I can find out 
the height by solving for height:

   height = volume / (pi * radius^2)

But no matter what dimensions they have, if the box and the bowl can 
hold the same amount of stuff, they have the same volume. The next 
time you're in a store like Lechter's or Williams and Sonoma, take a 
look at the various measuring cups they have for sale.  They'll all 
have different shapes, but what they'll have in common is that, no 
matter what the shape, a measuring cup marked '1 cup' will hold 
231/16 cubic inches of water, sugar, flour, or anything else you fill 
it with, even if the measuring cup is shaped like a box.  

So if you know how many cups of rice a box will hold, you _already_ 
know the volume of the box, although you may want to convert it to 
different units, like cubic inches, or cubic centimeters, or liters, 
or acre-feet, or whatever.  

It's sort of like this: Suppose I tell you that the length of a 
certain room is exactly 16 times the length of a particular shoe box.  
If you have the shoe box, you can measure it in inches, or 
centimeters, or whatever other units you prefer; and then you can 
multiply by 16 to get the length of the room. So if I tell you how 
many shoeboxes fit along one wall of the room, you 'know' the length 
of the room and you just have to convert it to units that you like 
better. 

I hope this helps.  Write back if you'd like to talk more
about this, or anything else.

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/