If you are looking to add some more outdoor living and entertaining space to your concrete patio, you may want to consider extending it. While pouring large slabs of concrete require a lot of manpower--much of it with experience--small extensions to patios are something that most do-it-yourself homeowners can handle.
Mark the perimeter of your planned patio extension on the ground using landscaping paint. Excavate this area to a depth that is 9 inches below the level of the current patio. This accounts for a 5-inch concrete slab and 4 inches for a gravel base.
Compact the soil in the excavated area firmly using a hand tamper or a heavy duty plate compactor. Add a 4-inch layer of finely crushed gravel and level it with a rake.
Build a form for the concrete slab by using 2-foot by 6-foot lumber to line the sides of the excavated area, laying them on edge on top of the gravel, and nailing them together at the corners. Set these 2x6's so that they are level and at the same height as the current patio. Hammer wooden stakes into the soil on the outside of the frame and right up against the form to provide support when the concrete is poured.
Mix the concrete per the manufacturer's directions if you are doing it yourself. Pour the concrete into the form and screed off the excess. This is done by using a 2x4 piece of lumber that is long enough to span the width of the form and resting it on the edges of the 2x6 form. Then have two people grasp the 2x4 on either end and drag it the length of the form along the edge of the 2x6. This will level and remove the excess concrete, and also reveal any low spots that need more concrete. Add more concrete to these spots and screed again.
Smooth and finish the surface of the slab with trowels. Keep the slab moist by watering it down, and allow it to cure undisturbed for at least four days, and a full week if possible so that the slab reaches maximum strength before being subjected to stress.