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Using a ruler and protractor can help. Find the halfway point on one of the sides. Take a compass and find one of the corners on the opposite side from where you took the halfway point. With the needle on the halfway point, place the arm at the opposite corner. Rotate the compass until the arm is in line with the side where you took the halfway point. This spot will now be the corner for the golden ratio outline rectangle. Using a ruler, extend your square into a rectangle with the point you found as one of its corners. This new rectangle can be used as a basis for drawing your spiral.
Draw a square with perfectly equal sides. Find the halfway point. Align a compass to the opposite corner. Extend the line. Draw the new rectangle.