Make a Phrygian-style pileus
To make a Phrygian-style pileus, fold the fabric in half. The fold should be up and down to your left.
The bottom should be the cut edge and be at a 90 degree angle from the fold —a straight line.
Measure your head around, use half that number, and mark it (B) on the cut edge, measuring from the bottom corner of the fold (A).
Add 2" or so to the number, and mark it up from the fold, measuring up from the bottom corner of the fold (C)
Now, for the tricky part. The third side of the triangle must be an outward curve for most of the way, but make another point 2" up from the (B) point on the bottom cut line. This is (D). Draw a line straight up between B and D, then an outward curve from D to C. This will be the back of the pileus.
Depending on what kind of material you are using, draw a cutting line outside the original line. Leather and Polar fleece needs a very little extra, wool needs more, 1/3 to ½ seam allowance.
Cut the fabric on the outside line, sew on the inside line: use pins to be sure it's big enough; and there it is!
Romans didn't like to sew; that's obvious from the edges of extant clothing. If you used a fabric that will unravel, just fold up a tiny bit of the bottom edge up underneath. If you use something that won't unravel, then leave the bottom edge alone.
Put the hat on, with the fold in front, and pull the tip forward until it folds front. Io Saturnalia!
— gratias to Merlinia Ambrosia for the written instructions!