Cut a template of your desired rug shape using butcher paper or other large paper. Make sure the shape you draw will fit on your carpet remnant.
Turn the carpet piece over to the back side. Position the template on the back of the carpet and trace the template shape with a permanent marker onto the carpet back. Remove and discard the template.
Put the carpet back side up on an old board or other work surface suitable for cutting. Cut the carpet following the outline you drew with the marker.
Attach instant carpet binding trim according to manufacturer's directions. Instant carpet binding provides a ready-made professionally bound edge with an adhesive strip attached. You can put it on the raw edge of a rug to prevent unraveling and save you the expense of hiring a rug binder. Instructions will likely direct you to do the following tasks: turn your rug with raw edges right side up. Remove about 12 inches of the adhesive backing from an instant carpet binding trim.
Place the instant binding product so the trim is touching the cut edge of the rug and the adhesive part sticks to the back of the rug. Pull the adhesive backing off a little at time as you work your way around the rug attaching the binding. Press the adhesive to the back of the rug firmly and keep the binding flush against the rug edge. Make small cuts in the adhesive as you go around curves or corners so the adhesive pieces can overlap in the back and the trim follows cleanly around any curves or corners.
Put a thin line of hot glue between the binding trim and the cut rug edge. Press the two together and hold until the glue is set. Continue hot-gluing the trim to the rug all the way around the rug to make sure trim is securely attached.