π Day has always been a special day for me, from my earliest days. In fact, I’ve never figured out whether I was so eager to celebrate my first π Day that I jumped the gun and sent my mom into labor early, or whether I just wanted be sure to experience all 24 hours of my first π Day. Whichever it was, I’ve certainly enjoyed and celebrated π Day ever since, and it’s been even more fun for me since I got involved with Sketchpad and had the opportunity to come up creative, animated decorations and diversions.
So I’m sharing here a few of my favorite circle dissections. They’re all good ways to discover that the circle’s area is given by either the half of the product of the circumference and the radius (for the dissections that yield a triangle) or by the product of half of the circumference and the radius (for the dissections that yield a rectangle). Further, increasing the number of cuts in these dissections leads to thinking about limits; the only way to turn a circle into a triangle or rectangle is to chop it into tiny, tiny sectors — but how tiny do they have to be?
I hope your students enjoy these, and that they lead to interesting discussions both about how each dissection relates the area of a circle to its radius and circumference, and about the reasoning behind the dissection: What happened to the curvature of the circle’s edge? Can you really base a logical argument on increasing the number of sectors without limit?
(Press the buttons below the Web Sketchpad model to look at one or another of the four models. For the explosion, reassembling doesn’t always work after an explosion, so you may have to use the Refresh button in your browser.)