One aspect of flooring installations is how to hide the termination of the flooring material against walls, under and around doorways and against areas such as a fireplace hearth. If you want the termination point to appear seamless, you have to install the material underneath the edge of a vertical area -- for example, installing baseboards around the edge of the room to hide the gap where the floor material meets the wall. For an existing hearth, however, the only way to make room for the flooring material is to undercut the hearth so you can slide the new flooring into place and have the termination edge out of sight.
Determine what type of material your hearth is made out of and pick up an appropriate blade when you are renting the undercutting saw at your local home improvement store.
Measure the thickness of your flooring material and add any extra for padding or underlayment. For example, with tile floors you have the thickness of the tile plus the thickness of the underlayment plus roughly 1/8 inch for the compressed thickness of the thinset mortar. Hardwood is generally the thickness of the material only, unless you have padding.
Set the height of the blade on the undercut machine to the thickness you need for your flooring type. Double-check this measurement before cutting.
Put on your safety gear. Position yourself on your hands and knees in front of the hearth. Place the machine on the floor with the blade up against the hearth. Hold the machine in both hands and turn it on. Bury the blade into the hearth and slowly push the machine down the length of the entire hearth, cutting as you go.
Pry loose the cut-off section with a hammer and chisel. Press the chisel into the gap left behind by the saw blade of the undercut saw. Pry downward to snap free the cut-off section you need to remove, leaving behind a gap for your flooring material to slide into place. Tap the chisel with a hammer if necessary to help break off the pieces.