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### 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/
```
Associated Topics:
High School Geometry
High School Higher-Dimensional Geometry
High School Practical Geometry

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