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Step-by-step build — 4 steps. Also available as a print sheet.
Measure the clearance under the bed first — this design needs at least 8" for 1x8 sides; more clearance lets you use 1x10. Then measure the footprint you want the trundle to occupy. These numbers drive every cut, so get them before you touch the saw.
Cut the plywood bottom to your figured width and to the bed length minus about 1-1/2" so it rolls in and out freely. In the source example that's 70-1/2" long by 38-1/2" wide. Keep the cut square.
Fasten the side boards around the platform, screwing into the ends of adjacent sides as well as into the platform edge so the corners lock. Pocket screws give the cleanest look; 2" screws and glue also hold. Keep it square.
Screw a caster to each of the four corners and two more in the center of the span — the center wheels carry the middle so a mattress and sleeper won't bow the bottom. Fill screw holes, sand, and finish so it glides on your floor. Confirm it rolls under the bed with clearance to spare.