Tutorial: half-square triangle version

At Quilt Camp, I was shown a new, slick way to make half-square triangles. The gals told me it was not their original idea, so I went onto the internet looking for a source. Found one on YouTube: Missouri Star Quilt Company's version.

I was asked (as the resident math teacher) to figure out how to tell what finished size would result from various original fabric squares.  The ladies had a pair of 5" starting blocks, which cut down to 4 half-sqaure triangle squares that were slightly larger than 3".  Meaning, when adjusted to 3", the resulting finished squares would be 2.5".  I blithely replied that I saw a pattern: would the finished half-square triangle blocks be one-half the size (more or less) of the uncut originals?

I was incorrect, but had to think about the math involved.  I woke up in the middle of the night (bleary-early Sunday morning, to be precise) knowing the process to determine the finished size when using any size squares to start with.  I have heard there are conversion tables available online, but you can do these basic calculations yourself.  Don't fear the math: you have the power!

The Process: what size will I get when I start with a xxx-inch square?

     ** Take the size of your original squares and subtract 1/2" (since you are sewing 1/4" around the outsides)
     ** Take this new number and divide by the square root of 2 (yes, the square root of 2!  see below for the math/nerdy reason for this number)
     ** This is the side length of each of your 4 new half-square triangle squares; subtract 1/2" for seams, and you will have the finished size.  Yay!
     ** EXAMPLE: starting with a 10" square, subtract 1/2" to give 9-1/2"; divide by square root of 2 and get 6.72"; subtract another 1/2" to finish out at 6.22".  (This is just a hair short of a 6-1/4" finished half-square.)

The backwards process: what size should I start with to get a xxx-inch finished block?
     ** Beginning with your desired finished size, add 1/2" for seam allowances
     ** Since we are working backwards, MULTIPLY this number by the square root of 2 (are you curious yet about why you use this number?  see below; I promise it won't hurt you to check out a little remedial geometry)
     ** Finally, ADD another 1/2" for seams.  Round up if you must (those square-root-2 calculations always end up with some wonky decimals), and this is the size of the original squares you should cut.
     ** EXAMPLE: wanting a 4" finished half-square block, add 1/2" to give 4-1/2"; multiply by square root of 2 and get 6.36"; add another 1/2" to yield 6.86".  (This means you should cut 6-7/8", because that is the next closest fractional amount to .86".)

Don't fear the math: a little geometry for you

The calculations for half-square triangles are based on right triangles; specifically, isosceles right triangles (you know--the ones you get when you cut a square on the diagonal: two sides the same length on either side of a right angle, and a longer hypotenuse)

The great thing about right triangles is the whole "a-squared plus b-squared equals c-squared" relationship between the side lengths.  This allows us to calculate any missing side lengths by just knowing the other sides.

 But the relationship between side lengths (the Pythagorean Theorem) is simplified in an isosceles right triangle, since sides a and b are both the same length.

Consider an isosceles right triangle with legs of length 10".  The hypotenuse of such a triangle is approximately 14.1", which is equivalent to 10 times the square root of 2.

Or, how about an isosceles right triangle with hypotenuse of length 4" (this would be the diagonal of my half-square triangle).  The side lengths of this triangle are approximately 2.8" each, which is equivalent to 4 divided by the square root of 2.

And now, the part that makes mathematics more powerful than individual calculations: taking these concrete examples and generalizing them into the side-length relationships in any isosceles right triangle.  Noticing how the square root of 2 played a part in both of the above examples, we go back to side lengths of a and a hypotenuse of length c.

See!  This math stuff isn't quite so hard.  And when you understand a bit of math, you feel powerful.