The Method Using Regular Polygons

1. The Method Using Regular Polygons

Archimedes of Syracuse, in Sicily, (287-212 BCE) was one of the greatest mathematicians, sometimes compared to Carl Gauss and Isaac Newton. Moreover, Archimedes wrote mathematics in a style that remains surprisingly readable even today. One of his most important achievements was the method of approximating the area of the unit circle using inscribed and circumscribed polygons, now known as the classical method of exhaustion.

Archimedes realized that the area of a regular polygon of n sides inscribed in a unit circle is smaller than the area of the circle, which in turn is smaller than the area of a regular polygon of n sides circumscribed about the circle. As n approaches infinity, the two polygonal areas should approach the area of the unit circle. In the following subsections, we will derive Archimedes method.

1.1. Inscribed Regular Polygon

First we determine n points (P₁, P₂, ..., P_{n − 1}, P_n) equally spaced on circumference of the unit circle. These points determine a regular polygon inscribed on the unit circle, as shown in Figure 1a.

Figure 1a. The inscribed polygon

Thus $Image: Equation1.gif$ so . $Image: Equation2.gif$ . Using the law of sines, $Image: Equation3.gif$ .

The area of this polygon is n times the area of triangle, since n triangles make up this polygon. So the formula for the area of the regular inscribed polygon is simply

$Image: Equation4.gif$

Using the fact that $Image: Equation14.gif$ , one of the most famous limits in calculus, it is easy to show that $Image: Equation6.gif$ . If the students have not yet been taught the basic limit, we can ask Maple for the answer:

> limit((1/2)*n*sin(2*Pi/n),n=infinity);

1.2 Circumscribed Regular Polygon

Next we determine n points (P₁, P₂, ..., P_{n − 1}, P_n) on corners of a regular polygon, each side of which is tangent to the unit circle, as shown in Figure 1b.

Figure 1b. The circumscribed polygon

As before, $Image: Equation1.gif$ so . $Image: Equation2.gif$ . We must calculate the area A(P_nOP₁) of the triangle shown in Figure 1c.

Figure 1c. A triangle formed by the circumscribed polygon

From basic trigonometry, $Image: Equation7.gif$ . Using the law of sines again we have,

$Image: Equation8.gif$

Since the triangles are congruent, the area of the polygon is n times the area of triangle P_nOP₁ triangle. Thus the formula for the area of the regular circumscribed polygon is simply

$Image: Equation9.gif$

We can use basic limits to show $Image: Equation10.gif$ . Again, we can ask Maple to verify the answer:

> limit((n/2)*(1/cos(Pi/n))^2*sin(2*Pi/n),n=infinity);

So π is the limit of the areas of the inscribed regular polygons and the circumscribed regular polygons as the number of side n tends to infinity.

1.3. The Maplet

The Polygon Method Maplet illustrates the area of the unit circle as the limit of the areas of the inscribed and circumscribed regular polygons. The program is written in Maple 9 and is published in Application Center at the Maplesoft web site. A snapshot of the Maplet is given below in Figure 1d.

Maplet Instructions

First, click on "Start" button. An inscribed triangle and a circumscribed triangle appear in the graph regions, and the areas of the triangles are given in the textboxes. Clicking on the Increase-n button increases n (the number of sides) by 1, and clicking on the Decrease-n decreases n by 1. Again, the inscribed and circumscribed regular polygons are shown in the graph regions, and the corresponding areas are given in the textboxes.

Figure 1d. The Polygon Method Maplet

Selecting Table from the menu bar gives a table of the areas of the inscribed and circumscribed regular polygons, for all values of n from 3 to 1000.

Figure 1e. The table of approximations

Now download and run the Polygon Maplet and experiment yourself.