Associated Topics || Dr. Math Home || Search Dr. Math

### Numerically Equal Volumes and Surface Areas

```
Date: 06/04/2001 at 18:35:02
From: Irene
Subject: Geometry

Find all rectangular solids with integral dimensions, the volumes and
surface areas of which are numerically equal.
```

```
Date: 06/06/2001 at 02:29:20
From: Doctor Greenie
Subject: Re: Geometry

Hi, Irene - that was a curious problem and I had fun with it!

Given a rectangular solid with length l, width w, and height h:

volume = lwh
and
total surface area = 2lw+2lh+2wh

You are looking for all integer triples (l,w,h) for which the volume
and surface area are numerically equal.

So you have

lwh = 2lw+2lh+2wh

Solving this equation for l in terms of w and h:

lwh-2lw-2lh = 2wh
l(wh-2w-2h) = 2wh

and so we have

2wh
l = --------  (1)
wh-2w-2h

In looking for integer triples (l,w,h) that satisfy this equation, we
can assume that h <= w <= l, because a 3x4x5 rectangular solid is
indistinguishable from a 5x3x4 or a 4x3x5 rectangular solid.  And of
course, h, w, and l are all positive integers.

Since I have decided to call my shortest dimension h, I will try
different integer values for h; for each choice of the value for h, I
will look for integer values of w that are greater than or equal to h
and for which equation (1) will give integer values of l greater than
or equal to w.

h=1: Equation (1) becomes

l = 2w/(w-2w-2) = 2w/(-w-2)

In this expression, the denominator is always negative; so there are
no solutions with h=1.

h=2: Equation (1) becomes

l = 4w/(2w-2w-4) = 4w/(-4)

In this expression the denominator is again always negative; so there
are no solutions with h=2.

h=3: Equation (1) becomes

l = 6w/(3w-2w-6) = 6w/(w-6)

In this expression, the denominator is positive for values w>6; so we
can look for solutions with h=3 and w>6:

h   w   l
--------------------
3   7   42/1 = 42
3   8   48/2 = 24
3   9   54/3 = 18
3  10   60/4 = 15
3  11   66/5 (not an integer)
3  12   72/6 = 12

For larger values of w, l would be less than w; so there are no
further solutions with h=3.

h=4: Equation (1) becomes

l = 8w/(4w-2w-8) = 8w/(2w-8)

In this expression, the denominator is positive for value w>4; so we
can look for solutions with h=4 and w>4:

h   w   l
--------------------
4   5   40/2 = 20
4   6   48/4 = 12
4   7   56/6 (not an integer)
4   8   64/8 = 8

For larger values of w, l would be less than w; so there are no
further solutions with h=4.

h=5: Equation (1) becomes

l = 10w/(5w-2w-10) = 10w/(3w-10)

In this expression, the denominator is positive for value w>=4;
however, w>=h; so we can look for solutions with h=5 and w>=5:

h   w   l
--------------------
5   5   50/5 = 10
5   6   60/8 (not an integer)
5   7   70/11 (not an integer)

For larger values of w, l would be less than w; so there are no
further solutions with h=5.

h=6: Equation (1) becomes

l = 12w/(6w-2w-12) = 12w/(4w-12)

In this expression, the denominator is positive for value w>3;
however, w>=h; so we can look for solutions with h=3 and (since w>=h)
w >=6:

h   w   l
--------------------
6   6   72/12 = 6

For larger values of w, l would be less than w; so there are no
further solutions with h=6.

There are no additional solutions with h>6. For h>6, and with w>=h,
equation (1) gives values of l which are less than w; but our
requirement is that l>=w.

So we have the following set of solutions (h,w,l) to this problem; and
this is the complete set of solutions:

(3,7,42)
(3,8,24)
(3,9,18)
(3,10,15)
(3,12,12)
(4,5,20)
(4,6,12)
(4,8,8)
(5,5,10)
(6,6,6)

- Doctor Greenie, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Geometry
High School Higher-Dimensional Geometry
High School Number Theory

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search