Half Square Triangle formula

My quilting genius, and very crafty, dear wife asked me a simple question: How do I calculate the size of a "half square triangle"?

"What is a Half Square Triangle?" I asked?

This is "sewing magic" where you start with two square pieces of fabric, sewn together with a 1/4" seam around all edges. Then you cut that square into 4 pieces, across the diagonals of that square. You then unfold the "triangle" along the sewn seam, now you have 4 smaller squares.

See the picture below of what I'm describing:


The question my wife asked: For a desired "magic square" size of say 4", which is one of these four "trianular bits" unfolded along the seam line: What is the larger fabric square size to use?

Hmm, this is a question for the Numberphile YouTube Channel, I thought. This involves some rather serious use of the Pythagorian theorem. In the spirit of Numberphile, I broke out some Teletype Paper - similar to the traditional brown Butcher Paper all maths is scribbled out on Numberphile. I then proceeded to "dazzle" my wife with my 'mastery' of Algebra and Geometry.

This turns out to be a rather tricky maths problem. Cutting a square along the diagonal, then unfolding along the sewn seam to reveal four smaller squares, which are rotated 90° from the larger "parent square" is a bit of a mind bending problem to solve.

After an hour attempting to understand the original problem, and drawing out pictures of the cut squares transformation on paper, it was obvious the Pythagorian Theorm is required to answer this question.

My wife had two prototype half squares constructed. Once I understood the problem, the answer rather quickly was drawn out as a picture on Teletype paper. Applying Geometry, this equation solves the original ask:



HST = Square Edge length "magic square"
FullSquare = Starting square edge length


Example: Desired Half Square Triangle size is 4 inches. What is the size of the starting FullSquare?

Applying the above formula, let's solve this equation one step at at time.

The desired Half Square Triangle size is 4". For this equation HST = 4.



Now square that answer of 8:



Divide 64 by 2:


Now take the square root of this resultant mess:


This is 5 11/32 inches. Almost done. Just add 1/2" to this result and you now have the size of the parent square:



This is close enough to round up to 5 28/32, or 5 7/8".

See, Easy Peasey!