I decided to use my favorite IKEA fabric and plain old felt as a backing (to be gentle on my newly built IKEA coffeetable).
You can cut a template out of cardboard or just cut your square freehand. I cut the felt first because it's the size the coaster will be in the end. Then I placed the felt square over the patterned fabric, positioning it as centrally as possible. I left about 3/4" border while cutting around the felt. The top/patterned fabric will be hemmed, but the felt, of course, doesn't need hemming.
Then I pressed the 3/4" hem around the patterned fabric with the iron.
fold over the corners of the patterned fabric after pressing
line up the felt with the pressed patterned piece and pin, keeping the corners folded as you go
sew it on the frontside, as close to the edge as possible
once you're done with the outside border hem, clip your threads and sew a square in the middle (to keep your coaster flat and reinforce the shape). I just did it freehand, but you could mark the outline or corners with chalk.
remove pins and press with iron to smooth out pin marks
time for a tea party!