Swimming Pool Volume

Date: 02/28/2002 at 11:16:54
From: David Strong
Subject: Cubic feet vs gallons.

What formula would I use to calculate the volume of a swimming pool 
135' x 70' with different depths of 3', 5', 8' and 11'?  

Thanks,
DS

Date: 02/28/2002 at 12:36:27
From: Doctor Peterson
Subject: Re: Cubic feet vs gallons.

Hi, David.

It's not quite clear exactly what shape the pool is; I presume it 
doesn't just drop off suddenly from one depth to another, but slopes 
down from one level to another. Suppose the cross-section looks like 
this, with straight lines within each section:

        a        b          c         d
    +-------+---------+-----------+------+
    |r      |         |           |      |
    |       |s        |           |      |
    +       |         |t          |      |
        \   |         |           |u     |v
            +_____    |           |      |
                  ----+           |      |
                          \       |      |
                              \   |      |
                                  +------+

The area of this cross-section will be the sum of four trapezoids, 
each of whose areas is its "height" (width in this case) times the 
average of its two "bases" (depths in this case). The volume of the 
pool is the product of that area and the width perpendicular to this 
cross-section.

So the volume will be

    V = w[a(r+s)/2 + b(s+t)/2 + c(t+u)/2 + d(u+v)/2]

This can be rewritten as

    V = w/2 * [ar + (a+b)s + (b+c)t + (c+d)u + dv]

If the measurements are all in feet, this will give cubic feet. To 
convert to gallons, multiply by 12^3 = 1728 to get cubic inches, and 
divide by 231 cubic inches per gallon.

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