Calculating How Much a Tablecloth Will Overhang the Table

Date: 07/27/2006 at 14:31:07
From: Juanita
Subject: Measurements

Will a tablecloth that is 54" x 126" hang to the floor when put on a 
table that is 8' x 3'?

Date: 07/28/2006 at 14:12:19
From: Doctor Ian
Subject: Re: Measurements

Hi Juanita,

Let's convert everything to inches, to make things easier.  If we
center the cloth on the table, we have this situation, as viewed from
above.  

                 126
      +------------------------+
      |                        |
      |   +----------------+   |
      |   |       96       |   |
      |   |             36 |   | 54
      |   |                |   |
      |   A----------------+   |
      |                        |
      B------------------------+

The inner rectangle is the table.  The outer rectangle is the cloth.

If any part will touch the floor, it would be the point labeled B. 

So we want to know if the distance from A to B is shorter than the
distance from the table surface to the floor.  

Does that make sense? 

How can we find that?  First, we find the overhang in each direction:

                 126
      +------------------------+
      |                        |
      |   +----------------+   |
      |   |       96       |   |
      |   |             36 |   | 54
      |   |                |   |
      |   A----------------+   |
      |                        |
      B---C--------------------+

The distance from A to C is half the excess cloth in that direction:

   AC = (54 - 36)/2

      = 18/2

      = 9

The distance from C to B is half the excess cloth in that direction:

   CB = (126 - 96)/2

      = 30/2

      = 15

Because ACB is a right triangle, and we have the two legs, we can use
the Pythagorean Theorem, 

  Pythagorean theorem
    http://mathforum.org/dr.math/faq/faq.pythagorean.html 

to find the hypotenuse, which is the distance AB:

   (AB)^2 = (AC)^2 + (CB)^2

   (AB)^2 = 9^2 + 15^2
   
   (AB)^2 = 306
              ___
       AB = \/306

       AB = 17.5 (approximately)

So in this case, if your table is more than 18 inches above the floor,
the cloth will hang above the floor.  But you can use the same
procedure to figure it out for any combination of table and cloth.

Does this help? 

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