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Step-by-step build — 5 steps. Also available as a print sheet.
Rip the plywood into two strips 16" wide and one strip 15-1/4" wide, then crosscut into eight pieces all 28-1/2" long. You need four 15-1/4" pieces, so rip one of the 16" pieces down to 15-1/4". Accurate rips here are what make the finished base look like a solid column.
Drill 3/4" pocket holes along one edge of each 15-1/4" wide piece. Join each narrow piece to a 16" wide piece at a right angle to form an L — build four identical leg sets. Keep the outside faces flush; that's what hides the seam.
Stand the four L-legs around the center so they wrap into a column, and tie them together at the ends with the 1x4s using 1-1/4" brad nails and glue. Check the column is square and sits flat before the glue sets.
Gusset the inside joints with the 2"-wide plywood scraps, glued and pinned with 1-1/4" brad nails. This is a big table's whole footing — the reinforcement is cheap insurance against racking.
Fasten mending plates across the top of the column to receive the tabletop. If your top is only 3/4" thick at the attachment points, use scrap 3/4" plywood cleats instead of plates so a screw can't blow through.
Sand and finish to match your top. The round or square top is a separate build; attach it so a wide solid top can still move seasonally without cracking.