Intermediate Level

Areas

In Area of a circle we illustrate the determination of the area of a circle by its decomposition in an increasing number of circular sectors, recreating the process used by Archimedes (287-212 BC) to prove the relation $$P = 2A$$ between the perimeter of the unit circumference ($$P$$) and the area of the unit circle ($$A = \pi$$).

In the present application, using the areas of regular polygons inscribed and circumscribed to a unit circle, we construct a frame of $$\pi$$ by sequences of rational numbers, thus obtaining approximations by excess and by defect of this irrational number.

We recreate the demonstration of Archimedes. He began by using hexagons inscribed and circumscribed to the unit circle, doubling successively the number of sides. When he reached a 96-sided polygons, he found that the area $$\pi$$ of the unit circle ranged between $$3 + {10\over71}$$ and $$3 + {1\over7}$$, which leads to an approximation of $$\pi$$ with two exact decimal places.