Diagonally Pieced Backing

One day, while browsing blogs, I found a reference to diagonally pieced backings.  A neat way to stretch the width of your fabric, without have to buy a ton of extra fabric and piece 2 vertical panels.  You simply fold your fabric diagonally from corner to corner.  Cut along this line, to get 2 triangles.  Then, you slide one triangle up and out, to create a wider fabric sheet.  Carefully line up the edges with this new offset, and sew them back together again (see the first diagram below).  In my case, my fabric was 1" too narrow, and 6" too long - so I went on faith, cut and sewed, and came up with a backing that fit.  But that won't work all the time :)  So, I decided I needed to know more. 

First, I did some more digging into where this idea came from.  I found this link which provides a formula you can use, to determine how long a piece of fabric you need in order to piece it diagonally and come out ahead. And this site even does the calculations for you.  Note that if your quilt is more than 50% wider than your fabric, you are better off buying a piece of fabric twice the length of your quilt, and piecing with a vertical seam. But if your quilt is, for example, 90 inches, you could piece 2 panels (80" width), and then diagonally stretch that, without having to buy a third length of fabric.

That's all fine and good, and I will certainly remember this technique for those backings that are just *that much* too narrow.  But I needed more.  Where did the formula come from, and why does it work (I am an engineer, after all).  So I started drawing pictures and doing calculations.  And if you aren't a math fan, you may want to stop reading here.  But please come back later, for a more "quilty" post :)

Remeber that you always want your backing to be larger than the quilt, by 2-6 inches, depending on your quilting plans.  Lq is length of quilt including the extra backing length you want; Wq is width of quilt including backing - plus one inch for the seam.  Lf is length of fabric you need to buy.  Wf is useable width of your fabric.

First, I noticed that when you slide the fabric, you get a little triangle on 2 corners.  So I pulled out my Pythagorean Theorem, and tried to make this all make sense, with a² + b² = c².  But that just turned into a big mess, so I threw in the towel on that approach, and worked John's equation backwards. Apparently he was telling me that area A equals area B.

That took a lot of convincing, but I drew more pictures, crossed out some regions, and decided that , low and behold, he was right.  And in fact, these 2 areas are also equal (see left and right diagrams). That means:

  Lq * Wf equals Lf ( 2Wf - Wq )

That led me to an even simpler version of John's equation:
Lf = ( Lq*Wf ) / (2Wf - Wq)

And thus, I now have a simple equation for calculating my backs, and I know why it works (which increases my faith in the process by leaps and bounds).  Just remember to add your overhang to the quilt size, and subtract about 1" from your fabric width to account for the seam (Wq is width of quilt, including the excess you want on the backing, and Lq is the length of your quilt, including excess).