WE have a structure that has four posts makeing a rectangle that is 22
week wire by 12 feet long. on top of the rectangle is a curved roof
that is made from a tubes bent into a coninuous arc, the tube length
is 28 feet. The tubes fit into the posts across the 22 foot dimention.
If I want to run a roop diagonally and over the roof how much length
will I need? What is the formula?

Hi, Lingcod!

Please let me know if I'm missing some crucial aspect of this.

You know the length of the tubing in the arc is 28', so the actual
dimensions we need to consider for this would be 28'x12'. You say the
tubes fit into the poles on either end, but not how much of the tube
is in the pole. I'm assuming for this calculation that the tube ends
are exposed and 28' is the full measure. If it's not, adjust the value
for the length accordingly in the formula.

The pitch of the roof wouldn't be relevant as long as it's continuous
(no dips or peaks) and has only one arc (straight on the short ends),
and your rectangle has perfect right angles in the corners. This seems
to be what you've described, so to determine the diagonal length of
the roof, you'd just need the hypotenuse of the new rectangle. You'd
use the Pythagorean Formula:

a squared + b squared = c squared.

OR

12x12 + 28x28 = 928

The square root of 928 is 30.46 (rounded to two digits), so you'd need
approximately 30.46' of rope to go from corner to corner flush.

I tested this with a piece of paper and a string, securing the string
while the paper was flat, then bending the paper in various arcs, and
the string always remained flush with the paper, never loose, never
too tight.

SEARCH TERMS: measure hypotenuse

ONLINE REFERENCE: http://library.thinkquest.org/20991/geo/stri.html#pythagoras


***

Some things that occurred to me while I was answering this: 

You would need more if you need to tie the rope. To determine how much
you need to tie it, I'd need to know the diameter of the rope, and the
diameter of the poles and how you're planning to secure it (i.e.
looped around the pole, knotted, pegged to the ground). Also, it'd be
unlikely you would have the rope going directly across the corner of
the rectangle, so you may have a few extra inches to play with,
depending on how you are attaching it.

The nature of the rope would affect its fit over time as well, as it
is exposed to the elements. The initial measure would be accurate, but
it may tighten or loosen with age and exposure.

You may be interested in information sites such as the following,
which outline care and use issues for rope, as well as safe working
loads:

http://www.boatwashington.org/knots_rope_care.htm
http://www.ropecord.com/

SEARCH TERMS: rope condition exposure weather

great answer