I've been enjoying some mitered square projects recently (here's the last one I did). Sometimes, when making right triangles, rectangles, or squares, we need to extrapolate the diagonal measurement from the horizontal, or vice versa. Fortunately for us, the ancients figured out how to do that about 3800 years ago.
In my recent projects, the diagonal measurement of a mitered square needs to be the width of my hand, because I'm making mittens. My hand is about 7 inches around, meaning the diagonal width of the square needs to be 3.5 inches.
The Pythagorean theorem says that if I add together the squared lengths of two sides of a right triangle, rectangle, or square, then the sum is the square of the diagonal measurement (the hypotenuse).
A2 +B2 = C2
If I'm making a square, and I know that my diagonal measurement is 3.5, then I can square that to get 12.25, and then divide the square in half to get the squares of the two sides. In this case, A2 and B2 are each 6.125, and the square root of each is approximately 2.5. Fortunately, yarn math doesn't need to be terribly precise. If I came up with an even number, I would add or subtract one, since mitered squares need a center stitch.
If I were making a square by starting with the stitches for one side, I would multiply my gauge by 2.5, and cast on or chain that number. Since I am doing mitered squares, I double the length of the square (2.5 x 2 = 5), and then multiply that by my gauge. In this case, my gauge is 5 stitches per inch, and 5 x 5 = 25, so I cast on 25 to start my square.
As you can see in these pictures, my gauge was a little looser than anticipated, so the swatch is a little bigger than 2.5 x 2.5, and, therefor, the diagonal measurement is a bit more than 3.5.
If I wanted to make a triangular shawl that was, say, 72 inches along the long edge (the hypotenuse), and planned to use a stitch with an approximately square gauge the math would work the same way. 722 = 5,184. Half of 5,184 is 2.592, which is the square of the length of each short side of the shawl. The square root of 2,592 is about 51 inches. If I was to start the shawl by casting on for one of the short ends, then I multiply my gague by 51, and that's my starting number of stitches. Then I decrease down along one side to create the other two sides of the shawl. Alternately, I can start at the corner between the two short sides and increase out until the short sides are 51 inches long, and I don't have to worry about counting stitches or figuring out how to measure something gathered onto a circular needle or that is longer than my measuring tape.
And if you don't want to do the math yourself, you can use a right triangle calculator here.
The Pythagorean Theorem is one of those memorable little formulae we all had to learn in school but probably figured we'd never use, but it really is handy for needlework.