Turtle quilt: underestimating the diagonal

After planning out my turtle quilt, I broke down the pattern and assigned each turtle as one, 9 inch unit:

As you can see from the diagram, each turtle is made up of 4 smaller, 4.5" squares. Accounting for a 1/4" seam allowance around each square, I cut 4" squares for the smaller green triangles and 5" squares to make the larger green triangles. WHICH COMES TO MY FIRST MISTAKE!

You see this square? 

So it makes perfectly good sense to leave a 1/4" seam allowance on either side (meaning you cut a 5" square) if you want to piece a square with a final dimension of 4.5". BUT (and yes, unfortunately this took me a while to realize and figure out) this doesn't work with triangles. As per the diagram above, I needed to piece two triangles together to give me a square with a final dimension of 4.5". So, I cut myself a 5" square and then cut it across the diagonal. It look me a lot of staring before I realized...

*GASP*... oh the horror (and mostly just ashamed that I didn't realize it sooner) that I forgot to account for the seam allowance on the diagonal. But okay, okay, I'll just fix it my adding another 1/4" and cut a 5.25" square, right? HERE COMES MY SECOND MISTAKE

Having forgotten about basic triangle geometry, I forgot to account for the fact that a 1/4" seam allowance on the edge does not equal to a 1/4" seam allowance in the diagonal direction (as indicated by the orange line). Oh dear, oh dear, why did I not start with simple squares for my first quilt! So, after doing some calculations (and rounding), I figured out that I needed to add an additional 1/8" to account for the seam allowance on the diagonal.

*Whew* that was a lot of figuring out and recalling old geometry. To conclude for today's post, if you want to piece a square out of two triangles with an edge of 4.5", you have to add 1/4"+1/4" (the regular seam allowance) +1/4"+1/8" (for the diagonal) and cut a square of 5 3/8".