Draw two lines connecting the opposite corners of the square, forming a large "X."
Set the point of the compass on one of the corners of the square and open it to the center of the "X."
Use the compass to draw an imaginary semi-circle. Carefully mark the two points where the circle crosses the edge of the square.
Move the point of the compass to the next corner and repeat the process. Continue until there are a total of eight marks on the edge of the square.
Use the ruler to connect the two marks at each corner of the square with a straight line to form a triangle. Using the appropriate cutting implement for the material the square is made of, cut away the corners along the lines. The resulting shape will be a regular octagon.