Pi = 4?
This Demonstration shows a perversion of Archimedes' "method of exhaustion" to determine the value of . The definition of is the circumference of a circle of diameter 1. Start with such a circle inscribed in a square. The perimeter of the square is clearly 4 times the diameter: . Next, indent the corners of the square, as shown by selecting step . The perimeter of the distorted square is clearly unchanged. Thus we can write . Continue this process by further indentations of the square by selecting . The perimeter of the resulting polygon remains the same: . It appears that . The perimeter of the polygon clearly approaches the circumference of the circle. After seven or eight steps, we can hardly distinguish the two on the scale of the graphic even with magnification. The apparent conclusion is that . But didn't we learn in school that ?[more]
The fallacy comes from violating the definition of arc length: the approximating polygonal curves should have all their vertices on the circle.
Incidentally, the same procedure would be valid if we were seeking the area of the circle.[less]
 "The Diagonal Paradox," The Freaky World of Math (Nov 17, 2011). freakymath.blogspot.com/2011/11/diagonal-paradox.html.