Limits: Approaching Pi

Since \(\pi\) is an irrational number, we cannot write down its exact decimal or fractional value. We can, however, define it in terms of circles. If \(r\) is the radius of a circle and \(C\) is its circumference, then \[ \pi = \frac{C}{2r} \] We can also view \(\pi\) in terms of the area \(A\) of a circle: \[ \pi = \frac{A}{r^2} \] It's the latter formula that we'll use here. Suppose we have a circle with radius 1. Its area would then be equal to \[ \pi = \frac{A}{1^2} = A \] So if we can approximate the area of a circle with radius 1, we'll have an approximation of \(\pi\) itself. To do so, we'll draw two regular, convex polygons: the largest polygon we can fit inside the circle, and the smallest polygon we can fit outside the circle. Each polygon will have the same number of sides. Because we already have convenient formulas for the area of polygons, we can calculate the inner and outer polygon areas. The true value of \(\pi\) will lie somewhere in between. As the number of sides of the polygon increases, they begin to "fit" closer to the circle itself, and we begin to see a better and better approximation of \(\pi\).

In calculus terminology, the value of \(\pi\) is the limit of the area \(A(P_n)\) of a regular convex polygon with \(n\) sides and radius 1 as the value of \(n\) increases towards infinity: \[ \pi = \lim_{n \to \infty} A(P_n) \]

Some helpful geometry terms and formulas, all applying to regular convex polygons:

• The radius \(R\) of a polygon is the distance from its center point to one of its vertices (corners).
• The apothem \(t\) of a polygon is the distance from its center point to the center of one of its edges. It is associated to the radius of the polygon by the formula \[ R = \frac{t}{\cos(\pi/n)} \] where \(n\) is the number of sides of the polygon.
• The perimeter \(p\) of a polygon is the sum of the lengths of its edges. The length \(s\) of each side can be calculated using the formula \[ s = 2R\sin(\pi/n) \] where again \(n\) is the number of sides of the polygon.
• The area of a polygon can be found by the formula \[ A = \frac{pt}{2} \] where \(p\) is the perimeter of the polygon and \(t\) is the apothem.

Change the number of sides in the polygon, which can be any integer value between 3 and 500. Information about the inner and outer polygons will be calculated and displayed. Note that as the number of sides increases, the area of both polygons grows closer to \(\pi\) while their perimeter grows closer to \(2\pi\).

This applet was created using the p5 JavaScript library and was last updated September 2019.