Make it square
February 3rd, 2006 by SplineGuy
I came across a fairly interesting problem in my reading this week and thought I’d share it with you.
I’m sure we all know that if we are trying to build something square (or rectangular) and we want to get the angles at each of the corners as close to 90 degrees as possible, the rule-of-thumb for “square-ness” is to check the diagonals. If they are the same length, then you have managed a rectangle, otherwise you are off and need to move one side so as to make the diagonals match. So here’s the question, how far does one move it?
Say for example, you have decided to lay out a foundation for a storage shed and you want the building to be 12 ft x 16 ft. You lay out your boundaries and measure the diagonals. We, of course, are assuming that the lengths each side is as accurate as necessary. I measure the two diagonals and they have a difference of 9 in (or 0.75 ft). How far do I need to move, say, the long side, to make it match?
Naturally, the most common approach at this point is the trial and error method. Try until you get it right, but using just a bit of trigonometry and geometry, we can derive a formula for the answer. Since it’s late and I’m tired, I’ll spare you the derivation but maybe I’ll come back in a day or two and post it in a comment. If you beg, it may happen sooner.
At any rate, here it is: Determine the difference of your two diagonals, call that 
. Then, take the average of the two diagonals, call that 
. Finally, let 
be the length of the side you are not moving. You will move the other side a distance of 
.
So for our example, consider that we had 
and assume that we know 
. We are going to move the long side, so 
. We will move the long side a distance of 
.
So, just a question: If it needs to be 30 ft x 40 ft and the difference of the 2 diagonals is 1 ft. and the average is 50 ft. How far would I need to move the long side?
