Extend a tape measure 16 inches straight out from one corner of the fireplace opening and make a mark on the floor. Repeat on the other side of the opening.
Draw a line connecting the two marks on the floor. Draw a line connecting each mark with the corner at the fireplace opening. This rectangle outlines your hearth. Measure the diagonals of the rectangle. The outline is square if the measurements match.
Transfer the outline measurements to a sheet of backer board. Cut the sheet to fit with a circular saw.
Place the backer board over the outline. Screw the backer board to the plywood. Place screws in the corners and in three rows across the board every 4 or 5 inches.
Measure the fireplace opening. Find its center point. Place a tack or nail at its center. Wrap a string around the tack. Extend it straight out past the backer board and place another tack in the plywood. Wrap the string around the tack.
Measure the hearth from front to back along the string line. Find the center point and make a mark on the backer board.
Tack another string line, perpendicular to the first one, that passes over the center mark. This will divide the backer board into four sections.
Fit the tiles over the backer board to determine the layout. Start at the front of the hearth at the center and work outward. Keep a space of about 1/8 of an inch between the tiles. Mark any tile that needs to be cut. Avoid leaving a tile less than 1 inch wide.
Draw lines between the gaps in the tile.
Place your tiles on a board in the same pattern as the layout. Remove any tiles that need cutting. Cut them to size with a tile cutter. Place them back on the board in their proper position.
Use a chalk-line tool, which unrolls a cord that leaves a line of chalk, to outline the tile pattern. Ensure that the chalk covers the lines you made when laying out the tiles.
Spread a layer of thinset mortar over a portion of the backer board with a notched-edge trowel.
Apply the tiles in the mortar guided by the chalk lines. Use 1/8-inch spacers to leave the appropriate gap between tiles. Start at the center and work your way out to the edges. Check each section for level when you finish.
Finish a small area, then apply more mortar and continue. Thinset mortar has a working time of about a half hour.
Wait 24 hours after tiling to apply the grout.