First of all, the navigators of old conveniently had devices like the astrolabe to measure this angle, and these are kind of hard to find on a desert island. A protractor would do, but even math geeks like me do not carry one in my wallet. We could use the degree markings on a magnetic compass if we had one. Tom Hanks had nothing like this on the island, which means we will have to use old fashioned geometry. Our second and more devastating problem is that if we really are stranded somewhere in the Fiji Islands, we would be in the Southern Hemisphere, and Polaris is nowhere to be seen in the southern sky.
Lets start with the Northern Sky procedure.
Tie a rock or heavy object to the other end like a pendulum, but you do not want it to swing, we just want to use it as a plumb bob. With everything perfectly aligned, make a mark on the paper where the north star lines up. Now all we have to do is find the angle between the line from the mark to the corner and the bottom edge of the paper. Since we do not have a protractor, we will have to measure this angle a different way. One method is by geometrical construction.