Measure the length and width of the area rug. Don't go by the stated measurements -- the actual size of the rug may vary considerably.
Cut a thick, nonslip rug pad 1 inch smaller than the rug in each direction. This allows the rug to rest over the edges of the rug pad, 1/2 inch in each direction, helping to keep the edges and corners flat to reduce curling and tripping hazards. Center the rug pad on top of the rug, and check that it's even and flat. Position the sofa, and check that there are no ripples. Have an assistant lift one end of the sofa to allow you to straighten the rug, if needed.
Cut two 12-inch lengths of carpet tape for each corner of the rug. Remove the backing from one piece of tape at a time. Lift a corner of the rug, and apply one piece of carpet tape 1/2 inch from each edge on the corner; begin at the corner, and extend one piece of tape horizontally near the rug's edge and the other vertically. Set the corner down, and smooth it flat with your hands.
Press the corner of the rug against the corner of the rug pad with both hands to make the carpet tape stick to the rug pad. Repeat with each corner.