The Journal of Online
Mathematics and Its Applications, Volume 7 (2007)

Maplets for the Area of the Unit Circle, Muharrem
Aktümen and Ahmet Kaçar

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.

First we determine `n` points (`P`_{1}, `P`_{2}, ...,
`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 so .. Using the law of sines, .

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

Using the fact that , one of the most famous limits in calculus, it is easy to show that . 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);

π

Next we determine `n` points (`P`_{1}, `P`_{2}, ...,
`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, so .. We must calculate the area `A`(`P`_{n}`O``P`_{1})
of the triangle shown in Figure
1c.

Figure 1c. A triangle formed by the circumscribed polygon

From basic trigonometry, . Using the law of sines again we have,

Since the triangles are congruent, the area of the polygon is `n`
times the area of triangle `P`_{n}`O``P`_{1}
triangle. Thus the formula for the area of the regular circumscribed polygon
is simply

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

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

π

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.

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.

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.