Initially the green inner square is half the area of the white outer square.

Click the **Start** button. (If the animation is too small for your screen, click the **Large** button.)

The inner square will rotate and expand. It will stop when its area exceeds the circle's area.

Now use the **-1** button to back up a few times, until the A side length equals 50 (or for large animations, 100). That number is one-eighth of the side of the outer square.

Constructing one-eighth is easy. Bisect one side, then bisect the right segment, and then bisect THAT right segment. Voilá! Do that on all four sides of the square, then connect the points at the one-eighth marks, and you have a square that has 99.47% of the area of the circle. This calculation is easily verified using the Pythagorean Theorem: A^{2} + B^{2} = C^{2}.

Here's a fun fact: When a square and circle of equal area have the same midpoint (as they do in this Web app), the four yellow corners of the square stick out, and four blue trimmings of the circle stick out. Since the overlapping areas are equal, then the four parts of the square that don't overlap are equal in area to the four trimmings of the circle that don't overlap. Or, one corner equals one trimming, strange as that is to the eye.

Scott Pendleton

scott@scottpendleton.us