Trim the fibers to create a 1/2-inch-wide border, free of carpet fibers, all the way around a carpet remnant with scissors. Remove as many of the loose fibers as you can to expose the carpet backing.
Place the binding tape snugged up against the untrimmed carpet fibers on the rug remnant. Thread the carpet needle with heavy-duty thread in a matching color to your binding. Hand sew the top of the binding tape all the way around the carpet remnant using a whip stitch. Overlap the beginning of the binding tape by 1/2 inch with the end of the binding tape after you've finished the sewing the binding tape to the top of the carpet. Cut the excess away.
Fold the carpet binding tape under the edge of the rug remnant allowing a fold to extend 1/4 inch past the edge of the carpet backing.
Sew the back edge of the carpet tape onto the backing of the carpet using the same whip stitch.
Iron the fold line flat.