Chapter 4.2, Problem 21MS

Pie are squared. The circumference of a circle of radius r is 2itr. Suppose you have a round pie, and 1000 of your best friends come over for dessert. You divide the circular pie into 1000 equal-size pieces by cutting from the center. Then, before serving them, you rearrange the pieces by putting the first piece with the curved part up, the next piece right next to it with curved edge down, the next one with curved side up, alternating until all 1000 pieces are arranged. The shape you have constructed is almost an exact rectangle except that its top and bottom edges are each made of 500 just slightly curved tiny segm ents that came from the edge of the pie. How long is the rectangle? How wide is it? What is its area? Why is this story a convincing demonstration that the formula for the area of a circle is correct?