So I looked at pictures of newspaper-related things and saw newsie hats online. And then I reverse-engineered them to come up with this newsie hat approximation. It's pretty cool!
Step 2: Do some measurements and math
Measure around your fabric circle.
Calculate (fabric - head) / 6 = the base of the triangle you will cut.
Calculate head / 6 = the spacing between triangle bases.
Mark around the perimeter of the circle. Put little exes on the bits that are going to be cut out.
Get a smaller circle (still bigger than your head) and trace it in the center of the fabric dot. Draw triangles that take up the x'd bases and have points on the inner circle.
Cut them out.
Step 4: Decorative stitching
Get a straightedge and trace across the top of the hat, opposite corner to opposite corner. They should all cross in the same place in the middle.
Sew offset from both sides of these lines to make decorative seam-like stitching.
Step 6: Cut out the brim
Step 7: Sew the brim
Iron the raw edges inside and sew the cloth over to completely encapsulate the cardboard.