Place the two pieces of 18-by-19-by-3/4-inch plywood flat on a worktable. These are the sides of the toy box. Make two separate marks 3/8-inch from one of the 18-inch sides of each piece, lay the straightedge on the marks and draw lines on the plywood.
Measure from one of the 19-inch sides and make marks on the lines at 2, 9 1/2 and 16 inches. Drill 3/16-inch pilot holes on the marks followed by countersink holes on top of the pilot holes.
Apply wood glue to the 19-inch edges of the 19-by-36-by-3/4-inch plywood. This is the bottom of the toy box. Place it between the sides and secure the pieces to each other through the pilot holes with the drywall screws. Clean up the excess glue with a damp cloth.
Put the two pieces of 18-by-37 1/2-by-3/4-inch plywood on the worktable. These are the front and back of the toy box. Make two separate marks 3/8-inch from the 18-inch sides of each piece and four marks 3/8-inch from one of the 37 1/2-inch sides.
Lay the straightedge on the marks and draw lines as before. Measure from one of the 37 1/2-inch sides and make marks on the lines at 2, 9 and 16 inches. From one of the 18-inch sides, make marks on the lines at 2, 13, 24 and 35 1/2-inches.
Drill pilot holes and countersink holes as before. Apply glue to the edges of the sides and bottom of the toy box facing up. Secure one of the 18-by-37 1/2-by-3/4-inch plywood to the sides and bottom of the toy box with the drywall screws.
Turn the toy box over and secure the remaining 18-by-37 1/2-by-3/4-inch plywood in the same manner. Clean up excess glue as before.
Install the piano hinge on the 20-by-37 1/2-by-3/4-inch plywood. This is the lid of the toy box. Secure the lid to the toy box via the piano hinge.