Dear Ms. Cowan, 

Re: The number of angles in a circle.

    Let Sn, n >= 3, be a regular polygon with n edges. The number of angles in Sn is n. As n tends to infinity the sequence (Sn) goes to a circle. Therefore the number of angles in a circle is infinity. This point of view goes back 2,500 years to the time of Archimedes in Greece, and to even earlier times in China and India. This construction was employed in antiquity to bound the value of pi with rational numbers, both above and below. 

    Anna discussed her homework with me --- after having completed it all on her own. She had identified one of the quadrilaterals as the one figure with the most number of angles. However, after I presented her with the argument as above, and after we had drawn a few pictures, we agreed that a circle has more angles than any finitely-generated polygon. 

    It might interest you to know than in our own times we use different, power series, methods to approximate transcendental numbers like pi. Power series methods are said to date back in recorded history to Madhava (ca. 1340-1425) and Nilakantha (ca. 1450-1550) in Kerala, India (See, e.g., page 28 in "A Radical Approach to Real Analysis," David Bressoud, Mathematical Association of America, 1994). Nilakantha calculated pi with the beautiful series pi / 4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - ... , which series was later independently rediscovered by each of Leibniz, Gregory, and Newton. This series falls out almost trivially when we integrate over the first quadrant of a unit circle centered at the origin. And that gives us the area pi / 4.

    The first person known to have made widespread and powerful use of power series was Isaac Newton (1642-1727). Modern computer programs for pi often use power series due to Srinivas Ramanujan (1887-1920). E.g., Borwein, Borwein, and Bailey, "Ramanujan, Modular Equations, and Approximations to Pi, or How to Compute one Billion Digits of Pi," The American Mathematical Monthly, vol. 96, no. 3, March 1989, pp. 201-219. 

Sincerely,
Arun Kumar